MATHEMATICS

1062 - Mathematics

ISBN $81-7450-634-9$$81-7450-634-9$81-7450-634-981-7450-634-9$81-7450-634-9$

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PD 700T SU

(C) National Council of Educational Research and Training, 2006,2022

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Publication Team

Foreword

Rationalisation of Content in the Textbooks

• Overlapping with similar content included in other subject areas in the same class
• Similar content included in the lower or higher class in the same subject
• Difficulty level
• Content, which is easily accessible to students without much interventions from teachers and can be learned by children through self-learning or peer-learning
• Content, which is irrelevant in the present context

Preface

• The matter needs to be linked to what the child has studied before, and to her experiences.
• The language used in the book, including that for 'word problems', must be clear, simple and unambiguous.
• Concepts/processes should be introduced through situations from the children's environment.
• For each concept/process give several examples and exercises, but not of the same kind. This ensures that the children use the concept/process again and again, but in varying contexts. Here 'several' should be within reason, not overloading the child.
• Encourage the children to see, and come out with, diverse solutions to problems.
• As far as possible, give the children motivation for results used.
• All proofs need to be given in a non-didactic manner, allowing the learner to see the flow of reason. The focus should be on proofs where a short and clear argument reinforces mathematical thinking and reasoning.
• Whenever possible, more than one proof is to be given.
• Proofs and solutions need to be used as vehicles for helping the learner develop a clear and logical way of expressing her arguments.
• All geometric constructions should be accompanied by an analysis of the construction and a proof for the steps taken to do the required construction. Accordingly, the children would be trained to do the same while doing constructions.
• Add such small anecdotes, pictures, cartoons and historical remarks at several places which the children would find interesting.
• Include optional exercises for the more interested learners. These would not be tested in the examinations.
• Give answers to all exercises, and solutions/hints for those that the children may require.
• Whenever possible, propagate constitutional values.

Textbook Development Committee

Chairperson, Advisory Group in Science and Mathematics

Chief Advisors

Chief Coordinator

Members

Acknowledgements

Contents

1. Real Numbers ..... 1
   1.1 Introduction ..... 1
   1.2 The Fundamental Theorem of Arithmetic ..... 2
   1.3 Revisiting Irrational Numbers ..... 6
   1.4 Summary ..... 9
2. Polynomials ..... 10
   2.1 Introduction ..... 10
   2.2 Geometrical Meaning of the Zeroes of a Polynomial ..... 11
   2.3 Relationship between Zeroes and Coefficients of a Polynomial ..... 18
   2.4 Summary ..... 23
3. Pair of Linear Equations in Two Variables ..... 24
   3.1 Introduction ..... 24
   3.2 Graphical Method of Solution of a Pair of Linear Equations ..... 25
   3.3 Algebraic Methods of Solving a Pair of Linear Equations ..... 30
   3.3.1 Substitution Method ..... 30
   3.3.2 Elimination Method ..... 34
   3.4 Summary ..... 37
4. Quadratic Equations ..... 38
   4.1 Introduction ..... 38
   4.2 Quadratic Equations ..... 39
   4.3 Solution of a Quadratic Equation by Factorisation ..... 42
   4.4 Nature of Roots ..... 44
   4.5 Summary ..... 47
5. Arithmetic Progressions ..... 49
   5.1 Introduction ..... 49
   5.2 Arithmetic Progressions ..... 51
   $5.3n$$5.3n$5.3 n5.3 n$5.3n$th Term of an AP ..... 56
   5.4 Sum of First $n$$n$nn$n$ Terms of an AP ..... 63
   5.5 Summary ..... 72
6. Triangles ..... 73
   6.1 Introduction ..... 73
   6.2 Similar Figures ..... 74
   6.3 Similarity of Triangles ..... 79
   6.4 Criteria for Similarity of Triangles ..... 85
   6.5 Summary ..... 97
7. Coordinate Geometry ..... 99
   7.1 Introduction ..... 99
   7.2 Distance Formula ..... 100
   7.3 Section Formula ..... 106
   7.4 Summary ..... 112
8. Introduction to Trigonometry ..... 113
   8.1 Introduction ..... 113
   8.2 Trigonometric Ratios ..... 114
   8.3 Trigonometric Ratios of Some Specific Angles ..... 121
   8.4 Trigonometric Identities ..... 128
   8.5 Summary ..... 132
9. Some Applications of Trigonometry ..... 133
   9.1 Heights and Distances ..... 133
   9.2 Summary ..... 143
10. Circles ..... 144
    10.1 Introduction ..... 144
    10.2 Tangent to a Circle ..... 145
    10.3 Number of Tangents from a Point on a Circle ..... 147
    10.4 Summary ..... 153
11. Areas Related to Circles ..... 154
    11.1 Areas of Sector and Segment of a Circle ..... 154
    11.2 Summary ..... 160
12. Surface Areas and Volumes ..... 161
    12.1 Introduction ..... 161
    12.2 Surface Area of a Combination of Solids ..... 162
    12.3 Volume of a Combination of Solids ..... 167
    12.4 Summary ..... 170
13. Statistics ..... 171
    13.1 Introduction ..... 171
    13.2 Mean of Grouped Data ..... 171
    13.3 Mode of Grouped Data ..... 183
    13.4 Median of Grouped Data ..... 188
    13.5 Summary ..... 200
14. Probability ..... 202
    14.1 Probability — A Theoretical Approach ..... 202
    14.2 Summary ..... 217
    Appendix A1 : Proofs in Mathematics ..... 218
    A1.1 Introduction ..... 218
    A1.2 Mathematical Statements Revisited ..... 218
    A1.3 Deductive Reasoning ..... 221
    A1.4 Conjectures, Theorems, Proofs and Mathematical Reasoning ..... 223
    A1.5 Negation of a Statement ..... 228
    A1.6 Converse of a Statement ..... 231
    A1.7 Proof by Contradiction ..... 234
    A1.8 Summary ..... 238
    Appendix A2 : Mathematical Modelling ..... 239
    A2.1 Introduction ..... 239
    A2.2 Stages in Mathematical Modelling ..... 240
    A2.3 Some Illustrations ..... 244
    A2.4 Why is Mathematical Modelling Important? ..... 248
    A2.5 Summary ..... 249
    Answers/Hints ..... 250

Constitution of India

Fundamental Duties

REAL NUMBERS

1

1.1 Introduction

1.2 The Fundamental Theorem of Arithmetic

EXERCISE 1.1

1. Express each number as a product of its prime factors:
   (i) 140
   (ii) 156
   (iii) 3825
   (iv) 5005
   (v) 7429
2. Find the $\mathrm{L}\mathrm{C}\mathrm{M}$$\mathrm{L}\mathrm{C}\mathrm{M}$LCM\mathrm{LCM}$\mathrm{L}\mathrm{C}\mathrm{M}$ and $\mathrm{H}\mathrm{C}\mathrm{F}$$\mathrm{H}\mathrm{C}\mathrm{F}$HCF\mathrm{HCF}$\mathrm{H}\mathrm{C}\mathrm{F}$ of the following pairs of integers and verify that $\mathrm{L}\mathrm{C}\mathrm{M}\times \mathrm{H}\mathrm{C}\mathrm{F}=$$\mathrm{L}\mathrm{C}\mathrm{M}\times \mathrm{H}\mathrm{C}\mathrm{F}=$LCMxxHCF=\mathrm{LCM} \times \mathrm{HCF}=$\mathrm{L}\mathrm{C}\mathrm{M}\times \mathrm{H}\mathrm{C}\mathrm{F}=$ product of the two numbers.
   (i) 26 and 91
   (ii) 510 and 92
   (iii) 336 and 54
3. Find the LCM and HCF of the following integers by applying the prime factorisation method.
   (i) 12,15 and 21
   (ii) 17,23 and 29
   (iii) 8,9 and 25
4. Given that $\mathrm{HCF}(306,657)=9$$\mathrm{HCF}(306,657)=9$HCF(306,657)=9\operatorname{HCF}(306,657)=9$\mathrm{HCF}(306,657)=9$, find $\mathrm{LCM}(306,657)$$\mathrm{LCM}(306,657)$LCM(306,657)\operatorname{LCM}(306,657)$\mathrm{LCM}(306,657)$.
5. Check whether $6^n$$6^n$6^(n)6^{n}$6^n$ can end with the digit 0 for any natural number $n$$n$nn$n$.
6. Explain why $7\times 11\times 13+13$$7\times 11\times 13+13$7xx11 xx13+137 \times 11 \times 13+13$7\times 11\times 13+13$ and $7\times 6\times 5\times 4\times 3\times 2\times 1+5$$7\times 6\times 5\times 4\times 3\times 2\times 1+5$7xx6xx5xx4xx3xx2xx1+57 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1+5$7\times 6\times 5\times 4\times 3\times 2\times 1+5$ are composite numbers.
7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the
   same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

1.3 Revisiting Irrational Numbers

• the sum or difference of a rational and an irrational number is irrational and
• the product and quotient of a non-zero rational and irrational number is irrational.

EXERCISE 1.2

1. Prove that $\sqrt{5}$$\sqrt{5}$sqrt5\sqrt{5}$\sqrt{5}$ is irrational.
2. Prove that $3+2\sqrt{5}$$3+2\sqrt{5}$3+2sqrt53+2 \sqrt{5}$3+2\sqrt{5}$ is irrational.
3. Prove that the following are irrationals :
   (i) $\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$(1)/(sqrt2)\frac{1}{\sqrt{2}}$\frac{1}{\sqrt{2}}$
   (ii) $7\sqrt{5}$$7\sqrt{5}$7sqrt57 \sqrt{5}$7\sqrt{5}$
   (iii) $6+\sqrt{2}$$6+\sqrt{2}$6+sqrt26+\sqrt{2}$6+\sqrt{2}$

1.4 Summary

1. The Fundamental Theorem of Arithmetic :

2. If $p$$p$pp$p$ is a prime and $p$$p$pp$p$ divides $a^2$$a^2$a^(2)a^{2}$a^2$, then $p$$p$pp$p$ divides $a$$a$aa$a$, where $a$$a$aa$a$ is a positive integer.
3. To prove that $\sqrt{2},\sqrt{3}$$\sqrt{2},\sqrt{3}$sqrt2,sqrt3\sqrt{2}, \sqrt{3}$\sqrt{2},\sqrt{3}$ are irrationals.

ANOTE TO THE READER

PolyNOMIALS

2.1 Introduction

2.2 Geometrical Meaning of the Zeroes of a Polynomial

Solution :

EXERCISE 2.1

1. The graphs of $y=p(x)$$y=p(x)$y=p(x)y=p(x)$y=p(x)$ are given in Fig. 2.10 below, for some polynomials $p(x)$$p(x)$p(x)p(x)$p(x)$. Find the number of zeroes of $p(x)$$p(x)$p(x)p(x)$p(x)$, in each case.

2.3 Relationship between Zeroes and Coefficients of a Polynomial

EXERCISE 2.2

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
   (i) $x^2-2x-8$$x^2-2x-8$x^(2)-2x-8x^{2}-2 x-8$x^2-2x-8$
   (ii) $4s^2-4s+1$$4s^2-4s+1$4s^(2)-4s+14 s^{2}-4 s+1$4s^2-4s+1$
   (iii) $6x^2-3-7x$$6x^2-3-7x$6x^(2)-3-7x6 x^{2}-3-7 x$6x^2-3-7x$
   (iv) $4u^2+8u$$4u^2+8u$4u^(2)+8u4 u^{2}+8 u$4u^2+8u$
   (v) $t^2-15$$t^2-15$t^(2)-15t^{2}-15$t^2-15$
   (vi) $3x^2-x-4$$3x^2-x-4$3x^(2)-x-43 x^{2}-x-4$3x^2-x-4$
2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
   (i) $\frac{1}{4},-1$$\frac{1}{4},-1$(1)/(4),-1\frac{1}{4},-1$\frac{1}{4},-1$
   (ii) $\sqrt{2},\frac{1}{3}$$\sqrt{2},\frac{1}{3}$sqrt2,(1)/(3)\sqrt{2}, \frac{1}{3}$\sqrt{2},\frac{1}{3}$
   (iii) $0,\sqrt{5}$$0,\sqrt{5}$0,sqrt50, \sqrt{5}$0,\sqrt{5}$
   (iv) 1,1
   (v) $-\frac{1}{4},\frac{1}{4}$$-\frac{1}{4},\frac{1}{4}$-(1)/(4),(1)/(4)-\frac{1}{4}, \frac{1}{4}$-\frac{1}{4},\frac{1}{4}$
   (vi) 4,1

2.4 Summary

1. Polynomials of degrees 1,2 and 3 are called linear, quadratic and cubic polynomials respectively.
2. A quadratic polynomial in $x$$x$xx$x$ with real coefficients is of the form $a{x}^2+bx+c$$a{x}^2+bx+c$ax^(2)+bx+ca x^{2}+b x+c$a{x}^2+bx+c$, where $a,b,c$$a,b,c$a,b,ca, b, c$a,b,c$ are real numbers with $a\ne 0$$a\ne 0$a!=0a \neq 0$a\ne 0$.
3. The zeroes of a polynomial $p(x)$$p(x)$p(x)p(x)$p(x)$ are precisely the $x$$x$xx$x$-coordinates of the points, where the graph of $y=p(x)$$y=p(x)$y=p(x)y=p(x)$y=p(x)$ intersects the $x$$x$xx$x$-axis.
4. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.
5. If $\alpha$$\alpha$alpha\alpha$\alpha$ and $\beta$$\beta$beta\beta$\beta$ are the zeroes of the quadratic polynomial $a{x}^2+bx+c$$a{x}^2+bx+c$ax^(2)+bx+ca x^{2}+b x+c$a{x}^2+bx+c$, then

6. If $\alpha ,\beta ,\gamma$$\alpha ,\beta ,\gamma$alpha,beta,gamma\alpha, \beta, \gamma$\alpha ,\beta ,\gamma$ are the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$$a{x}^3+b{x}^2+cx+d$ax^(3)+bx^(2)+cx+da x^{3}+b x^{2}+c x+d$a{x}^3+b{x}^2+cx+d$, then

3.1 Introduction

3.2 Graphical Method of Solution of a Pair of Linear Equations

EXERCISE 3.1

1. Form the pair of linear equations in the following problems, and find their solutions graphically.

2. On comparing the ratios $\frac{a_1}{a_2},\frac{b_1}{b_2}$$\frac{a_1}{a_2},\frac{b_1}{b_2}$(a_(1))/(a_(2)),(b_(1))/(b_(2))\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$\frac{a_1}{a_2},\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$$\frac{c_1}{c_2}$(c_(1))/(c_(2))\frac{c_{1}}{c_{2}}$\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
   (i) $5x-4y+8=0$$5x-4y+8=0$5x-4y+8=05 x-4 y+8=0$5x-4y+8=0$
   (ii) $9x+3y+12=0$$9x+3y+12=0$9x+3y+12=09 x+3 y+12=0$9x+3y+12=0$
   $7x+6y-9=0$$7x+6y-9=0$7x+6y-9=07 x+6 y-9=0$7x+6y-9=0$
   $18x+6y+24=0$$18x+6y+24=0$18 x+6y+24=018 x+6 y+24=0$18x+6y+24=0$
   (iii) $6x-3y+10=0$$6x-3y+10=0$6x-3y+10=06 x-3 y+10=0$6x-3y+10=0$
   $2x-y+9=0$$2x-y+9=0$2x-y+9=02 x-y+9=0$2x-y+9=0$
3. On comparing the ratios $\frac{a_1}{a_2},\frac{b_1}{b_2}$$\frac{a_1}{a_2},\frac{b_1}{b_2}$(a_(1))/(a_(2)),(b_(1))/(b_(2))\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$\frac{a_1}{a_2},\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$$\frac{c_1}{c_2}$(c_(1))/(c_(2))\frac{c_{1}}{c_{2}}$\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent.
   (i) $3x+2y=5;2x-3y=7$$3x+2y=5;2x-3y=7$3x+2y=5;quad2x-3y=73 x+2 y=5 ; \quad 2 x-3 y=7$3x+2y=5;2x-3y=7$
   (ii) $2x-3y=8;4x-6y=9$$2x-3y=8;4x-6y=9$2x-3y=8;4x-6y=92 x-3 y=8 ; 4 x-6 y=9$2x-3y=8;4x-6y=9$
   (iii) $\frac{3}{2}x+\frac{5}{3}y=7;9x-10y=14$$\frac{3}{2}x+\frac{5}{3}y=7;9x-10y=14$(3)/(2)x+(5)/(3)y=7;9x-10 y=14\frac{3}{2} x+\frac{5}{3} y=7 ; 9 x-10 y=14$\frac{3}{2}x+\frac{5}{3}y=7;9x-10y=14$
   (iv) $5x-3y=11;-10x+6y=-22$$5x-3y=11;-10x+6y=-22$5x-3y=11;-10 x+6y=-225 x-3 y=11 ;-10 x+6 y=-22$5x-3y=11;-10x+6y=-22$
   (v) $\frac{4}{3}x+2y=8;2x+3y=12$$\frac{4}{3}x+2y=8;2x+3y=12$(4)/(3)x+2y=8;2x+3y=12\frac{4}{3} x+2 y=8 ; 2 x+3 y=12$\frac{4}{3}x+2y=8;2x+3y=12$
4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

5. Half the perimeter of a rectangular garden, whose length is $4\mathrm{m}$$4\mathrm{m}$4m4 \mathrm{~m}$4\mathrm{m}$ more than its width, is $36\mathrm{m}$$36\mathrm{m}$36m36 \mathrm{~m}$36\mathrm{m}$. Find the dimensions of the garden.
6. Given the linear equation $2x+3y-8=0$$2x+3y-8=0$2x+3y-8=02 x+3 y-8=0$2x+3y-8=0$, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
   (i) intersecting lines
   (ii) parallel lines

7. Draw the graphs of the equations $x-y+1=0$$x-y+1=0$x-y+1=0x-y+1=0$x-y+1=0$ and $3x+2y-12=0$$3x+2y-12=0$3x+2y-12=03 x+2 y-12=0$3x+2y-12=0$. Determine the coordinates of the vertices of the triangle formed by these lines and the $x$$x$xx$x$-axis, and shade the triangular region.

3.3 Algebraic Methods of Solving a Pair of Linear Equations

Solution :

EXERCISE 3.2

1. Solve the following pair of linear equations by the substitution method.
   (i) $x+y=14$$x+y=14$x+y=14x+y=14$x+y=14$
   (ii) $s-t=3$$s-t=3$s-t=3s-t=3$s-t=3$
   $x-y=4$$x-y=4$x-y=4x-y=4$x-y=4$
   $\frac{s}{3}+\frac{t}{2}=6$$\frac{s}{3}+\frac{t}{2}=6$(s)/(3)+(t)/(2)=6\frac{s}{3}+\frac{t}{2}=6$\frac{s}{3}+\frac{t}{2}=6$
   (iii) $3x-y=3$$3x-y=3$3x-y=33 x-y=3$3x-y=3$
   (iv) $0.2x+0.3y=1.3$$0.2x+0.3y=1.3$0.2 x+0.3 y=1.30.2 x+0.3 y=1.3$0.2x+0.3y=1.3$
   $9x-3y=9$$9x-3y=9$9x-3y=99 x-3 y=9$9x-3y=9$
   $0.4x+0.5y=2.3$$0.4x+0.5y=2.3$0.4 x+0.5 y=2.30.4 x+0.5 y=2.3$0.4x+0.5y=2.3$
   (v) $\sqrt{2}x+\sqrt{3}y=0$$\sqrt{2}x+\sqrt{3}y=0$sqrt2x+sqrt3y=0\sqrt{2} x+\sqrt{3} y=0$\sqrt{2}x+\sqrt{3}y=0$
   (vi) $\frac{3x}{2}-\frac{5y}{3}=-2$$\frac{3x}{2}-\frac{5y}{3}=-2$(3x)/(2)-(5y)/(3)=-2\frac{3 x}{2}-\frac{5 y}{3}=-2$\frac{3x}{2}-\frac{5y}{3}=-2$
   $\sqrt{3}x-\sqrt{8}y=0$$\sqrt{3}x-\sqrt{8}y=0$sqrt3x-sqrt8y=0\sqrt{3} x-\sqrt{8} y=0$\sqrt{3}x-\sqrt{8}y=0$
   $\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$$\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$(x)/(3)+(y)/(2)=(13)/(6)\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$
2. Solve $2x+3y=11$$2x+3y=11$2x+3y=112 x+3 y=11$2x+3y=11$ and $2x-4y=-24$$2x-4y=-24$2x-4y=-242 x-4 y=-24$2x-4y=-24$ and hence find the value of ' $m$$m$mm$m$ ' for which $y=mx+3$$y=mx+3$y=mx+3y=m x+3$y=mx+3$.
3. Form the pair of linear equations for the following problems and find their solution by substitution method.

3.3.2 Elimination Method

Remarks :

1. The method used in solving the example above is called the elimination method, because we eliminate one variable first, to get a linear equation in one variable.

2. You could also have used the substitution, or graphical method, to solve this problem. Try doing so, and see which method is more convenient.

Solution :

EXERCISE 3.3

1. Solve the following pair of linear equations by the elimination method and the substitution method :
   (i) $x+y=5$$x+y=5$x+y=5x+y=5$x+y=5$ and $2x-3y=4$$2x-3y=4$2x-3y=42 x-3 y=4$2x-3y=4$
   (ii) $3x+4y=10$$3x+4y=10$3x+4y=103 x+4 y=10$3x+4y=10$ and $2x-2y=2$$2x-2y=2$2x-2y=22 x-2 y=2$2x-2y=2$
   (iii) $3x-5y-4=0$$3x-5y-4=0$3x-5y-4=03 x-5 y-4=0$3x-5y-4=0$ and $9x=2y+7$$9x=2y+7$9x=2y+79 x=2 y+7$9x=2y+7$
   (iv) $\frac{x}{2}+\frac{2y}{3}=-1$$\frac{x}{2}+\frac{2y}{3}=-1$(x)/(2)+(2y)/(3)=-1\frac{x}{2}+\frac{2 y}{3}=-1$\frac{x}{2}+\frac{2y}{3}=-1$ and $x-\frac{y}{3}=3$$x-\frac{y}{3}=3$x-(y)/(3)=3x-\frac{y}{3}=3$x-\frac{y}{3}=3$
2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:

3.4 Summary

1. A pair of linear equations in two variables can be represented, and solved, by the:

2. Graphical Method :

3. Algebraic Methods : We have discussed the following methods for finding the solution(s) of a pair of linear equations :

4. If a pair of linear equations is given by $a_{1}x+b_{1}y+c_1=0$$a_{1}x+b_{1}y+c_1=0$a_(1)x+b_(1)y+c_(1)=0a_{1} x+b_{1} y+c_{1}=0$a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$$a_{2}x+b_{2}y+c_2=0$a_(2)x+b_(2)y+c_(2)=0a_{2} x+b_{2} y+c_{2}=0$a_{2}x+b_{2}y+c_2=0$, then the following situations can arise :

5. There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so that they are reduced to a pair of linear equations.

Quadratic Equations

4.1 Introduction

4.2 Quadratic Equations

Solution :

Solution :

EXERCISE 4.1

1. Check whether the following are quadratic equations :
   (i) $(x+1{)}^2=2(x-3)$$(x+1{)}^2=2(x-3)$(x+1)^(2)=2(x-3)(x+1)^{2}=2(x-3)$(x+1{)}^2=2(x-3)$
   (ii) $x^2-2x=(-2)(3-x)$$x^2-2x=(-2)(3-x)$x^(2)-2x=(-2)(3-x)x^{2}-2 x=(-2)(3-x)$x^2-2x=(-2)(3-x)$
   (iii) $(x-2)(x+1)=(x-1)(x+3)$$(x-2)(x+1)=(x-1)(x+3)$(x-2)(x+1)=(x-1)(x+3)(x-2)(x+1)=(x-1)(x+3)$(x-2)(x+1)=(x-1)(x+3)$
   (iv) $(x-3)(2x+1)=x(x+5)$$(x-3)(2x+1)=x(x+5)$(x-3)(2x+1)=x(x+5)(x-3)(2 x+1)=x(x+5)$(x-3)(2x+1)=x(x+5)$
   (v) $(2x-1)(x-3)=(x+5)(x-1)$$(2x-1)(x-3)=(x+5)(x-1)$(2x-1)(x-3)=(x+5)(x-1)(2 x-1)(x-3)=(x+5)(x-1)$(2x-1)(x-3)=(x+5)(x-1)$
   (vi) $x^2+3x+1=(x-2{)}^2$$x^2+3x+1=(x-2{)}^2$x^(2)+3x+1=(x-2)^(2)x^{2}+3 x+1=(x-2)^{2}$x^2+3x+1=(x-2{)}^2$
   (vii) $(x+2{)}^3=2x(x^2-1)$$(x+2{)}^3=2x\left(x^2-1\right)$(x+2)^(3)=2x(x^(2)-1)(x+2)^{3}=2 x\left(x^{2}-1\right)$(x+2{)}^3=2x(x^2-1)$
   (viii) $x^3-4x^2-x+1=(x-2{)}^3$$x^3-4x^2-x+1=(x-2{)}^3$x^(3)-4x^(2)-x+1=(x-2)^(3)x^{3}-4 x^{2}-x+1=(x-2)^{3}$x^3-4x^2-x+1=(x-2{)}^3$
2. Represent the following situations in the form of quadratic equations :

4.3 Solution of a Quadratic Equation by Factorisation

EXERCISE 4.2

1. Find the roots of the following quadratic equations by factorisation:
   (i) $x^2-3x-10=0$$x^2-3x-10=0$x^(2)-3x-10=0x^{2}-3 x-10=0$x^2-3x-10=0$
   (ii) $2x^2+x-6=0$$2x^2+x-6=0$2x^(2)+x-6=02 x^{2}+x-6=0$2x^2+x-6=0$
   (iii) $\sqrt{2}{x}^2+7x+5\sqrt{2}=0$$\sqrt{2}{x}^2+7x+5\sqrt{2}=0$sqrt2x^(2)+7x+5sqrt2=0\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$\sqrt{2}{x}^2+7x+5\sqrt{2}=0$
   (iv) $2x^2-x+\frac{1}{8}=0$$2x^2-x+\frac{1}{8}=0$2x^(2)-x+(1)/(8)=02 x^{2}-x+\frac{1}{8}=0$2x^2-x+\frac{1}{8}=0$
   (v) $100x^2-20x+1=0$$100x^2-20x+1=0$100x^(2)-20 x+1=0100 x^{2}-20 x+1=0$100x^2-20x+1=0$
2. Solve the problems given in Example 1.
3. Find two numbers whose sum is 27 and product is 182 .
4. Find two consecutive positive integers, sum of whose squares is 365 .
5. The altitude of a right triangle is $7\mathrm{c}\mathrm{m}$$7\mathrm{c}\mathrm{m}$7cm7 \mathrm{~cm}$7\mathrm{c}\mathrm{m}$ less than its base. If the hypotenuse is $13\mathrm{c}\mathrm{m}$$13\mathrm{c}\mathrm{m}$13cm13 \mathrm{~cm}$13\mathrm{c}\mathrm{m}$, find the other two sides.
6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ 90 , find the number of articles produced and the cost of each article.

4.4 Nature of Roots

EXERCISE 4.3

1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
   (i) $2x^2-3x+5=0$$2x^2-3x+5=0$2x^(2)-3x+5=02 x^{2}-3 x+5=0$2x^2-3x+5=0$
   (ii) $3x^2-4\sqrt{3}x+4=0$$3x^2-4\sqrt{3}x+4=0$3x^(2)-4sqrt3x+4=03 x^{2}-4 \sqrt{3} x+4=0$3x^2-4\sqrt{3}x+4=0$
   (iii) $2x^2-6x+3=0$$2x^2-6x+3=0$2x^(2)-6x+3=02 x^{2}-6 x+3=0$2x^2-6x+3=0$
2. Find the values of $k$$k$kk$k$ for each of the following quadratic equations, so that they have two equal roots.
   (i) $2x^2+kx+3=0$$2x^2+kx+3=0$2x^(2)+kx+3=02 x^{2}+k x+3=0$2x^2+kx+3=0$
   (ii) $kx(x-2)+6=0$$kx(x-2)+6=0$kx(x-2)+6=0k x(x-2)+6=0$kx(x-2)+6=0$
3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800{\mathrm{m}}^2$$800{\mathrm{m}}^2$800m^(2)800 \mathrm{~m}^{2}$800{\mathrm{m}}^2$ ? If so, find its length and breadth.
4. Is the following situation possible? If so, determine their present ages.

5. Is it possible to design a rectangular park of perimeter $80\mathrm{m}$$80\mathrm{m}$80m80 \mathrm{~m}$80\mathrm{m}$ and area $400{\mathrm{m}}^2$$400{\mathrm{m}}^2$400m^(2)400 \mathrm{~m}^{2}$400{\mathrm{m}}^2$ ? If so, find its length and breadth.

4.5 Summary

1. A quadratic equation in the variable $x$$x$xx$x$ is of the form $a{x}^2+bx+c=0$$a{x}^2+bx+c=0$ax^(2)+bx+c=0a x^{2}+b x+c=0$a{x}^2+bx+c=0$, where $a,b,c$$a,b,c$a,b,ca, b, c$a,b,c$ are real numbers and $a\ne 0$$a\ne 0$a!=0a \neq 0$a\ne 0$.
2. A real number $\alpha$$\alpha$alpha\alpha$\alpha$ is said to be a root of the quadratic equation $a{x}^2+bx+c=0$$a{x}^2+bx+c=0$ax^(2)+bx+c=0a x^{2}+b x+c=0$a{x}^2+bx+c=0$, if $a{\alpha }^2+b\alpha +c=0$$a{\alpha }^2+b\alpha +c=0$aalpha^(2)+b alpha+c=0a \alpha^{2}+b \alpha+c=0$a{\alpha }^2+b\alpha +c=0$. The zeroes of the quadratic polynomial $a{x}^2+bx+c$$a{x}^2+bx+c$ax^(2)+bx+ca x^{2}+b x+c$a{x}^2+bx+c$ and the roots of the quadratic equation $a{x}^2+bx+c=0$$a{x}^2+bx+c=0$ax^(2)+bx+c=0a x^{2}+b x+c=0$a{x}^2+bx+c=0$ are the same.
3. If we can factorise $a{x}^2+bx+c,a\ne 0$$a{x}^2+bx+c,a\ne 0$ax^(2)+bx+c,a!=0a x^{2}+b x+c, a \neq 0$a{x}^2+bx+c,a\ne 0$, into a product of two linear factors, then the roots of the quadratic equation $a{x}^2+bx+c=0$$a{x}^2+bx+c=0$ax^(2)+bx+c=0a x^{2}+b x+c=0$a{x}^2+bx+c=0$ can be found by equating each factor to zero.
4. Quadratic formula: The roots of a quadratic equation $a{x}^2+bx+c=0$$a{x}^2+bx+c=0$ax^(2)+bx+c=0a x^{2}+b x+c=0$a{x}^2+bx+c=0$ are given by $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$(-b+-sqrt(b^(2)-4ac))/(2a)\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, provided $b^2-4ac\ge 0$$b^2-4ac\ge 0$b^(2)-4ac >= 0b^{2}-4 a c \geq 0$b^2-4ac\ge 0$.
5. A quadratic equation $a{x}^2+bx+c=0$$a{x}^2+bx+c=0$ax^(2)+bx+c=0a x^{2}+b x+c=0$a{x}^2+bx+c=0$ has

Arithmetic Progressions

5.1 Introduction

2.2 Arithmetic Progressions

EXERCISE 5.1

1. In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?

2. Write first four terms of the AP, when the first term $a$$a$aa$a$ and the common difference $d$$d$dd$d$ are given as follows:

3. For the following APs, write the first term and the common difference:
   (i) $3,1,-1,-3,\dots$$3,1,-1,-3,\dots$3,1,-1,-3,dots3,1,-1,-3, \ldots$3,1,-1,-3,\dots$
   (ii) $-5,-1,3,7,\dots$$-5,-1,3,7,\dots$-5,-1,3,7,dots-5,-1,3,7, \ldots$-5,-1,3,7,\dots$
   (iii) $\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{13}{3},\dots$$\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{13}{3},\dots$(1)/(3),(5)/(3),(9)/(3),(13)/(3),dots\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \ldots$\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{13}{3},\dots$
   (iv) $0.6,1.7,2.8,3.9,\dots$$0.6,1.7,2.8,3.9,\dots$0.6,1.7,2.8,3.9,dots0.6,1.7,2.8,3.9, \ldots$0.6,1.7,2.8,3.9,\dots$
4. Which of the following are APs ? If they form an AP, find the common difference $d$$d$dd$d$ and write three more terms.
   (i) $2,4,8,16,\dots$$2,4,8,16,\dots$2,4,8,16,dots2,4,8,16, \ldots$2,4,8,16,\dots$
   (ii) $2,\frac{5}{2},3,\frac{7}{2},\dots$$2,\frac{5}{2},3,\frac{7}{2},\dots$2,(5)/(2),3,(7)/(2),dots2, \frac{5}{2}, 3, \frac{7}{2}, \ldots$2,\frac{5}{2},3,\frac{7}{2},\dots$
   (iii) $-1.2,-3.2,-5.2,-7.2,\dots$$-1.2,-3.2,-5.2,-7.2,\dots$-1.2,-3.2,-5.2,-7.2,dots-1.2,-3.2,-5.2,-7.2, \ldots$-1.2,-3.2,-5.2,-7.2,\dots$
   (iv) $-10,-6,-2,2,\dots$$-10,-6,-2,2,\dots$-10,-6,-2,2,dots-10,-6,-2,2, \ldots$-10,-6,-2,2,\dots$
   (v) $3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2},\dots$$3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2},\dots$3,3+sqrt2,3+2sqrt2,3+3sqrt2,dots3,3+\sqrt{2}, 3+2 \sqrt{2}, 3+3 \sqrt{2}, \ldots$3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2},\dots$
   (vi) $0.2,0.22,0.222,0.2222,\dots$$0.2,0.22,0.222,0.2222,\dots$0.2,0.22,0.222,0.2222,dots0.2,0.22,0.222,0.2222, \ldots$0.2,0.22,0.222,0.2222,\dots$
   (vii) $0,-4,-8,-12,\dots$$0,-4,-8,-12,\dots$0,-4,-8,-12,dots0,-4,-8,-12, \ldots$0,-4,-8,-12,\dots$
   (viii) $-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\dots$$-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\dots$-(1)/(2),-(1)/(2),-(1)/(2),-(1)/(2),dots-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}, \ldots$-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\dots$
   (ix) $1,3,9,27,\dots$$1,3,9,27,\dots$1,3,9,27,dots1,3,9,27, \ldots$1,3,9,27,\dots$
   (x) $a,2a,3a,4a,\dots$$a,2a,3a,4a,\dots$a,2a,3a,4a,dotsa, 2 a, 3 a, 4 a, \ldots$a,2a,3a,4a,\dots$
   (xi) $a,a^2,a^3,a^4,\dots$$a,a^2,a^3,a^4,\dots$a,a^(2),a^(3),a^(4),dotsa, a^{2}, a^{3}, a^{4}, \ldots$a,a^2,a^3,a^4,\dots$
   (xii) $\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},\cdots$$\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},\cdots$sqrt2,sqrt8,sqrt18,sqrt32,cdots\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \cdots$\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},\cdots$
   (xiii) $\sqrt{3},\sqrt{6},\sqrt{9},\sqrt{12},\dots$$\sqrt{3},\sqrt{6},\sqrt{9},\sqrt{12},\dots$sqrt3,sqrt6,sqrt9,sqrt12,dots\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots$\sqrt{3},\sqrt{6},\sqrt{9},\sqrt{12},\dots$
   (xiv) $1^2,3^2,5^2,7^2,\dots$$1^2,3^2,5^2,7^2,\dots$1^(2),3^(2),5^(2),7^(2),dots1^{2}, 3^{2}, 5^{2}, 7^{2}, \ldots$1^2,3^2,5^2,7^2,\dots$
   (xv) $1^2,5^2,7^2,73,\dots$$1^2,5^2,7^2,73,\dots$1^(2),5^(2),7^(2),73,dots1^{2}, 5^{2}, 7^{2}, 73, \ldots$1^2,5^2,7^2,73,\dots$

$5.3n$$5.3n$5.3 n5.3 n$5.3n$th Term of an AP

Alternative Solution :

EXERCISE 2.2

1. Fill in the blanks in the following table, given that $a$$a$aa$a$ is the first term, $d$$d$dd$d$ the common difference and $a_n$$a_n$a_(n)a_{n}$a_n$ the $n$$n$nn$n$th term of the AP:

2. Choose the correct choice in the following and justify :

3. In the following APs, find the missing terms in the boxes :

4. Which term of the AP : $3,8,13,18,\dots$$3,8,13,18,\dots$3,8,13,18,dots3,8,13,18, \ldots$3,8,13,18,\dots$,is 78 ?
5. Find the number of terms in each of the following APs :
   (i) $7,13,19,\dots ,205$$7,13,19,\dots ,205$7,13,19,dots,2057,13,19, \ldots, 205$7,13,19,\dots ,205$
   (ii) $18,15\frac{1}{2},13,\dots ,-47$$18,15\frac{1}{2},13,\dots ,-47$18,15(1)/(2),13,dots,-4718,15 \frac{1}{2}, 13, \ldots,-47$18,15\frac{1}{2},13,\dots ,-47$
6. Check whether -150 is a term of the AP : $11,8,5,2\dots$$11,8,5,2\dots$11,8,5,2dots11,8,5,2 \ldots$11,8,5,2\dots$
7. Find the 31 st term of an AP whose 11 th term is 38 and the 16th term is 73.
8. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
9. If the $3rd$$3rd$3rd3 r d$3rd$ and the 9 th terms of an AP are 4 and -8 respectively, which term of this AP is zero?
10. The 17th term of an AP exceeds its 10 th term by 7 . Find the common difference.
11. Which term of the AP : $3,15,27,39,\dots$$3,15,27,39,\dots$3,15,27,39,dots3,15,27,39, \ldots$3,15,27,39,\dots$ will be 132 more than its 54 th term?
12. Two APs have the same common difference. The difference between their 100th terms is 100 , what is the difference between their 1000th terms?
13. How many three-digit numbers are divisible by 7 ?
14. How many multiples of 4 lie between 10 and 250 ?
15. For what value of $n$$n$nn$n$, are the $n$$n$nn$n$th terms of two APs: $63,65,67,\dots$$63,65,67,\dots$63,65,67,dots63,65,67, \ldots$63,65,67,\dots$ and $3,10,17,\dots$$3,10,17,\dots$3,10,17,dots3,10,17, \ldots$3,10,17,\dots$ equal?
16. Determine the AP whose third term is 16 and the 7 th term exceeds the 5 th term by 12 .
17. Find the 20th term from the last term of the AP : $3,8,13,\dots ,253$$3,8,13,\dots ,253$3,8,13,dots,2533,8,13, \ldots, 253$3,8,13,\dots ,253$.
18. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
19. Subba Rao started work in 1995 at an annual salary of ₹ 5000 and received an increment of ₹ 200 each year. In which year did his income reach ₹ 7000 ?
20. Ramkali saved ₹ 5 in the first week of a year and then increased her weekly savings by ₹ 1.75 . If in the $n$$n$nn$n$th week, her weekly savings become ₹ 20.75 , find $n$$n$nn$n$.

5.4 Sum of First $n$$n$nn$n$ Terms of an AP

Remarks:

1. In this case, the sum of the first 4 terms $=$$=$==$=$ the sum of the first 13 terms $=78$$=78$=78=78$=78$.
2. Two answers are possible because the sum of the terms from 5th to 13 th will be zero. This is because $a$$a$aa$a$ is positive and $d$$d$dd$d$ is negative, so that some terms will be positive and some others negative, and will cancel out each other.

Solution :

EXERCISE 5.3

1. Find the sum of the following APs:
   (i) $2,7,12,\dots$$2,7,12,\dots$2,7,12,dots2,7,12, \ldots$2,7,12,\dots$, to 10 terms.
   (ii) $-37,-33,-29,\dots$$-37,-33,-29,\dots$-37,-33,-29,dots-37,-33,-29, \ldots$-37,-33,-29,\dots$, to 12 terms.
   (iii) $0.6,1.7,2.8,\dots$$0.6,1.7,2.8,\dots$0.6,1.7,2.8,dots0.6,1.7,2.8, \ldots$0.6,1.7,2.8,\dots$, to 100 terms.
   (iv) $\frac{1}{15},\frac{1}{12},\frac{1}{10},\dots$$\frac{1}{15},\frac{1}{12},\frac{1}{10},\dots$(1)/(15),(1)/(12),(1)/(10),dots\frac{1}{15}, \frac{1}{12}, \frac{1}{10}, \ldots$\frac{1}{15},\frac{1}{12},\frac{1}{10},\dots$, to 11 terms.
2. Find the sums given below :
   (i) $7+10\frac{1}{2}+14+\dots +84$$7+10\frac{1}{2}+14+\dots +84$7+10(1)/(2)+14+dots+847+10 \frac{1}{2}+14+\ldots+84$7+10\frac{1}{2}+14+\dots +84$
   (ii) $34+32+30+\dots +10$$34+32+30+\dots +10$34+32+30+dots+1034+32+30+\ldots+10$34+32+30+\dots +10$
   (iii) $-5+(-8)+(-11)+\dots +(-230)$$-5+(-8)+(-11)+\dots +(-230)$-5+(-8)+(-11)+dots+(-230)-5+(-8)+(-11)+\ldots+(-230)$-5+(-8)+(-11)+\dots +(-230)$
3. In an AP:

4. How many terms of the AP : $9,17,25,\dots$$9,17,25,\dots$9,17,25,dots9,17,25, \ldots$9,17,25,\dots$ must be taken to give a sum of 636 ?
5. The first term of an $\mathrm{A}\mathrm{P}$$\mathrm{A}\mathrm{P}$AP\mathrm{AP}$\mathrm{A}\mathrm{P}$ is 5 , the last term is 45 and the sum is 400 . Find the number of terms and the common difference.
6. The first and the last terms of an $\mathrm{A}\mathrm{P}$$\mathrm{A}\mathrm{P}$AP\mathrm{AP}$\mathrm{A}\mathrm{P}$ are 17 and 350 respectively. If the common difference is 9 , how many terms are there and what is their sum?
7. Find the sum of first 22 terms of an AP in which $d=7$$d=7$d=7d=7$d=7$ and 22 nd term is 149 .
8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289 , find the sum of first $n$$n$nn$n$ terms.
10. Show that $a_1,a_2,\dots ,a_n,\dots$$a_1,a_2,\dots ,a_n,\dots$a_(1),a_(2),dots,a_(n),dotsa_{1}, a_{2}, \ldots, a_{n}, \ldots$a_1,a_2,\dots ,a_n,\dots$ form an AP where $a_n$$a_n$a_(n)a_{n}$a_n$ is defined as below :
    (i) $a_n=3+4n$$a_n=3+4n$a_(n)=3+4na_{n}=3+4 n$a_n=3+4n$
    (ii) $a_n=9-5n$$a_n=9-5n$a_(n)=9-5na_{n}=9-5 n$a_n=9-5n$

11. If the sum of the first $n$$n$nn$n$ terms of an $\mathrm{A}\mathrm{P}$$\mathrm{A}\mathrm{P}$AP\mathrm{AP}$\mathrm{A}\mathrm{P}$ is $4n-n^2$$4n-n^2$4n-n^(2)4 n-n^{2}$4n-n^2$, what is the first term (that is ${\mathrm{S}}_1$${\mathrm{S}}_1$S_(1)\mathrm{S}_{1}${\mathrm{S}}_1$ )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the $n$$n$nn$n$th terms.
12. Find the sum of the first 40 positive integers divisible by 6 .
13. Find the sum of the first 15 multiples of 8 .
14. Find the sum of the odd numbers between 0 and 50 .
15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day, etc., the penalty for each succeeding day being ₹ 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
16. A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹ 20 less than its preceding prize, find the value of each of the prizes.
17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?
18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii $0.5\mathrm{c}\mathrm{m},1.0\mathrm{c}\mathrm{m},1.5\mathrm{c}\mathrm{m},2.0\mathrm{c}\mathrm{m},\dots$$0.5\mathrm{c}\mathrm{m},1.0\mathrm{c}\mathrm{m},1.5\mathrm{c}\mathrm{m},2.0\mathrm{c}\mathrm{m},\dots$0.5cm,1.0cm,1.5cm,2.0cm,dots0.5 \mathrm{~cm}, 1.0 \mathrm{~cm}, 1.5 \mathrm{~cm}, 2.0 \mathrm{~cm}, \ldots$0.5\mathrm{c}\mathrm{m},1.0\mathrm{c}\mathrm{m},1.5\mathrm{c}\mathrm{m},2.0\mathrm{c}\mathrm{m},\dots$ as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? $($$\left(\right.$(:}\left(\right.$($ Take $\pi =\frac{22}{7})$$\left.\pi =\frac{22}{7}\right)${: pi=(22)/(7))\left.\pi=\frac{22}{7}\right)$\pi =\frac{22}{7})$

20. In a potato race, a bucket is placed at the starting point, which is $5\mathrm{m}$$5\mathrm{m}$5m5 \mathrm{~m}$5\mathrm{m}$ from the first potato, and the other potatoes are placed $3\mathrm{m}$$3\mathrm{m}$3m3 \mathrm{~m}$3\mathrm{m}$ apart in a straight line. There are ten potatoes in the line (see Fig. 5.6).

EXERCISE 5.4 (Optional)

1. Which term of the AP: $121,117,113,\dots$$121,117,113,\dots$121,117,113,dots121,117,113, \ldots$121,117,113,\dots$, is its first negative term?

2. The sum of the third and the seventh terms of an AP is 6 and their product is 8 . Find the sum of first sixteen terms of the AP.
3. A ladder has rungs $25\mathrm{c}\mathrm{m}$$25\mathrm{c}\mathrm{m}$25cm25 \mathrm{~cm}$25\mathrm{c}\mathrm{m}$ apart. (see Fig. 5.7). The rungs decrease uniformly in length from $45\mathrm{c}\mathrm{m}$$45\mathrm{c}\mathrm{m}$45cm45 \mathrm{~cm}$45\mathrm{c}\mathrm{m}$ at the bottom to $25\mathrm{c}\mathrm{m}$$25\mathrm{c}\mathrm{m}$25cm25 \mathrm{~cm}$25\mathrm{c}\mathrm{m}$ at the top. If the top and the bottom rungs are $2\frac{1}{2}\mathrm{m}$$2\frac{1}{2}\mathrm{m}$2(1)/(2)m2 \frac{1}{2} \mathrm{~m}$2\frac{1}{2}\mathrm{m}$ apart, what is the length of the wood required for the rungs?

4. The houses of a row are numbered consecutively from 1 to 49 . Show that there is a value of $x$$x$xx$x$ such that the sum of the numbers of the houses preceding the house numbered $x$$x$xx$x$ is equal to the sum of the numbers of the houses following it. Find this value of $x$$x$xx$x$.

5. A small terrace at a football ground comprises of 15 steps each of which is $50\mathrm{m}$$50\mathrm{m}$50m50 \mathrm{~m}$50\mathrm{m}$ long and built of solid concrete.

5.5 Summary

1. An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number $d$$d$dd$d$ to the preceding term, except the first term. The fixed number $d$$d$dd$d$ is called the common difference.

2. A given list of numbers $a_1,a_2,a_3,\dots$$a_1,a_2,a_3,\dots$a_(1),a_(2),a_(3),dotsa_{1}, a_{2}, a_{3}, \ldots$a_1,a_2,a_3,\dots$ is an AP, if the differences $a_2-a_1,a_3-a_2$$a_2-a_1,a_3-a_2$a_(2)-a_(1),a_(3)-a_(2)a_{2}-a_{1}, a_{3}-a_{2}$a_2-a_1,a_3-a_2$, $a_4-a_3,\dots$$a_4-a_3,\dots$a_(4)-a_(3),dotsa_{4}-a_{3}, \ldots$a_4-a_3,\dots$, give the same value, i.e., if $a_{k+1}-a_k$$a_{k+1}-a_k$a_(k+1)-a_(k)a_{k+1}-a_{k}$a_{k+1}-a_k$ is the same for different values of $k$$k$kk$k$.
3. In an AP with first term $a$$a$aa$a$ and common difference $d$$d$dd$d$, the $n$$n$nn$n$th term (or the general term) is given by $a_n=a+(n-1)d$$a_n=a+(n-1)d$a_(n)=a+(n-1)da_{n}=a+(n-1) d$a_n=a+(n-1)d$.
4. The sum of the first $n$$n$nn$n$ terms of an AP is given by :

5. If $l$$l$ll$l$ is the last term of the finite AP, say the $n$$n$nn$n$th term, then the sum of all terms of the AP is given by :

A NOTE TO THE READER

6.1 Introduction

6.2 Similar Figures

EXERCISE 6.1

1. Fill in the blanks using the correct word given in brackets :

2. Give two different examples of pair of

3. State whether the following quadrilaterals are similar or not:

6.3 Similarity of Triangles

Solution :

EXERCISE 6.2

1. In Fig. 6.17, (i) and (ii), $\mathrm{D}\mathrm{E}\parallel \mathrm{B}\mathrm{C}$$\mathrm{D}\mathrm{E}\parallel \mathrm{B}\mathrm{C}$DE||BC\mathrm{DE} \| \mathrm{BC}$\mathrm{D}\mathrm{E}\parallel \mathrm{B}\mathrm{C}$. Find $\mathrm{E}\mathrm{C}$$\mathrm{E}\mathrm{C}$EC\mathrm{EC}$\mathrm{E}\mathrm{C}$ in (i) and $\mathrm{A}\mathrm{D}$$\mathrm{A}\mathrm{D}$AD\mathrm{AD}$\mathrm{A}\mathrm{D}$ in (ii).

3. In Fig. 6.18, if $\mathrm{L}\mathrm{M}\parallel \mathrm{C}\mathrm{B}$$\mathrm{L}\mathrm{M}\parallel \mathrm{C}\mathrm{B}$LM||CB\mathrm{LM} \| \mathrm{CB}$\mathrm{L}\mathrm{M}\parallel \mathrm{C}\mathrm{B}$ and $\mathrm{L}\mathrm{N}\parallel \mathrm{C}\mathrm{D}$$\mathrm{L}\mathrm{N}\parallel \mathrm{C}\mathrm{D}$LN||CD\mathrm{LN} \| \mathrm{CD}$\mathrm{L}\mathrm{N}\parallel \mathrm{C}\mathrm{D}$, prove that

4. In Fig. 6.19, $\mathrm{D}\mathrm{E}\parallel \mathrm{A}\mathrm{C}$$\mathrm{D}\mathrm{E}\parallel \mathrm{A}\mathrm{C}$DE||AC\mathrm{DE} \| \mathrm{AC}$\mathrm{D}\mathrm{E}\parallel \mathrm{A}\mathrm{C}$ and $\mathrm{D}\mathrm{F}\parallel \mathrm{A}\mathrm{E}$$\mathrm{D}\mathrm{F}\parallel \mathrm{A}\mathrm{E}$DF||AE\mathrm{DF} \| \mathrm{AE}$\mathrm{D}\mathrm{F}\parallel \mathrm{A}\mathrm{E}$. Prove that

5. In Fig. 6.20, $DE\parallel OQ$$DE\parallel OQ$DE||OQD E \| O Q$DE\parallel OQ$ and $DF\parallel OR$$DF\parallel OR$DF||ORD F \| O R$DF\parallel OR$. Show that $\mathrm{E}\mathrm{F}\parallel \mathrm{Q}\mathrm{R}$$\mathrm{E}\mathrm{F}\parallel \mathrm{Q}\mathrm{R}$EF||QR\mathrm{EF} \| \mathrm{QR}$\mathrm{E}\mathrm{F}\parallel \mathrm{Q}\mathrm{R}$.
6. In Fig. 6.21, $\mathrm{A},\mathrm{B}$$\mathrm{A},\mathrm{B}$A,B\mathrm{A}, \mathrm{B}$\mathrm{A},\mathrm{B}$ and $\mathrm{C}$$\mathrm{C}$C\mathrm{C}$\mathrm{C}$ are points on $\mathrm{O}\mathrm{P},\mathrm{O}\mathrm{Q}$$\mathrm{O}\mathrm{P},\mathrm{O}\mathrm{Q}$OP,OQ\mathrm{OP}, \mathrm{OQ}$\mathrm{O}\mathrm{P},\mathrm{O}\mathrm{Q}$ and $OR$$OR$ORO R$OR$ respectively such that $AB\parallel PQ$$AB\parallel PQ$AB||PQA B \| P Q$AB\parallel PQ$ and $AC\parallel PR$$AC\parallel PR$AC||PRA C \| P R$AC\parallel PR$. Show that $BC\parallel QR$$BC\parallel QR$BC||QRB C \| Q R$BC\parallel QR$.
7. Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).
8. Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).
9. $ABCD$$ABCD$ABCDA B C D$ABCD$ is a trapezium in which $AB\parallel DC$$AB\parallel DC$AB||DCA B \| D C$AB\parallel DC$ and its diagonals intersect each other at the point $O$$O$OO$O$. Show that $\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}$$\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}$(AO)/(BO)=(CO)/(DO)\frac{\mathrm{AO}}{\mathrm{BO}}=\frac{\mathrm{CO}}{\mathrm{DO}}$\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}$.

10. The diagonals of a quadrilateral $ABCD$$ABCD$ABCDA B C D$ABCD$ intersect each other at the point $O$$O$OO$O$ such that $\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}\cdot$$\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}\cdot$(AO)/(BO)=(CO)/(DO)*\frac{\mathrm{AO}}{\mathrm{BO}}=\frac{\mathrm{CO}}{\mathrm{DO}} \cdot$\frac{\mathrm{A}\mathrm{O}}{\mathrm{B}\mathrm{O}}=\frac{\mathrm{C}\mathrm{O}}{\mathrm{D}\mathrm{O}}\cdot$ Show that $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ABCD\mathrm{ABCD}$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a trapezium.

6.4 Criteria for Similarity of Triangles

EXERCISE 6.3

1. State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :

2. In Fig. $6.35,\mathrm{\Delta }\mathrm{O}\mathrm{D}\mathrm{C}\sim \mathrm{\Delta }\mathrm{O}\mathrm{B}\mathrm{A},\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{C}={125}^{\circ }$$6.35,\mathrm{\Delta }\mathrm{O}\mathrm{D}\mathrm{C}\sim \mathrm{\Delta }\mathrm{O}\mathrm{B}\mathrm{A},\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{C}={125}^{\circ }$6.35,DeltaODC∼DeltaOBA,/_BOC=125^(@)6.35, \Delta \mathrm{ODC} \sim \Delta \mathrm{OBA}, \angle \mathrm{BOC}=125^{\circ}$6.35,\mathrm{\Delta }\mathrm{O}\mathrm{D}\mathrm{C}\sim \mathrm{\Delta }\mathrm{O}\mathrm{B}\mathrm{A},\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{C}={125}^{\circ }$ and $\mathrm{\angle }\mathrm{C}\mathrm{D}\mathrm{O}={70}^{\circ }$$\mathrm{\angle }\mathrm{C}\mathrm{D}\mathrm{O}={70}^{\circ }$/_CDO=70^(@)\angle \mathrm{CDO}=70^{\circ}$\mathrm{\angle }\mathrm{C}\mathrm{D}\mathrm{O}={70}^{\circ }$. Find $\mathrm{\angle }\mathrm{D}\mathrm{O}\mathrm{C},\mathrm{\angle }\mathrm{D}\mathrm{C}\mathrm{O}$$\mathrm{\angle }\mathrm{D}\mathrm{O}\mathrm{C},\mathrm{\angle }\mathrm{D}\mathrm{C}\mathrm{O}$/_DOC,/_DCO\angle \mathrm{DOC}, \angle \mathrm{DCO}$\mathrm{\angle }\mathrm{D}\mathrm{O}\mathrm{C},\mathrm{\angle }\mathrm{D}\mathrm{C}\mathrm{O}$ and $\mathrm{\angle }\mathrm{O}\mathrm{A}\mathrm{B}$$\mathrm{\angle }\mathrm{O}\mathrm{A}\mathrm{B}$/_OAB\angle \mathrm{OAB}$\mathrm{\angle }\mathrm{O}\mathrm{A}\mathrm{B}$.
3. Diagonals $AC$$AC$ACA C$AC$ and $BD$$BD$BDB D$BD$ of a trapezium $ABCD$$ABCD$ABCDA B C D$ABCD$ with $\mathrm{A}\mathrm{B}\parallel \mathrm{D}\mathrm{C}$$\mathrm{A}\mathrm{B}\parallel \mathrm{D}\mathrm{C}$AB||DC\mathrm{AB} \| \mathrm{DC}$\mathrm{A}\mathrm{B}\parallel \mathrm{D}\mathrm{C}$ intersect each other at the point $\mathrm{O}$$\mathrm{O}$O\mathrm{O}$\mathrm{O}$. Using a similarity criterion for two

4. In Fig. 6.36, $\frac{\mathrm{Q}\mathrm{R}}{\mathrm{Q}\mathrm{S}}=\frac{\mathrm{Q}\mathrm{T}}{\mathrm{P}\mathrm{R}}$$\frac{\mathrm{Q}\mathrm{R}}{\mathrm{Q}\mathrm{S}}=\frac{\mathrm{Q}\mathrm{T}}{\mathrm{P}\mathrm{R}}$(QR)/(QS)=(QT)/(PR)\frac{\mathrm{QR}}{\mathrm{QS}}=\frac{\mathrm{QT}}{\mathrm{PR}}$\frac{\mathrm{Q}\mathrm{R}}{\mathrm{Q}\mathrm{S}}=\frac{\mathrm{Q}\mathrm{T}}{\mathrm{P}\mathrm{R}}$ and $\mathrm{\angle }1=\mathrm{\angle }2$$\mathrm{\angle }1=\mathrm{\angle }2$/_1=/_2\angle 1=\angle 2$\mathrm{\angle }1=\mathrm{\angle }2$. Show that $\mathrm{△}\mathrm{P}\mathrm{Q}\mathrm{S}\sim \mathrm{\Delta }\mathrm{T}\mathrm{Q}\mathrm{R}$$\mathrm{△}\mathrm{P}\mathrm{Q}\mathrm{S}\sim \mathrm{\Delta }\mathrm{T}\mathrm{Q}\mathrm{R}$/_\PQS∼DeltaTQR\triangle \mathrm{PQS} \sim \Delta \mathrm{TQR}$\mathrm{△}\mathrm{P}\mathrm{Q}\mathrm{S}\sim \mathrm{\Delta }\mathrm{T}\mathrm{Q}\mathrm{R}$.
5. $\mathrm{S}$$\mathrm{S}$S\mathrm{S}$\mathrm{S}$ and $\mathrm{T}$$\mathrm{T}$T\mathrm{T}$\mathrm{T}$ are points on sides $\mathrm{P}\mathrm{R}$$\mathrm{P}\mathrm{R}$PR\mathrm{PR}$\mathrm{P}\mathrm{R}$ and $\mathrm{Q}\mathrm{R}$$\mathrm{Q}\mathrm{R}$QR\mathrm{QR}$\mathrm{Q}\mathrm{R}$ of $\mathrm{△}\mathrm{P}\mathrm{Q}\mathrm{R}$$\mathrm{△}\mathrm{P}\mathrm{Q}\mathrm{R}$/_\PQR\triangle \mathrm{PQR}$\mathrm{△}\mathrm{P}\mathrm{Q}\mathrm{R}$ such that $\mathrm{\angle }\mathrm{P}=\mathrm{\angle }\mathrm{R}\mathrm{T}\mathrm{S}$$\mathrm{\angle }\mathrm{P}=\mathrm{\angle }\mathrm{R}\mathrm{T}\mathrm{S}$/_P=/_RTS\angle \mathrm{P}=\angle \mathrm{RTS}$\mathrm{\angle }\mathrm{P}=\mathrm{\angle }\mathrm{R}\mathrm{T}\mathrm{S}$. Show that $\mathrm{\Delta }\mathrm{R}\mathrm{P}\mathrm{Q}\sim \mathrm{\Delta }\mathrm{R}\mathrm{T}\mathrm{S}$$\mathrm{\Delta }\mathrm{R}\mathrm{P}\mathrm{Q}\sim \mathrm{\Delta }\mathrm{R}\mathrm{T}\mathrm{S}$DeltaRPQ∼DeltaRTS\Delta \mathrm{RPQ} \sim \Delta \mathrm{RTS}$\mathrm{\Delta }\mathrm{R}\mathrm{P}\mathrm{Q}\sim \mathrm{\Delta }\mathrm{R}\mathrm{T}\mathrm{S}$.

11. In Fig. $6.40,\mathrm{E}$$6.40,\mathrm{E}$6.40,E6.40, \mathrm{E}$6.40,\mathrm{E}$ is a point on side $\mathrm{C}\mathrm{B}$$\mathrm{C}\mathrm{B}$CB\mathrm{CB}$\mathrm{C}\mathrm{B}$ produced of an isosceles triangle $\mathrm{A}\mathrm{B}\mathrm{C}$$\mathrm{A}\mathrm{B}\mathrm{C}$ABC\mathrm{ABC}$\mathrm{A}\mathrm{B}\mathrm{C}$ with $AB=AC$$AB=AC$AB=ACA B=A C$AB=AC$. If $AD\perp BC$$AD\perp BC$AD _|_ BCA D \perp B C$AD\perp BC$ and $EF\perp AC$$EF\perp AC$EF _|_ ACE F \perp A C$EF\perp AC$, prove that $\mathrm{△}\mathrm{A}\mathrm{B}\mathrm{D}\sim \mathrm{\Delta }\mathrm{E}\mathrm{C}\mathrm{F}$$\mathrm{△}\mathrm{A}\mathrm{B}\mathrm{D}\sim \mathrm{\Delta }\mathrm{E}\mathrm{C}\mathrm{F}$/_\ABD∼DeltaECF\triangle \mathrm{ABD} \sim \Delta \mathrm{ECF}$\mathrm{△}\mathrm{A}\mathrm{B}\mathrm{D}\sim \mathrm{\Delta }\mathrm{E}\mathrm{C}\mathrm{F}$.
12. Sides $AB$$AB$ABA B$AB$ and $BC$$BC$BCB C$BC$ and median $AD$$AD$ADA D$AD$ of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}$$\mathrm{A}\mathrm{B}\mathrm{C}$ABC\mathrm{ABC}$\mathrm{A}\mathrm{B}\mathrm{C}$ are respectively proportional to sides PQ and QR and median PM of $\mathrm{△}$$\mathrm{△}$/_\\triangle$\mathrm{△}$ PQR (see Fig. 6.41). Show that

14. Sides $AB$$AB$ABA B$AB$ and $AC$$AC$ACA C$AC$ and median $AD$$AD$ADA D$AD$ of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}$$\mathrm{A}\mathrm{B}\mathrm{C}$ABC\mathrm{ABC}$\mathrm{A}\mathrm{B}\mathrm{C}$ are respectively proportional to sides PQ and PR and median PM of another triangle PQR.

15. A vertical pole of length $6\mathrm{m}$$6\mathrm{m}$6m6 \mathrm{~m}$6\mathrm{m}$ casts a shadow $4\mathrm{m}$$4\mathrm{m}$4m4 \mathrm{~m}$4\mathrm{m}$ long on the ground and at the same time a tower casts a shadow $28\mathrm{m}$$28\mathrm{m}$28m28 \mathrm{~m}$28\mathrm{m}$ long. Find the height of the tower.
16. If $AD$$AD$ADA D$AD$ and $PM$$PM$PMP M$PM$ are medians of triangles $ABC$$ABC$ABCA B C$ABC$ and $PQR$$PQR$PQRP Q R$PQR$, respectively where

6.5 Summary

1. Two figures having the same shape but not necessarily the same size are called similar figures.
2. All the congruent figures are similar but the converse is not true.
3. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion).
4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion).
7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion).
8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion).
9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion).

A Note To the Reader

CoORDINATE GEOMETRY

7

7.1 Introduction

7.2 Distance Formula

Remarks :

1. In particular, the distance of a point $\mathrm{P}(x,y)$$\mathrm{P}(x,y)$P(x,y)\mathrm{P}(x, y)$\mathrm{P}(x,y)$ from the origin $\mathrm{O}(0,0)$$\mathrm{O}(0,0)$O(0,0)\mathrm{O}(0,0)$\mathrm{O}(0,0)$ is given by

2. We can also write, $\mathrm{P}\mathrm{Q}=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$$\mathrm{P}\mathrm{Q}=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}$PQ=sqrt((x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2))\mathrm{PQ}=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$\mathrm{P}\mathrm{Q}=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$. (Why?)

EXERCISE 7.1

1. Find the distance between the following pairs of points :
   (i) $(2,3),(4,1)$$(2,3),(4,1)$(2,3),(4,1)(2,3),(4,1)$(2,3),(4,1)$
   (ii) $(-5,7),(-1,3)$$(-5,7),(-1,3)$(-5,7),(-1,3)(-5,7),(-1,3)$(-5,7),(-1,3)$
   (iii) $(a,b),(-a,-b)$$(a,b),(-a,-b)$(a,b),(-a,-b)(a, b),(-a,-b)$(a,b),(-a,-b)$
2. Find the distance between the points $(0,0)$$(0,0)$(0,0)(0,0)$(0,0)$ and $(36,15)$$(36,15)$(36,15)(36,15)$(36,15)$. Can you now find the distance between the two towns A and B discussed in Section 7.2.
3. Determine if the points $(1,5),(2,3)$$(1,5),(2,3)$(1,5),(2,3)(1,5),(2,3)$(1,5),(2,3)$ and $(-2,-11)$$(-2,-11)$(-2,-11)(-2,-11)$(-2,-11)$ are collinear.
4. Check whether $(5,-2),(6,4)$$(5,-2),(6,4)$(5,-2),(6,4)(5,-2),(6,4)$(5,-2),(6,4)$ and $(7,-2)$$(7,-2)$(7,-2)(7,-2)$(7,-2)$ are the vertices of an isosceles triangle.
5. In a classroom, 4 friends are seated at the points A, B, C and $D$$D$DD$D$ as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, "Don't you think ABCD is a square?" Chameli disagrees. Using distance formula, find which of them is correct.
6. Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

7. Find the point on the $x$$x$xx$x$-axis which is equidistant from $(2,-5)$$(2,-5)$(2,-5)(2,-5)$(2,-5)$ and $(-2,9)$$(-2,9)$(-2,9)(-2,9)$(-2,9)$.
8. Find the values of $y$$y$yy$y$ for which the distance between the points $\mathrm{P}(2,-3)$$\mathrm{P}(2,-3)$P(2,-3)\mathrm{P}(2,-3)$\mathrm{P}(2,-3)$ and $\mathrm{Q}(10,y)$$\mathrm{Q}(10,y)$Q(10,y)\mathrm{Q}(10, y)$\mathrm{Q}(10,y)$ is 10 units.
9. If $\mathrm{Q}(0,1)$$\mathrm{Q}(0,1)$Q(0,1)\mathrm{Q}(0,1)$\mathrm{Q}(0,1)$ is equidistant from $\mathrm{P}(5,-3)$$\mathrm{P}(5,-3)$P(5,-3)\mathrm{P}(5,-3)$\mathrm{P}(5,-3)$ and $\mathrm{R}(x,6)$$\mathrm{R}(x,6)$R(x,6)\mathrm{R}(x, 6)$\mathrm{R}(x,6)$, find the values of $x$$x$xx$x$. Also find the distances $QR$$QR$QRQ R$QR$ and PR.
10. Find a relation between $x$$x$xx$x$ and $y$$y$yy$y$ such that the point $(x,y)$$(x,y)$(x,y)(x, y)$(x,y)$ is equidistant from the point $(3,6)$$(3,6)$(3,6)(3,6)$(3,6)$ and $(-3,4)$$(-3,4)$(-3,4)(-3,4)$(-3,4)$.

7.3 Section Formula

EXERCISE 7.2

1. Find the coordinates of the point which divides the join of $(-1,7)$$(-1,7)$(-1,7)(-1,7)$(-1,7)$ and $(4,-3)$$(4,-3)$(4,-3)(4,-3)$(4,-3)$ in the ratio $2:3$$2:3$2:32: 3$2:3$.
2. Find the coordinates of the points of trisection of the line segment joining $(4,-1)$$(4,-1)$(4,-1)(4,-1)$(4,-1)$ and $(-2,-3)$$(-2,-3)$(-2,-3)(-2,-3)$(-2,-3)$.
3. To conduct Sports Day activities, in your rectangular shaped school ground $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ABCD\mathrm{ABCD}$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$, lines have been drawn with chalk powder at a distance of $1\mathrm{m}$$1\mathrm{m}$1m1 \mathrm{~m}$1\mathrm{m}$ each. 100 flower pots have been placed at a distance of $1\mathrm{m}$$1\mathrm{m}$1m1 \mathrm{~m}$1\mathrm{m}$ from each other along $\mathrm{A}\mathrm{D}$$\mathrm{A}\mathrm{D}$AD\mathrm{AD}$\mathrm{A}\mathrm{D}$, as shown in Fig. 7.12. Niharika runs $\frac{1}{4}$$\frac{1}{4}$(1)/(4)\frac{1}{4}$\frac{1}{4}$ th the distance $\mathrm{A}\mathrm{D}$$\mathrm{A}\mathrm{D}$AD\mathrm{AD}$\mathrm{A}\mathrm{D}$ on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$$\frac{1}{5}$(1)/(5)\frac{1}{5}$\frac{1}{5}$ th the distance $\mathrm{A}\mathrm{D}$$\mathrm{A}\mathrm{D}$AD\mathrm{AD}$\mathrm{A}\mathrm{D}$ on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

4. Find the ratio in which the line segment joining the points $(-3,10)$$(-3,10)$(-3,10)(-3,10)$(-3,10)$ and $(6,-8)$$(6,-8)$(6,-8)(6,-8)$(6,-8)$ is divided by $(-1,6)$$(-1,6)$(-1,6)(-1,6)$(-1,6)$.
5. Find the ratio in which the line segment joining $\mathrm{A}(1,-5)$$\mathrm{A}(1,-5)$A(1,-5)\mathrm{A}(1,-5)$\mathrm{A}(1,-5)$ and $\mathrm{B}(-4,5)$$\mathrm{B}(-4,5)$B(-4,5)\mathrm{B}(-4,5)$\mathrm{B}(-4,5)$ is divided by the $x$$x$xx$x$-axis. Also find the coordinates of the point of division.
6. If $(1,2),(4,y),(x,6)$$(1,2),(4,y),(x,6)$(1,2),(4,y),(x,6)(1,2),(4, y),(x, 6)$(1,2),(4,y),(x,6)$ and $(3,5)$$(3,5)$(3,5)(3,5)$(3,5)$ are the vertices of a parallelogram taken in order, find $x$$x$xx$x$ and $y$$y$yy$y$.
7. Find the coordinates of a point $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$, where $\mathrm{A}\mathrm{B}$$\mathrm{A}\mathrm{B}$AB\mathrm{AB}$\mathrm{A}\mathrm{B}$ is the diameter of a circle whose centre is $(2,-3)$$(2,-3)$(2,-3)(2,-3)$(2,-3)$ and $B$$B$BB$B$ is $(1,4)$$(1,4)$(1,4)(1,4)$(1,4)$.
8. If $A$$A$AA$A$ and $B$$B$BB$B$ are $(-2,-2)$$(-2,-2)$(-2,-2)(-2,-2)$(-2,-2)$ and $(2,-4)$$(2,-4)$(2,-4)(2,-4)$(2,-4)$, respectively, find the coordinates of $\mathrm{P}$$\mathrm{P}$P\mathrm{P}$\mathrm{P}$ such that $\mathrm{A}\mathrm{P}=\frac{3}{7}\mathrm{A}\mathrm{B}$$\mathrm{A}\mathrm{P}=\frac{3}{7}\mathrm{A}\mathrm{B}$AP=(3)/(7)AB\mathrm{AP}=\frac{3}{7} \mathrm{AB}$\mathrm{A}\mathrm{P}=\frac{3}{7}\mathrm{A}\mathrm{B}$ and $\mathrm{P}$$\mathrm{P}$P\mathrm{P}$\mathrm{P}$ lies on the line segment $\mathrm{A}\mathrm{B}$$\mathrm{A}\mathrm{B}$AB\mathrm{AB}$\mathrm{A}\mathrm{B}$.
9. Find the coordinates of the points which divide the line segment joining $\mathrm{A}(-2,2)$$\mathrm{A}(-2,2)$A(-2,2)\mathrm{A}(-2,2)$\mathrm{A}(-2,2)$ and $\mathrm{B}(2,8)$$\mathrm{B}(2,8)$B(2,8)\mathrm{B}(2,8)$\mathrm{B}(2,8)$ into four equal parts.
10. Find the area of a rhombus if its vertices are $(3,0),(4,5),(-1,4)$$(3,0),(4,5),(-1,4)$(3,0),(4,5),(-1,4)(3,0),(4,5),(-1,4)$(3,0),(4,5),(-1,4)$ and $(-2,-1)$$(-2,-1)$(-2,-1)(-2,-1)$(-2,-1)$ taken in order. $[$$\left[\right.$[:}\left[\right.$[$ Hint : Area of a rhombus $=\frac{1}{2}$$=\frac{1}{2}$=(1)/(2)=\frac{1}{2}$=\frac{1}{2}$ (product of its diagonals) $]$$]$]]$]$

7.4 Summary

1. The distance between $\mathrm{P}(x_1,y_1)$$\mathrm{P}\left(x_1,y_1\right)$P(x_(1),y_(1))\mathrm{P}\left(x_{1}, y_{1}\right)$\mathrm{P}(x_1,y_1)$ and $\mathrm{Q}(x_2,y_2)$$\mathrm{Q}\left(x_2,y_2\right)$Q(x_(2),y_(2))\mathrm{Q}\left(x_{2}, y_{2}\right)$\mathrm{Q}(x_2,y_2)$ is $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$$\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2))\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$.
2. The distance of a point $\mathrm{P}(x,y)$$\mathrm{P}(x,y)$P(x,y)\mathrm{P}(x, y)$\mathrm{P}(x,y)$ from the origin is $\sqrt{x^2+y^2}$$\sqrt{x^2+y^2}$sqrt(x^(2)+y^(2))\sqrt{x^{2}+y^{2}}$\sqrt{x^2+y^2}$.
3. The coordinates of the point $\mathrm{P}(x,y)$$\mathrm{P}(x,y)$P(x,y)\mathrm{P}(x, y)$\mathrm{P}(x,y)$ which divides the line segment joining the points $\mathrm{A}(x_1,y_1)$$\mathrm{A}\left(x_1,y_1\right)$A(x_(1),y_(1))\mathrm{A}\left(x_{1}, y_{1}\right)$\mathrm{A}(x_1,y_1)$ and $\mathrm{B}(x_2,y_2)$$\mathrm{B}\left(x_2,y_2\right)$B(x_(2),y_(2))\mathrm{B}\left(x_{2}, y_{2}\right)$\mathrm{B}(x_2,y_2)$ internally in the ratio $m_1:m_2$$m_1:m_2$m_(1):m_(2)m_{1}: m_{2}$m_1:m_2$ are $(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2})$$\left(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2}\right)$((m_(1)x_(2)+m_(2)x_(1))/(m_(1)+m_(2)),(m_(1)y_(2)+m_(2)y_(1))/(m_(1)+m_(2)))\left(\frac{m_{1} x_{2}+m_{2} x_{1}}{m_{1}+m_{2}}, \frac{m_{1} y_{2}+m_{2} y_{1}}{m_{1}+m_{2}}\right)$(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2})$.
4. The mid-point of the line segment joining the points $\mathrm{P}(x_1,y_1)$$\mathrm{P}\left(x_1,y_1\right)$P(x_(1),y_(1))\mathrm{P}\left(x_{1}, y_{1}\right)$\mathrm{P}(x_1,y_1)$ and $\mathrm{Q}(x_2,y_2)$$\mathrm{Q}\left(x_2,y_2\right)$Q(x_(2),y_(2))\mathrm{Q}\left(x_{2}, y_{2}\right)$\mathrm{Q}(x_2,y_2)$ is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$((x_(1)+x_(2))/(2),(y_(1)+y_(2))/(2))\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

ANOTE TO THE READER

INTRODUCTIONTO TRIGONOMETRY 8

There is perhaps nothing which so occupies the middle position of mathematics as trigonometry.

- J.F. Herbart (1890)

8.1 Introduction

1. Suppose the students of a school are visiting Qutub Minar. Now, if a student is looking at the top of the Minar, a right triangle can be imagined to be made, as shown in Fig 8.1. Can the student find out the height of the Minar, without actually measuring it?
2. Suppose a girl is sitting on the balcony of her house located on the bank of a river. She is looking down at a flower pot placed on a stair of a temple situated nearby on the other bank of the river. A right triangle is imagined to be made in this situation as shown in Fig.8.2. If you know the height at which the person is sitting, can you find the width of the river?

3. Suppose a hot air balloon is flying in the air. A girl happens to spot the balloon in the sky and runs to her mother to tell her about it. Her mother rushes out of the house to look at the balloon.Now when the girl had spotted the balloon intially it was at point $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$. When both the mother and daughter came out to see it, it had already

8.2 Trigonometric Ratios

EXERCISE 8.1

1. In $\mathrm{△}ABC$$\mathrm{△}ABC$/_\ABC\triangle A B C$\mathrm{△}ABC$, right-angled at $B,AB=24\mathrm{c}\mathrm{m},BC=7\mathrm{c}\mathrm{m}$$B,AB=24\mathrm{c}\mathrm{m},BC=7\mathrm{c}\mathrm{m}$B,AB=24cm,BC=7cmB, A B=24 \mathrm{~cm}, B C=7 \mathrm{~cm}$B,AB=24\mathrm{c}\mathrm{m},BC=7\mathrm{c}\mathrm{m}$. Determine :

2. In Fig. 8.13, find $\tan P-\cot R$$\tan P-\cot R$tan P-cot R\tan P-\cot R$\tan P-\cot R$.
3. If $\sin \mathrm{A}=\frac{3}{4}$$\sin \mathrm{A}=\frac{3}{4}$sin A=(3)/(4)\sin \mathrm{A}=\frac{3}{4}$\sin \mathrm{A}=\frac{3}{4}$, calculate $\cos \mathrm{A}$$\cos \mathrm{A}$cos A\cos \mathrm{A}$\cos \mathrm{A}$ and $\tan \mathrm{A}$$\tan \mathrm{A}$tan A\tan \mathrm{A}$\tan \mathrm{A}$.
4. Given $15\cot \mathrm{A}=8$$15\cot \mathrm{A}=8$15 cot A=815 \cot \mathrm{A}=8$15\cot \mathrm{A}=8$, find $\sin \mathrm{A}$$\sin \mathrm{A}$sin A\sin \mathrm{A}$\sin \mathrm{A}$ and $\sec \mathrm{A}$$\sec \mathrm{A}$sec A\sec \mathrm{A}$\sec \mathrm{A}$.
5. Given $\sec \theta =\frac{13}{12}$$\sec \theta =\frac{13}{12}$sec theta=(13)/(12)\sec \theta=\frac{13}{12}$\sec \theta =\frac{13}{12}$, calculate all other trigonometric ratios.

6. If $\mathrm{\angle }\mathrm{A}$$\mathrm{\angle }\mathrm{A}$/_A\angle \mathrm{A}$\mathrm{\angle }\mathrm{A}$ and $\mathrm{\angle }\mathrm{B}$$\mathrm{\angle }\mathrm{B}$/_B\angle \mathrm{B}$\mathrm{\angle }\mathrm{B}$ are acute angles such that $\cos \mathrm{A}=\cos \mathrm{B}$$\cos \mathrm{A}=\cos \mathrm{B}$cos A=cos B\cos \mathrm{A}=\cos \mathrm{B}$\cos \mathrm{A}=\cos \mathrm{B}$, then show that $\mathrm{\angle }\mathrm{A}=\mathrm{\angle }\mathrm{B}$$\mathrm{\angle }\mathrm{A}=\mathrm{\angle }\mathrm{B}$/_A=/_B\angle \mathrm{A}=\angle \mathrm{B}$\mathrm{\angle }\mathrm{A}=\mathrm{\angle }\mathrm{B}$.
7. If $\cot \theta =\frac{7}{8}$$\cot \theta =\frac{7}{8}$cot theta=(7)/(8)\cot \theta=\frac{7}{8}$\cot \theta =\frac{7}{8}$, evaluate : (i) $\frac{(1+\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(1-\cos \theta )}$$\frac{(1+\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(1-\cos \theta )}$((1+sin theta)(1-sin theta))/((1+cos theta)(1-cos theta))\frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}$\frac{(1+\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(1-\cos \theta )}$,

8. If $3\cot A=4$$3\cot A=4$3cot A=43 \cot A=4$3\cot A=4$, check whether $\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}={\cos }^{2}A-{\sin }^2\mathrm{A}$$\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}={\cos }^{2}A-{\sin }^2\mathrm{A}$(1-tan^(2)A)/(1+tan^(2)A)=cos^(2)A-sin^(2)A\frac{1-\tan ^{2} A}{1+\tan ^{2} A}=\cos ^{2} A-\sin ^{2} \mathrm{~A}$\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}={\cos }^{2}A-{\sin }^2\mathrm{A}$ or not.
9. In triangle $ABC$$ABC$ABCA B C$ABC$, right-angled at $B$$B$BB$B$, if $\tan A=\frac{1}{\sqrt{3}}$$\tan A=\frac{1}{\sqrt{3}}$tan A=(1)/(sqrt3)\tan A=\frac{1}{\sqrt{3}}$\tan A=\frac{1}{\sqrt{3}}$, find the value of:

10. In $\mathrm{△}PQR$$\mathrm{△}PQR$/_\PQR\triangle P Q R$\mathrm{△}PQR$, right-angled at $Q,PR+QR=25\mathrm{c}\mathrm{m}$$Q,PR+QR=25\mathrm{c}\mathrm{m}$Q,PR+QR=25cmQ, P R+Q R=25 \mathrm{~cm}$Q,PR+QR=25\mathrm{c}\mathrm{m}$ and $PQ=5\mathrm{c}\mathrm{m}$$PQ=5\mathrm{c}\mathrm{m}$PQ=5cmP Q=5 \mathrm{~cm}$PQ=5\mathrm{c}\mathrm{m}$. Determine the values of $\sin \mathrm{P},\cos \mathrm{P}$$\sin \mathrm{P},\cos \mathrm{P}$sin P,cos P\sin \mathrm{P}, \cos \mathrm{P}$\sin \mathrm{P},\cos \mathrm{P}$ and $\tan \mathrm{P}$$\tan \mathrm{P}$tan P\tan \mathrm{P}$\tan \mathrm{P}$.
11. State whether the following are true or false. Justify your answer.

8.3 Trigonometric Ratios of Some Specific Angles

Trigonometric Ratios of ${45}^{\circ }$${45}^{\circ }$45^(@)45^{\circ}${45}^{\circ }$

Trigonometric Ratios of ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ and ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$

Trigonometric Ratios of $0^{\circ }$$0^{\circ }$0^(@)0^{\circ}$0^{\circ }$ and ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$

EXERCISE 8.2

1. Evaluate the following :
   (i) $\sin {60}^{\circ }\cos {30}^{\circ }+\sin {30}^{\circ }\cos {60}^{\circ }$$\sin {60}^{\circ }\cos {30}^{\circ }+\sin {30}^{\circ }\cos {60}^{\circ }$sin 60^(@)cos 30^(@)+sin 30^(@)cos 60^(@)\sin 60^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 60^{\circ}$\sin {60}^{\circ }\cos {30}^{\circ }+\sin {30}^{\circ }\cos {60}^{\circ }$
   (ii) $2{\tan }^2{45}^{\circ }+{\cos }^2{30}^{\circ }-{\sin }^2{60}^{\circ }$$2{\tan }^2{45}^{\circ }+{\cos }^2{30}^{\circ }-{\sin }^2{60}^{\circ }$2tan^(2)45^(@)+cos^(2)30^(@)-sin^(2)60^(@)2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}$2{\tan }^2{45}^{\circ }+{\cos }^2{30}^{\circ }-{\sin }^2{60}^{\circ }$
   (iii) $\frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+\mathrm{cosec}{30}^{\circ }}$$\frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+\mathrm{cosec}{30}^{\circ }}$(cos 45^(@))/(sec 30^(@)+cosec 30^(@))\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$\frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+\mathrm{cosec}{30}^{\circ }}$
   (iv) $\frac{\sin {30}^{\circ }+\tan {45}^{\circ }-\mathrm{cosec}{60}^{\circ }}{\sec {30}^{\circ }+\cos {60}^{\circ }+\cot {45}^{\circ }}$$\frac{\sin {30}^{\circ }+\tan {45}^{\circ }-\mathrm{cosec}{60}^{\circ }}{\sec {30}^{\circ }+\cos {60}^{\circ }+\cot {45}^{\circ }}$(sin 30^(@)+tan 45^(@)-cosec 60^(@))/(sec 30^(@)+cos 60^(@)+cot 45^(@))\frac{\sin 30^{\circ}+\tan 45^{\circ}-\operatorname{cosec} 60^{\circ}}{\sec 30^{\circ}+\cos 60^{\circ}+\cot 45^{\circ}}$\frac{\sin {30}^{\circ }+\tan {45}^{\circ }-\mathrm{cosec}{60}^{\circ }}{\sec {30}^{\circ }+\cos {60}^{\circ }+\cot {45}^{\circ }}$
   (v) $\frac{5{\cos }^2{60}^{\circ }+4{\sec }^2{30}^{\circ }-{\tan }^2{45}^{\circ }}{{\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ }}$$\frac{5{\cos }^2{60}^{\circ }+4{\sec }^2{30}^{\circ }-{\tan }^2{45}^{\circ }}{{\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ }}$(5cos^(2)60^(@)+4sec^(2)30^(@)-tan^(2)45^(@))/(sin^(2)30^(@)+cos^(2)30^(@))\frac{5 \cos ^{2} 60^{\circ}+4 \sec ^{2} 30^{\circ}-\tan ^{2} 45^{\circ}}{\sin ^{2} 30^{\circ}+\cos ^{2} 30^{\circ}}$\frac{5{\cos }^2{60}^{\circ }+4{\sec }^2{30}^{\circ }-{\tan }^2{45}^{\circ }}{{\sin }^2{30}^{\circ }+{\cos }^2{30}^{\circ }}$
2. Choose the correct option and justify your choice :

3. If $\tan (A+B)=\sqrt{3}$$\tan (A+B)=\sqrt{3}$tan(A+B)=sqrt3\tan (A+B)=\sqrt{3}$\tan (A+B)=\sqrt{3}$ and $\tan (A-B)=\frac{1}{\sqrt{3}};0^{\circ }<A+B\le {90}^{\circ };A>B$$\tan (A-B)=\frac{1}{\sqrt{3}};0^{\circ }<A+B\le {90}^{\circ };A>B$tan(A-B)=(1)/(sqrt3);0^(@) < A+B <= 90^(@);A > B\tan (A-B)=\frac{1}{\sqrt{3}} ; 0^{\circ}<A+B \leq 90^{\circ} ; A>B$\tan (A-B)=\frac{1}{\sqrt{3}};0^{\circ }<A+B\le {90}^{\circ };A>B$, find $A$$A$AA$A$ and $B$$B$BB$B$.
4. State whether the following are true or false. Justify your answer.

8.4 Trigonometric Identities

Solution :

EXERCISE 8.3

1. Express the trigonometric ratios $\sin \mathrm{A},\sec \mathrm{A}$$\sin \mathrm{A},\sec \mathrm{A}$sin A,sec A\sin \mathrm{A}, \sec \mathrm{A}$\sin \mathrm{A},\sec \mathrm{A}$ and $\tan \mathrm{A}$$\tan \mathrm{A}$tan A\tan \mathrm{A}$\tan \mathrm{A}$ in terms of $\cot \mathrm{A}$$\cot \mathrm{A}$cot A\cot \mathrm{A}$\cot \mathrm{A}$.
2. Write all the other trigonometric ratios of $\mathrm{\angle }\mathrm{A}$$\mathrm{\angle }\mathrm{A}$/_A\angle \mathrm{A}$\mathrm{\angle }\mathrm{A}$ in terms of $\sec \mathrm{A}$$\sec \mathrm{A}$sec A\sec \mathrm{A}$\sec \mathrm{A}$.
3. Choose the correct option. Justify your choice.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
   (i) $(\mathrm{cosec}\theta -\cot \theta {)}^2=\frac{1-\cos \theta }{1+\cos \theta }$$(\mathrm{cosec}\theta -\cot \theta {)}^2=\frac{1-\cos \theta }{1+\cos \theta }$(cosec theta-cot theta)^(2)=(1-cos theta)/(1+cos theta)(\operatorname{cosec} \theta-\cot \theta)^{2}=\frac{1-\cos \theta}{1+\cos \theta}$(\mathrm{cosec}\theta -\cot \theta {)}^2=\frac{1-\cos \theta }{1+\cos \theta }$
   (ii) $\frac{\cos \mathrm{A}}{1+\sin \mathrm{A}}+\frac{1+\sin \mathrm{A}}{\cos \mathrm{A}}=2\sec \mathrm{A}$$\frac{\cos \mathrm{A}}{1+\sin \mathrm{A}}+\frac{1+\sin \mathrm{A}}{\cos \mathrm{A}}=2\sec \mathrm{A}$(cos A)/(1+sin A)+(1+sin A)/(cos A)=2sec A\frac{\cos \mathrm{A}}{1+\sin \mathrm{A}}+\frac{1+\sin \mathrm{A}}{\cos \mathrm{A}}=2 \sec \mathrm{A}$\frac{\cos \mathrm{A}}{1+\sin \mathrm{A}}+\frac{1+\sin \mathrm{A}}{\cos \mathrm{A}}=2\sec \mathrm{A}$
   (iii) $\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \mathrm{cosec}\theta$$\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \mathrm{cosec}\theta$(tan theta)/(1-cot theta)+(cot theta)/(1-tan theta)=1+sec theta cosec theta\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \operatorname{cosec} \theta$\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \mathrm{cosec}\theta$

8.5 Summary

1. In a right triangle $ABC$$ABC$ABCA B C$ABC$, right-angled at $B$$B$BB$B$,

3. If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of the angle can be easily determined.
4. The values of trigonometric ratios for angles $0^{\circ },{30}^{\circ },{45}^{\circ },{60}^{\circ }$$0^{\circ },{30}^{\circ },{45}^{\circ },{60}^{\circ }$0^(@),30^(@),45^(@),60^(@)0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}$0^{\circ },{30}^{\circ },{45}^{\circ },{60}^{\circ }$ and ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$.
5. The value of $\sin \mathrm{A}$$\sin \mathrm{A}$sin A\sin \mathrm{A}$\sin \mathrm{A}$ or $\cos \mathrm{A}$$\cos \mathrm{A}$cos A\cos \mathrm{A}$\cos \mathrm{A}$ never exceeds 1 , whereas the value of $\sec \mathrm{A}(0^{\circ }\le \mathrm{A}<{90}^{\circ })$$\sec \mathrm{A}\left(0^{\circ }\le \mathrm{A}<{90}^{\circ }\right)$sec A(0^(@) <= A < 90^(@))\sec \mathrm{A}\left(0^{\circ} \leq \mathrm{A}<90^{\circ}\right)$\sec \mathrm{A}(0^{\circ }\le \mathrm{A}<{90}^{\circ })$ or $\mathrm{cosec}\mathrm{A}(0^{\circ }<\mathrm{A}\le {90}^{\circ })$$\mathrm{cosec}\mathrm{A}\left(0^{\circ }<\mathrm{A}\le {90}^{\circ }\right)$cosec A(0^(@) < A <= 90^(@))\operatorname{cosec} \mathrm{A}\left(0^{\circ}<\mathrm{A} \leq 90^{\circ}\right)$\mathrm{cosec}\mathrm{A}(0^{\circ }<\mathrm{A}\le {90}^{\circ })$ is always greater than or equal to 1.
6. ${\sin }^2\mathrm{A}+{\cos }^2\mathrm{A}=1$${\sin }^2\mathrm{A}+{\cos }^2\mathrm{A}=1$sin^(2)A+cos^(2)A=1\sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}=1${\sin }^2\mathrm{A}+{\cos }^2\mathrm{A}=1$,

SONEAPPICATIONS OF TRIGONOMETRY 9

9.1 Heights and Distances

EXERCISE 9.1

1. A circus artist is climbing a $20\mathrm{m}$$20\mathrm{m}$20m20 \mathrm{~m}$20\mathrm{m}$ long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ (see Fig. 9.11).
2. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ with it. The distance between

3. A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of $1.5\mathrm{m}$$1.5\mathrm{m}$1.5m1.5 \mathrm{~m}$1.5\mathrm{m}$, and is inclined at an angle of ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ to the ground, whereas for elder children, she wants to have a steep slide at a height of $3\mathrm{m}$$3\mathrm{m}$3m3 \mathrm{~m}$3\mathrm{m}$, and inclined at an angle of ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ to the ground. What should be the length of the slide in each case?
4. The angle of elevation of the top of a tower from a point on the ground, which is $30\mathrm{m}$$30\mathrm{m}$30m30 \mathrm{~m}$30\mathrm{m}$ away from the foot of the tower, is ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$. Find the height of the tower.
5. A kite is flying at a height of $60\mathrm{m}$$60\mathrm{m}$60m60 \mathrm{~m}$60\mathrm{m}$ above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$. Find the length of the string, assuming that there is no slack in the string.
6. A $1.5\mathrm{m}$$1.5\mathrm{m}$1.5m1.5 \mathrm{~m}$1.5\mathrm{m}$ tall boy is standing at some distance from a $30\mathrm{m}$$30\mathrm{m}$30m30 \mathrm{~m}$30\mathrm{m}$ tall building. The angle of elevation from his eyes to the top of the building increases from ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ to ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ as he walks towards the building. Find the distance he walked towards the building.
7. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a $20\mathrm{m}$$20\mathrm{m}$20m20 \mathrm{~m}$20\mathrm{m}$ high building are ${45}^{\circ }$${45}^{\circ }$45^(@)45^{\circ}${45}^{\circ }$ and ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ respectively. Find the height of the tower.
8. A statue, $1.6\mathrm{m}$$1.6\mathrm{m}$1.6m1.6 \mathrm{~m}$1.6\mathrm{m}$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ and from the same point the angle of elevation of the top of the pedestal is ${45}^{\circ }$${45}^{\circ }$45^(@)45^{\circ}${45}^{\circ }$. Find the height of the pedestal.
9. The angle of elevation of the top of a building from the foot of the tower is ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ and the angle of elevation of the top of the tower from the foot of the building is ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$. If the tower is $50\mathrm{m}$$50\mathrm{m}$50m50 \mathrm{~m}$50\mathrm{m}$ high, find the height of the building.
10. Two poles of equal heights are standing opposite each other on either side of the road, which is $80\mathrm{m}$$80\mathrm{m}$80m80 \mathrm{~m}$80\mathrm{m}$ wide. From a point between them on the road, the angles of elevation of the top of the poles are ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ and ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$, respectively. Find the height of the poles and the distances of the point from the poles.
11. A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$. From another point $20\mathrm{m}$$20\mathrm{m}$20m20 \mathrm{~m}$20\mathrm{m}$ away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ (see Fig. 9.12). Find the height of the tower and the width of

12. From the top of a $7\mathrm{m}$$7\mathrm{m}$7m7 \mathrm{~m}$7\mathrm{m}$ high building, the angle of elevation of the top of a cable tower is ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ and the angle of depression of its foot is ${45}^{\circ }$${45}^{\circ }$45^(@)45^{\circ}${45}^{\circ }$. Determine the height of the tower.
13. As observed from the top of a $75\mathrm{m}$$75\mathrm{m}$75m75 \mathrm{~m}$75\mathrm{m}$ high lighthouse from the sea-level, the angles of depression of two ships are ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ and ${45}^{\circ }$${45}^{\circ }$45^(@)45^{\circ}${45}^{\circ }$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
14. A $1.2\mathrm{m}$$1.2\mathrm{m}$1.2m1.2 \mathrm{~m}$1.2\mathrm{m}$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2\mathrm{m}$$88.2\mathrm{m}$88.2m88.2 \mathrm{~m}$88.2\mathrm{m}$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$. After some time, the angle of elevation reduces to ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$ (see Fig. 9.13 ). Find the distance travelled by the balloon during the interval.

15. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of ${30}^{\circ }$${30}^{\circ }$30^(@)30^{\circ}${30}^{\circ }$, which is approaching the foot of the
    tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$. Find the time taken by the car to reach the foot of the tower from this point.

9.2 Summary

1. (i) The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

2. The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

CircLes

10.1 Introduction

10.2 Tangent to a Circle

Remarks

1. By theorem above, we can also conclude that at any point on a circle there can be one and only one tangent.
2. The line containing the radius through the point of contact is also sometimes called the 'normal' to the circle at the point.

EXERCISE 10.1

1. How many tangents can a circle have?
2. Fill in the blanks :

4. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

10.3 Number of Tangents from a Point on a Circle

Remarks

1. The theorem can also be proved by using the Pythagoras Theorem as follows:

2. Note also that $\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{Q}=\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{R}$$\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{Q}=\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{R}$/_OPQ=/_OPR\angle \mathrm{OPQ}=\angle \mathrm{OPR}$\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{Q}=\mathrm{\angle }\mathrm{O}\mathrm{P}\mathrm{R}$. Therefore, $\mathrm{O}\mathrm{P}$$\mathrm{O}\mathrm{P}$OP\mathrm{OP}$\mathrm{O}\mathrm{P}$ is the angle bisector of $\mathrm{\angle }\mathrm{Q}\mathrm{P}\mathrm{R}$$\mathrm{\angle }\mathrm{Q}\mathrm{P}\mathrm{R}$/_QPR\angle \mathrm{QPR}$\mathrm{\angle }\mathrm{Q}\mathrm{P}\mathrm{R}$, i.e., the centre lies on the bisector of the angle between the two tangents.

EXERCISE 10.2

1. From a point $\mathrm{Q}$$\mathrm{Q}$Q\mathrm{Q}$\mathrm{Q}$, the length of the tangent to a circle is $24\mathrm{c}\mathrm{m}$$24\mathrm{c}\mathrm{m}$24cm24 \mathrm{~cm}$24\mathrm{c}\mathrm{m}$ and the distance of $\mathrm{Q}$$\mathrm{Q}$Q\mathrm{Q}$\mathrm{Q}$ from the centre is $25\mathrm{c}\mathrm{m}$$25\mathrm{c}\mathrm{m}$25cm25 \mathrm{~cm}$25\mathrm{c}\mathrm{m}$. The radius of the circle is
   (A) $7\mathrm{c}\mathrm{m}$$7\mathrm{c}\mathrm{m}$7cm7 \mathrm{~cm}$7\mathrm{c}\mathrm{m}$
   (B) $12\mathrm{c}\mathrm{m}$$12\mathrm{c}\mathrm{m}$12cm12 \mathrm{~cm}$12\mathrm{c}\mathrm{m}$
   (C) $15\mathrm{c}\mathrm{m}$$15\mathrm{c}\mathrm{m}$15cm15 \mathrm{~cm}$15\mathrm{c}\mathrm{m}$
   (D) $24.5\mathrm{c}\mathrm{m}$$24.5\mathrm{c}\mathrm{m}$24.5cm24.5 \mathrm{~cm}$24.5\mathrm{c}\mathrm{m}$
2. In Fig. 10.11, if TP and $\mathrm{T}\mathrm{Q}$$\mathrm{T}\mathrm{Q}$TQ\mathrm{TQ}$\mathrm{T}\mathrm{Q}$ are the two tangents to a circle with centre $\mathrm{O}$$\mathrm{O}$O\mathrm{O}$\mathrm{O}$ so that $\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{Q}={110}^{\circ }$$\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{Q}={110}^{\circ }$/_POQ=110^(@)\angle \mathrm{POQ}=110^{\circ}$\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{Q}={110}^{\circ }$, then $\mathrm{\angle }\mathrm{P}\mathrm{T}\mathrm{Q}$$\mathrm{\angle }\mathrm{P}\mathrm{T}\mathrm{Q}$/_PTQ\angle \mathrm{PTQ}$\mathrm{\angle }\mathrm{P}\mathrm{T}\mathrm{Q}$ is equal to
   (A) ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$
   (B) ${70}^{\circ }$${70}^{\circ }$70^(@)70^{\circ}${70}^{\circ }$
   (C) ${80}^{\circ }$${80}^{\circ }$80^(@)80^{\circ}${80}^{\circ }$
   (D) ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$

3. If tangents $\mathrm{P}\mathrm{A}$$\mathrm{P}\mathrm{A}$PA\mathrm{PA}$\mathrm{P}\mathrm{A}$ and $\mathrm{P}\mathrm{B}$$\mathrm{P}\mathrm{B}$PB\mathrm{PB}$\mathrm{P}\mathrm{B}$ from a point $\mathrm{P}$$\mathrm{P}$P\mathrm{P}$\mathrm{P}$ to a circle with centre $\mathrm{O}$$\mathrm{O}$O\mathrm{O}$\mathrm{O}$ are inclined to each other at angle of ${80}^{\circ }$${80}^{\circ }$80^(@)80^{\circ}${80}^{\circ }$, then $\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{A}$$\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{A}$/_POA\angle \mathrm{POA}$\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{A}$ is equal to
   (A) ${50}^{\circ }$${50}^{\circ }$50^(@)50^{\circ}${50}^{\circ }$
   (B) ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$
   (C) ${70}^{\circ }$${70}^{\circ }$70^(@)70^{\circ}${70}^{\circ }$
   (D) ${80}^{\circ }$${80}^{\circ }$80^(@)80^{\circ}${80}^{\circ }$
4. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
5. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
6. The length of a tangent from a point $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$ at distance $5\mathrm{c}\mathrm{m}$$5\mathrm{c}\mathrm{m}$5cm5 \mathrm{~cm}$5\mathrm{c}\mathrm{m}$ from the centre of the circle is 4 $\mathrm{c}\mathrm{m}$$\mathrm{c}\mathrm{m}$cm\mathrm{cm}$\mathrm{c}\mathrm{m}$. Find the radius of the circle.
7. Two concentric circles are of radii $5\mathrm{c}\mathrm{m}$$5\mathrm{c}\mathrm{m}$5cm5 \mathrm{~cm}$5\mathrm{c}\mathrm{m}$ and $3\mathrm{c}\mathrm{m}$$3\mathrm{c}\mathrm{m}$3cm3 \mathrm{~cm}$3\mathrm{c}\mathrm{m}$. Find the length of the chord of the larger circle which touches the smaller circle.
8. A quadrilateral $ABCD$$ABCD$ABCDA B C D$ABCD$ is drawn to circumscribe a circle (see Fig. 10.12). Prove that

9. In Fig. 10.13, $\mathrm{X}\mathrm{Y}$$\mathrm{X}\mathrm{Y}$XY\mathrm{XY}$\mathrm{X}\mathrm{Y}$ and ${\mathrm{X}}^{\mathrm{\prime }}{\mathrm{Y}}^{\mathrm{\prime }}$${\mathrm{X}}^{\mathrm{\prime }}{\mathrm{Y}}^{\mathrm{\prime }}$X^(')Y^(')\mathrm{X}^{\prime} \mathrm{Y}^{\prime}${\mathrm{X}}^{\mathrm{\prime }}{\mathrm{Y}}^{\mathrm{\prime }}$ are two parallel tangents to a circle with centre $\mathrm{O}$$\mathrm{O}$O\mathrm{O}$\mathrm{O}$ and another tangent $AB$$AB$ABA B$AB$ with point of contact $C$$C$CC$C$ intersecting $XY$$XY$XYX Y$XY$ at $A$$A$AA$A$ and $X^{\mathrm{\prime }}{Y}^{\mathrm{\prime }}$$X^{\mathrm{\prime }}{Y}^{\mathrm{\prime }}$X^(')Y^(')X^{\prime} Y^{\prime}$X^{\mathrm{\prime }}{Y}^{\mathrm{\prime }}$ at $B$$B$BB$B$. Prove that $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}={90}^{\circ }$$\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}={90}^{\circ }$/_AOB=90^(@)\angle \mathrm{AOB}=90^{\circ}$\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{B}={90}^{\circ }$.
10. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
11. Prove that the parallelogram circumscribing a circle is a rhombus.
12. A triangle $\mathrm{A}\mathrm{B}\mathrm{C}$$\mathrm{A}\mathrm{B}\mathrm{C}$ABC\mathrm{ABC}$\mathrm{A}\mathrm{B}\mathrm{C}$ is drawn to circumscribe a circle of radius $4\mathrm{c}\mathrm{m}$$4\mathrm{c}\mathrm{m}$4cm4 \mathrm{~cm}$4\mathrm{c}\mathrm{m}$ such that the segments $\mathrm{B}\mathrm{D}$$\mathrm{B}\mathrm{D}$BD\mathrm{BD}$\mathrm{B}\mathrm{D}$ and DC into which $\mathrm{B}\mathrm{C}$$\mathrm{B}\mathrm{C}$BC\mathrm{BC}$\mathrm{B}\mathrm{C}$ is divided by the point of contact $D$$D$DD$D$ are of lengths $8\mathrm{c}\mathrm{m}$$8\mathrm{c}\mathrm{m}$8cm8 \mathrm{~cm}$8\mathrm{c}\mathrm{m}$ and $6\mathrm{c}\mathrm{m}$$6\mathrm{c}\mathrm{m}$6cm6 \mathrm{~cm}$6\mathrm{c}\mathrm{m}$ respectively (see Fig. 10.14). Find the sides AB and AC.
13. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

10.4 Summary

1. The meaning of a tangent to a circle.
2. The tangent to a circle is perpendicular to the radius through the point of contact.
3. The lengths of the two tangents from an external point to a circle are equal.

Areas Related to Circles 11

11.1 Areas of Sector and Segment of a Circle

EXERCISE 11.1

1. Find the area of a sector of a circle with radius $6\mathrm{c}\mathrm{m}$$6\mathrm{c}\mathrm{m}$6cm6 \mathrm{~cm}$6\mathrm{c}\mathrm{m}$ if angle of the sector is ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$.
2. Find the area of a quadrant of a circle whose circumference is $22\mathrm{c}\mathrm{m}$$22\mathrm{c}\mathrm{m}$22cm22 \mathrm{~cm}$22\mathrm{c}\mathrm{m}$.
3. The length of the minute hand of a clock is $14\mathrm{c}\mathrm{m}$$14\mathrm{c}\mathrm{m}$14cm14 \mathrm{~cm}$14\mathrm{c}\mathrm{m}$. Find the area swept by the minute hand in 5 minutes.
4. A chord of a circle of radius $10\mathrm{c}\mathrm{m}$$10\mathrm{c}\mathrm{m}$10cm10 \mathrm{~cm}$10\mathrm{c}\mathrm{m}$ subtends a right angle at the centre. Find the area of the corresponding: (i) minor segment (ii) major sector. (Use $\pi =3.14$$\pi =3.14$pi=3.14\pi=3.14$\pi =3.14$ )
5. In a circle of radius $21\mathrm{c}\mathrm{m}$$21\mathrm{c}\mathrm{m}$21cm21 \mathrm{~cm}$21\mathrm{c}\mathrm{m}$, an arc subtends an angle of ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ at the centre. Find:
   (i) the length of the arc
   (ii) area of the sector formed by the arc

6. A chord of a circle of radius $15\mathrm{c}\mathrm{m}$$15\mathrm{c}\mathrm{m}$15cm15 \mathrm{~cm}$15\mathrm{c}\mathrm{m}$ subtends an angle of ${60}^{\circ }$${60}^{\circ }$60^(@)60^{\circ}${60}^{\circ }$ at the centre. Find the areas of the corresponding minor and major segments of the circle.

7. A chord of a circle of radius $12\mathrm{c}\mathrm{m}$$12\mathrm{c}\mathrm{m}$12cm12 \mathrm{~cm}$12\mathrm{c}\mathrm{m}$ subtends an angle of ${120}^{\circ }$${120}^{\circ }$120^(@)120^{\circ}${120}^{\circ }$ at the centre. Find the area of the corresponding segment of the circle.

8. A horse is tied to a peg at one corner of a square shaped grass field of side $15\mathrm{m}$$15\mathrm{m}$15m15 \mathrm{~m}$15\mathrm{m}$ by means of a $5\mathrm{m}$$5\mathrm{m}$5m5 \mathrm{~m}$5\mathrm{m}$ long rope (see Fig. 11.8). Find

9. A brooch is made with silver wire in the form of a circle with diameter $35\mathrm{m}\mathrm{m}$$35\mathrm{m}\mathrm{m}$35mm35 \mathrm{~mm}$35\mathrm{m}\mathrm{m}$. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. 11.9. Find :

10. An umbrella has 8 ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius $45\mathrm{c}\mathrm{m}$$45\mathrm{c}\mathrm{m}$45cm45 \mathrm{~cm}$45\mathrm{c}\mathrm{m}$, find the area between the two consecutive ribs of the umbrella.
11. A car has two wipers which do not overlap. Each wiper has a blade of length $25\mathrm{c}\mathrm{m}$$25\mathrm{c}\mathrm{m}$25cm25 \mathrm{~cm}$25\mathrm{c}\mathrm{m}$ sweeping through an angle of ${115}^{\circ }$${115}^{\circ }$115^(@)115^{\circ}${115}^{\circ }$. Find the total area cleaned at each sweep of the blades.
12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle ${80}^{\circ }$${80}^{\circ }$80^(@)80^{\circ}${80}^{\circ }$ to a distance of $16.5\mathrm{k}\mathrm{m}$$16.5\mathrm{k}\mathrm{m}$16.5km16.5 \mathrm{~km}$16.5\mathrm{k}\mathrm{m}$. Find the area of the sea over which the ships are warned. (Use $\pi =3.14$$\pi =3.14$pi=3.14\pi=3.14$\pi =3.14$ )
13. A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is $28\mathrm{c}\mathrm{m}$$28\mathrm{c}\mathrm{m}$28cm28 \mathrm{~cm}$28\mathrm{c}\mathrm{m}$, find the cost of making the designs at the rate of ₹ 0.35 per cm${}^2$${}^2$^(2)^{2}${}^2$. (Use $\sqrt{3}=1.7$$\sqrt{3}=1.7$sqrt3=1.7\sqrt{3}=1.7$\sqrt{3}=1.7$ )

14. Tick the correct answer in the following:

11.2 Summary

1. Length of an arc of a sector of a circle with radius $r$$r$rr$r$ and angle with degree measure $\theta$$\theta$theta\theta$\theta$ is $\frac{\theta }{360}\times 2\pi r$$\frac{\theta }{360}\times 2\pi r$(theta)/(360)xx2pi r\frac{\theta}{360} \times 2 \pi r$\frac{\theta }{360}\times 2\pi r$.
2. Area of a sector of a circle with radius $r$$r$rr$r$ and angle with degree measure $\theta$$\theta$theta\theta$\theta$ is $\frac{\theta }{360}\times \pi r^2$$\frac{\theta }{360}\times \pi r^2$(theta)/(360)xx pir^(2)\frac{\theta}{360} \times \pi r^{2}$\frac{\theta }{360}\times \pi r^2$.
3. Area of segment of a circle

12.1 Introduction

12.2 Surface Area of a Combination of Solids

• CSA of other hemisphere

EXERCISE 12.1

1. 2 cubes each of volume $64{\mathrm{c}\mathrm{m}}^3$$64{\mathrm{c}\mathrm{m}}^3$64cm^(3)64 \mathrm{~cm}^{3}$64{\mathrm{c}\mathrm{m}}^3$ are joined end to end. Find the surface area of the resulting cuboid.
2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is $14\mathrm{c}\mathrm{m}$$14\mathrm{c}\mathrm{m}$14cm14 \mathrm{~cm}$14\mathrm{c}\mathrm{m}$ and the total height of the vessel is $13\mathrm{c}\mathrm{m}$$13\mathrm{c}\mathrm{m}$13cm13 \mathrm{~cm}$13\mathrm{c}\mathrm{m}$. Find the inner surface area of the vessel.
3. A toy is in the form of a cone of radius $3.5\mathrm{c}\mathrm{m}$$3.5\mathrm{c}\mathrm{m}$3.5cm3.5 \mathrm{~cm}$3.5\mathrm{c}\mathrm{m}$ mounted on a hemisphere of same radius. The total height of the toy is $15.5\mathrm{c}\mathrm{m}$$15.5\mathrm{c}\mathrm{m}$15.5cm15.5 \mathrm{~cm}$15.5\mathrm{c}\mathrm{m}$. Find the total surface area of the toy.
4. A cubical block of side $7\mathrm{c}\mathrm{m}$$7\mathrm{c}\mathrm{m}$7cm7 \mathrm{~cm}$7\mathrm{c}\mathrm{m}$ is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter $l$$l$ll$l$ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is $14\mathrm{m}\mathrm{m}$$14\mathrm{m}\mathrm{m}$14mm14 \mathrm{~mm}$14\mathrm{m}\mathrm{m}$ and the diameter of the capsule is $5\mathrm{m}\mathrm{m}$$5\mathrm{m}\mathrm{m}$5mm5 \mathrm{~mm}$5\mathrm{m}\mathrm{m}$. Find its surface area.

7. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are $2.1\mathrm{m}$$2.1\mathrm{m}$2.1m2.1 \mathrm{~m}$2.1\mathrm{m}$ and $4\mathrm{m}$$4\mathrm{m}$4m4 \mathrm{~m}$4\mathrm{m}$ respectively, and the slant height of the top is $2.8\mathrm{m}$$2.8\mathrm{m}$2.8m2.8 \mathrm{~m}$2.8\mathrm{m}$, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹ 500 per ${\mathrm{m}}^2$${\mathrm{m}}^2$m^(2)\mathrm{m}^{2}${\mathrm{m}}^2$. (Note that the base of the tent will not be covered with canvas.)
8. From a solid cylinder whose height is $2.4\mathrm{c}\mathrm{m}$$2.4\mathrm{c}\mathrm{m}$2.4cm2.4 \mathrm{~cm}$2.4\mathrm{c}\mathrm{m}$ and diameter $1.4\mathrm{c}\mathrm{m}$$1.4\mathrm{c}\mathrm{m}$1.4cm1.4 \mathrm{~cm}$1.4\mathrm{c}\mathrm{m}$, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest ${\mathrm{c}\mathrm{m}}^2$${\mathrm{c}\mathrm{m}}^2$cm^(2)\mathrm{cm}^{2}${\mathrm{c}\mathrm{m}}^2$.
9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is $10\mathrm{c}\mathrm{m}$$10\mathrm{c}\mathrm{m}$10cm10 \mathrm{~cm}$10\mathrm{c}\mathrm{m}$, and its base is of radius $3.5\mathrm{c}\mathrm{m}$$3.5\mathrm{c}\mathrm{m}$3.5cm3.5 \mathrm{~cm}$3.5\mathrm{c}\mathrm{m}$, find the total surface area of the article.

12.3 Volume of a Combination of Solids

EXERCISE 12.2

1. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $1\mathrm{c}\mathrm{m}$$1\mathrm{c}\mathrm{m}$1cm1 \mathrm{~cm}$1\mathrm{c}\mathrm{m}$ and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$$\pi$pi\pi$\pi$.
2. Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is $3\mathrm{c}\mathrm{m}$$3\mathrm{c}\mathrm{m}$3cm3 \mathrm{~cm}$3\mathrm{c}\mathrm{m}$ and its length is $12\mathrm{c}\mathrm{m}$$12\mathrm{c}\mathrm{m}$12cm12 \mathrm{~cm}$12\mathrm{c}\mathrm{m}$. If each cone has a height of $2\mathrm{c}\mathrm{m}$$2\mathrm{c}\mathrm{m}$2cm2 \mathrm{~cm}$2\mathrm{c}\mathrm{m}$, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
3. A gulab jamun, contains sugar syrup up to about $30\mathrm{\%}$$30\mathrm{\%}$30%30 \%$30\mathrm{\%}$ of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length $5\mathrm{c}\mathrm{m}$$5\mathrm{c}\mathrm{m}$5cm5 \mathrm{~cm}$5\mathrm{c}\mathrm{m}$ and diameter $2.8\mathrm{c}\mathrm{m}$$2.8\mathrm{c}\mathrm{m}$2.8cm2.8 \mathrm{~cm}$2.8\mathrm{c}\mathrm{m}$ (see Fig. 12.15).
4. A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are $15\mathrm{c}\mathrm{m}$$15\mathrm{c}\mathrm{m}$15cm15 \mathrm{~cm}$15\mathrm{c}\mathrm{m}$ by $10\mathrm{c}\mathrm{m}$$10\mathrm{c}\mathrm{m}$10cm10 \mathrm{~cm}$10\mathrm{c}\mathrm{m}$ by $3.5\mathrm{c}\mathrm{m}$$3.5\mathrm{c}\mathrm{m}$3.5cm3.5 \mathrm{~cm}$3.5\mathrm{c}\mathrm{m}$. The radius of each of the depressions is 0.5 $\mathrm{c}\mathrm{m}$$\mathrm{c}\mathrm{m}$cm\mathrm{cm}$\mathrm{c}\mathrm{m}$ and the depth is $1.4\mathrm{c}\mathrm{m}$$1.4\mathrm{c}\mathrm{m}$1.4cm1.4 \mathrm{~cm}$1.4\mathrm{c}\mathrm{m}$. Find the volume of wood in the entire stand (see Fig. 12.16).
5. A vessel is in the form of an inverted cone. Its height is $8\mathrm{c}\mathrm{m}$$8\mathrm{c}\mathrm{m}$8cm8 \mathrm{~cm}$8\mathrm{c}\mathrm{m}$ and the radius of its top, which is open, is $5\mathrm{c}\mathrm{m}$$5\mathrm{c}\mathrm{m}$5cm5 \mathrm{~cm}$5\mathrm{c}\mathrm{m}$. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius $0.5\mathrm{c}\mathrm{m}$$0.5\mathrm{c}\mathrm{m}$0.5cm0.5 \mathrm{~cm}$0.5\mathrm{c}\mathrm{m}$ are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

6. A solid iron pole consists of a cylinder of height $220\mathrm{c}\mathrm{m}$$220\mathrm{c}\mathrm{m}$220cm220 \mathrm{~cm}$220\mathrm{c}\mathrm{m}$ and base diameter $24\mathrm{c}\mathrm{m}$$24\mathrm{c}\mathrm{m}$24cm24 \mathrm{~cm}$24\mathrm{c}\mathrm{m}$, which is surmounted by another cylinder of height $60\mathrm{c}\mathrm{m}$$60\mathrm{c}\mathrm{m}$60cm60 \mathrm{~cm}$60\mathrm{c}\mathrm{m}$ and radius $8\mathrm{c}\mathrm{m}$$8\mathrm{c}\mathrm{m}$8cm8 \mathrm{~cm}$8\mathrm{c}\mathrm{m}$. Find the mass of the pole, given that $1{\mathrm{c}\mathrm{m}}^3$$1{\mathrm{c}\mathrm{m}}^3$1cm^(3)1 \mathrm{~cm}^{3}$1{\mathrm{c}\mathrm{m}}^3$ of iron has approximately $8\mathrm{g}$$8\mathrm{g}$8g8 \mathrm{~g}$8\mathrm{g}$ mass. (Use $\pi =3.14$$\pi =3.14$pi=3.14\pi=3.14$\pi =3.14$ )
7. A solid consisting of a right circular cone of height $120\mathrm{c}\mathrm{m}$$120\mathrm{c}\mathrm{m}$120cm120 \mathrm{~cm}$120\mathrm{c}\mathrm{m}$ and radius $60\mathrm{c}\mathrm{m}$$60\mathrm{c}\mathrm{m}$60cm60 \mathrm{~cm}$60\mathrm{c}\mathrm{m}$ standing on a hemisphere of radius $60\mathrm{c}\mathrm{m}$$60\mathrm{c}\mathrm{m}$60cm60 \mathrm{~cm}$60\mathrm{c}\mathrm{m}$ is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is $60\mathrm{c}\mathrm{m}$$60\mathrm{c}\mathrm{m}$60cm60 \mathrm{~cm}$60\mathrm{c}\mathrm{m}$ and its height is $180\mathrm{c}\mathrm{m}$$180\mathrm{c}\mathrm{m}$180cm180 \mathrm{~cm}$180\mathrm{c}\mathrm{m}$.
8. A spherical glass vessel has a cylindrical neck $8\mathrm{c}\mathrm{m}$$8\mathrm{c}\mathrm{m}$8cm8 \mathrm{~cm}$8\mathrm{c}\mathrm{m}$ long, $2\mathrm{c}\mathrm{m}$$2\mathrm{c}\mathrm{m}$2cm2 \mathrm{~cm}$2\mathrm{c}\mathrm{m}$ in diameter; the diameter of the spherical part is $8.5\mathrm{c}\mathrm{m}$$8.5\mathrm{c}\mathrm{m}$8.5cm8.5 \mathrm{~cm}$8.5\mathrm{c}\mathrm{m}$. By measuring the amount of water it holds, a child finds its volume to be $345{\mathrm{c}\mathrm{m}}^3$$345{\mathrm{c}\mathrm{m}}^3$345cm^(3)345 \mathrm{~cm}^{3}$345{\mathrm{c}\mathrm{m}}^3$. Check whether she is correct, taking the above as the inside measurements, and $\pi =3.14$$\pi =3.14$pi=3.14\pi=3.14$\pi =3.14$.

12.4 Summary

1. To determine the surface area of an object formed by combining any two of the basic solids, namely, cuboid, cone, cylinder, sphere and hemisphere.
2. To find the volume of objects formed by combining any two of a cuboid, cone, cylinder, sphere and hemisphere.

Statistics

13.1 Introduction

13.2 Mean of Grouped Data

• the step-deviation method will be convenient to apply if all the $d_i$$d_i$d_(i)d_{i}$d_i$ 's have a common factor.
• The mean obtained by all the three methods is the same.
• The assumed mean method and step-deviation method are just simplified forms of the direct method.
• The formula $\overline{x}=a+h\overline{u}$$\overline{x}=a+h\overline{u}$bar(x)=a+h bar(u)\bar{x}=a+h \bar{u}$\overline{x}=a+h\overline{u}$ still holds if $a$$a$aa$a$ and $h$$h$hh$h$ are not as given above, but are any non-zero numbers such that $u_i=\frac{x_i-a}{h}$$u_i=\frac{x_i-a}{h}$u_(i)=(x_(i)-a)/(h)u_{i}=\frac{x_{i}-a}{h}$u_i=\frac{x_i-a}{h}$.

Activity 2 :

1. Collect the marks obtained by all the students of your class in Mathematics in the latest examination conducted by your school. Form a grouped frequency distribution of the data obtained.
2. Collect the daily maximum temperatures recorded for a period of 30 days in your city. Present this data as a grouped frequency table.
3. Measure the heights of all the students of your class (in $\mathrm{c}\mathrm{m}$$\mathrm{c}\mathrm{m}$cm\mathrm{cm}$\mathrm{c}\mathrm{m}$ ) and form a grouped frequency distribution table of this data.

EXERCISE 13.1

1. A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

2. Consider the following distribution of daily wages of 50 workers of a factory.

3. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18 . Find the missing frequency $f$$f$ff$f$.

4. Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.

5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.

6. The table below shows the daily expenditure on food of 25 households in a locality.

7. To find out the concentration of ${\mathrm{S}\mathrm{O}}_2$${\mathrm{S}\mathrm{O}}_2$SO_(2)\mathrm{SO}_{2}${\mathrm{S}\mathrm{O}}_2$ in the air (in parts per million, i.e., $\mathrm{p}\mathrm{p}\mathrm{m}$$\mathrm{p}\mathrm{p}\mathrm{m}$ppm\mathrm{ppm}$\mathrm{p}\mathrm{p}\mathrm{m}$ ), the data was collected for 30 localities in a certain city and is presented below:

8. A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.

9. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

13.3 Mode of Grouped Data

Remarks :

1. In Example 6, the mode is less than the mean. But for some other problems it may be equal or more than the mean also.
2. It depends upon the demand of the situation whether we are interested in finding the average marks obtained by the students or the average of the marks obtained by most
   of the students. In the first situation, the mean is required and in the second situation, the mode is required.

EXERCISE 13.2

1. The following table shows the ages of the patients admitted in a hospital during a year:

2. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components :

3. The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure :

4. The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.

5. The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.

6. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data :

13.4 Median of Grouped Data

Table 13.14

Solution :

Remarks :

1. There is a empirical relationship between the three measures of central tendency:

2. The median of grouped data with unequal class sizes can also be calculated. However, we shall not discuss it here.

EXERCISE 13.3

1. The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

2. If the median of the distribution given below is 28.5 , find the values of $x$$x$xx$x$ and $y$$y$yy$y$.

3. A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.

4. The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table :

5. The following table gives the distribution of the life time of 400 neon lamps :

7. The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

13.5 Summary

1. The mean for grouped data can be found by :

2. The mode for grouped data can be found by using the formula:

3. The cumulative frequency of a class is the frequency obtained by adding the frequencies of all the classes preceding the given class.
4. The median for grouped data is formed by using the formula:

A NOTE TO THE Reader

Probability
14

• R.S. Woodward

14.1 Probability — A Theoretical Approach

Remarks :

1. An event having only one outcome of the experiment is called an elementary event. In Example 1, both the events E and F are elementary events. Similarly, in Example 2, all the three events, Y, B and R are elementary events.
2. In Example 1, we note that : $\mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})=1$$\mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})=1$P(E)+P(F)=1\mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})=1$\mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})=1$

In general, it is true that for an event $E$$E$EE$E$,

Let us answer (ii) :

EXERCISE 14.1

1. Complete the following statements:

2. Which of the following experiments have equally likely outcomes? Explain.

3. Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?
4. Which of the following cannot be the probability of an event?
   (A) $\frac{2}{3}$$\frac{2}{3}$(2)/(3)\frac{2}{3}$\frac{2}{3}$
   (B) -1.5
   (C) $15\mathrm{\%}$$15\mathrm{\%}$15%15 \%$15\mathrm{\%}$
   (D) 0.7
5. If $\mathrm{P}(\mathrm{E})=0.05$$\mathrm{P}(\mathrm{E})=0.05$P(E)=0.05\mathrm{P}(\mathrm{E})=0.05$\mathrm{P}(\mathrm{E})=0.05$, what is the probability of 'not $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$ '?
6. A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out

7. It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992 . What is the probability that the 2 students have the same birthday?
8. A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red? (ii) not red?
9. A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be
   (i) red ?
   (ii) white?
   (iii) not green?
10. A piggy bank contains hundred 50p coins, fifty ₹ 1 coins, twenty ₹ 2 coins and ten ₹ 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin (i) will be a $50\mathrm{p}$$50\mathrm{p}$50p50 \mathrm{p}$50\mathrm{p}$ coin? (ii) will not be a ₹ 5 coin?
11. Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish (see Fig. 14.4). What is the probability that the fish taken out is a male fish?
12. A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers

13. A die is thrown once. Find the probability of getting
    (i) a prime number;
    (ii) a number lying between 2 and 6 ;
    (iii) an odd number.
14. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting
    (i) a king of red colour
    (ii) a face card
    (iii) a red face card
    (iv) the jack of hearts
    (v) a spade
    (vi) the queen of diamonds
15. Five cards - the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random.

17. (i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?

18. A box contains 90 discs which are numbered from 1 to 90 . If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5 .
19. A child has a die whose six faces show the letters as given below:

21. A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that

22. Refer to Example 13. (i) Complete the following table:

23. A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.
24. A die is thrown twice. What is the probability that
    (i) 5 will not come up either time?
    (ii) 5 will come up at least once?

25. Which of the following arguments are correct and which are not correct? Give reasons for your answer.

14.2 Summary

1. The theoretical (classical) probability of an event $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$, written as $\mathrm{P}(\mathrm{E})$$\mathrm{P}(\mathrm{E})$P(E)\mathrm{P}(\mathrm{E})$\mathrm{P}(\mathrm{E})$, is defined as

2. The probability of a sure event (or certain event) is 1 .
3. The probability of an impossible event is 0 .
4. The probability of an event $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$ is a number $\mathrm{P}(\mathrm{E})$$\mathrm{P}(\mathrm{E})$P(E)\mathrm{P}(\mathrm{E})$\mathrm{P}(\mathrm{E})$ such that

5. An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1 .
6. For any event $\mathrm{E},\mathrm{P}(\mathrm{E})+\mathrm{P}(\overline{\mathrm{E}})=1$$\mathrm{E},\mathrm{P}(\mathrm{E})+\mathrm{P}(\overline{\mathrm{E}})=1$E,P(E)+P( bar(E))=1\mathrm{E}, \mathrm{P}(\mathrm{E})+\mathrm{P}(\overline{\mathrm{E}})=1$\mathrm{E},\mathrm{P}(\mathrm{E})+\mathrm{P}(\overline{\mathrm{E}})=1$, where $\overline{\mathrm{E}}$$\overline{\mathrm{E}}$bar(E)\overline{\mathrm{E}}$\overline{\mathrm{E}}$ stands for 'not $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$ '. $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$ and $\overline{\mathrm{E}}$$\overline{\mathrm{E}}$bar(E)\overline{\mathrm{E}}$\overline{\mathrm{E}}$ are called complementary events.

A NOTE TO THE READER

ANSWERS/HINTS

EXERCISE 1.1

1. (i) $2^2\times 5\times 7$$2^2\times 5\times 7$2^(2)xx5xx72^{2} \times 5 \times 7$2^2\times 5\times 7$
   (ii) $2^2\times 3\times 13$$2^2\times 3\times 13$2^(2)xx3xx132^{2} \times 3 \times 13$2^2\times 3\times 13$
   (iii) $3^2\times 5^2\times 17$$3^2\times 5^2\times 17$3^(2)xx5^(2)xx173^{2} \times 5^{2} \times 17$3^2\times 5^2\times 17$
   (iv) $5\times 7\times 11\times 13$$5\times 7\times 11\times 13$5xx7xx11 xx135 \times 7 \times 11 \times 13$5\times 7\times 11\times 13$
   (v) $17\times 19\times 23$$17\times 19\times 23$17 xx19 xx2317 \times 19 \times 23$17\times 19\times 23$
2. (i) $\mathrm{L}\mathrm{C}\mathrm{M}=182;\mathrm{H}\mathrm{C}\mathrm{F}=13$$\mathrm{L}\mathrm{C}\mathrm{M}=182;\mathrm{H}\mathrm{C}\mathrm{F}=13$LCM=182;HCF=13\mathrm{LCM}=182 ; \mathrm{HCF}=13$\mathrm{L}\mathrm{C}\mathrm{M}=182;\mathrm{H}\mathrm{C}\mathrm{F}=13$
   (ii) $\mathrm{L}\mathrm{C}\mathrm{M}=23460;\mathrm{H}\mathrm{C}\mathrm{F}=2$$\mathrm{L}\mathrm{C}\mathrm{M}=23460;\mathrm{H}\mathrm{C}\mathrm{F}=2$LCM=23460;HCF=2\mathrm{LCM}=23460 ; \mathrm{HCF}=2$\mathrm{L}\mathrm{C}\mathrm{M}=23460;\mathrm{H}\mathrm{C}\mathrm{F}=2$
   (iii) $\mathrm{L}\mathrm{C}\mathrm{M}=3024;\mathrm{H}\mathrm{C}\mathrm{F}=6$$\mathrm{L}\mathrm{C}\mathrm{M}=3024;\mathrm{H}\mathrm{C}\mathrm{F}=6$LCM=3024;HCF=6\mathrm{LCM}=3024 ; \mathrm{HCF}=6$\mathrm{L}\mathrm{C}\mathrm{M}=3024;\mathrm{H}\mathrm{C}\mathrm{F}=6$
3. (i) $\mathrm{L}\mathrm{C}\mathrm{M}=420;\mathrm{H}\mathrm{C}\mathrm{F}=3$$\mathrm{L}\mathrm{C}\mathrm{M}=420;\mathrm{H}\mathrm{C}\mathrm{F}=3$LCM=420;HCF=3\mathrm{LCM}=420 ; \mathrm{HCF}=3$\mathrm{L}\mathrm{C}\mathrm{M}=420;\mathrm{H}\mathrm{C}\mathrm{F}=3$
   (ii) $\mathrm{L}\mathrm{C}\mathrm{M}=11339;\mathrm{H}\mathrm{C}\mathrm{F}=1$$\mathrm{L}\mathrm{C}\mathrm{M}=11339;\mathrm{H}\mathrm{C}\mathrm{F}=1$LCM=11339;HCF=1\mathrm{LCM}=11339 ; \mathrm{HCF}=1$\mathrm{L}\mathrm{C}\mathrm{M}=11339;\mathrm{H}\mathrm{C}\mathrm{F}=1$
   (iii) $\mathrm{L}\mathrm{C}\mathrm{M}=1800;\mathrm{H}\mathrm{C}\mathrm{F}=1$$\mathrm{L}\mathrm{C}\mathrm{M}=1800;\mathrm{H}\mathrm{C}\mathrm{F}=1$LCM=1800;HCF=1\mathrm{LCM}=1800 ; \mathrm{HCF}=1$\mathrm{L}\mathrm{C}\mathrm{M}=1800;\mathrm{H}\mathrm{C}\mathrm{F}=1$
4. 22338
5. 36 minutes

1. (i) No zeroes
   (ii) 1
   (iii) 3
   (iv) 2
   (v) 4
   (vi) 3

EXERCISE 2.2

1. (i) $-2,4$$-2,4$-2,4-2,4$-2,4$
   (ii) $\frac{1}{2},\frac{1}{2}$$\frac{1}{2},\frac{1}{2}$(1)/(2),(1)/(2)\frac{1}{2}, \frac{1}{2}$\frac{1}{2},\frac{1}{2}$
   (iii) $-\frac{1}{3},\frac{3}{2}$$-\frac{1}{3},\frac{3}{2}$-(1)/(3),(3)/(2)-\frac{1}{3}, \frac{3}{2}$-\frac{1}{3},\frac{3}{2}$
   (v) $-\sqrt{15},\sqrt{15}$$-\sqrt{15},\sqrt{15}$-sqrt15,sqrt15-\sqrt{15}, \sqrt{15}$-\sqrt{15},\sqrt{15}$
   (vi) $-1,\frac{4}{3}$$-1,\frac{4}{3}$-1,(4)/(3)-1, \frac{4}{3}$-1,\frac{4}{3}$

2. (i) $4x^2-x-4$$4x^2-x-4$4x^(2)-x-44 x^{2}-x-4$4x^2-x-4$

EXERCISE 3.1

1. (i) Required pair of linear equations is

2. (i) Intersect at a point

3. (i) Consistent

4. (i) Consistent

6. One possible answer for the three parts:
   (i) $3x+2y-7=0$$3x+2y-7=0$3x+2y-7=03 x+2 y-7=0$3x+2y-7=0$
   (ii) $2x+3y-12=0$$2x+3y-12=0$2x+3y-12=02 x+3 y-12=0$2x+3y-12=0$
   (iii) $4x+6y-16=0$$4x+6y-16=0$4x+6y-16=04 x+6 y-16=0$4x+6y-16=0$
7. Vertices of the triangle are $(-1,0),(4,0)$$(-1,0),(4,0)$(-1,0),(4,0)(-1,0),(4,0)$(-1,0),(4,0)$ and $(2,3)$$(2,3)$(2,3)(2,3)$(2,3)$.

EXERCISE 3.2

1. (i) $x=9,y=5$$x=9,y=5$x=9,y=5x=9, y=5$x=9,y=5$

3. (i) $x-y=26,x=3y$$x-y=26,x=3y$x-y=26,x=3yx-y=26, x=3 y$x-y=26,x=3y$, where $x$$x$xx$x$ and $y$$y$yy$y$ are two numbers $(x>y);x=39,y=13$$(x>y);x=39,y=13$(x > y);x=39,y=13(x>y) ; x=39, y=13$(x>y);x=39,y=13$.

EXERCISE 3.3

1. (i) $x=\frac{19}{5},y=\frac{6}{5}$$x=\frac{19}{5},y=\frac{6}{5}$x=(19)/(5),y=(6)/(5)x=\frac{19}{5}, y=\frac{6}{5}$x=\frac{19}{5},y=\frac{6}{5}$
   (ii) $x=2,y=1$$x=2,y=1$x=2,y=1x=2, y=1$x=2,y=1$
   (iii) $x=\frac{9}{13},y=-\frac{5}{13}$$x=\frac{9}{13},y=-\frac{5}{13}$x=(9)/(13),y=-(5)/(13)x=\frac{9}{13}, y=-\frac{5}{13}$x=\frac{9}{13},y=-\frac{5}{13}$
   (iv) $x=2,y=-3$$x=2,y=-3$x=2,y=-3x=2, y=-3$x=2,y=-3$
2. (i) $x-y+2=0,2x-y-1=0$$x-y+2=0,2x-y-1=0$x-y+2=0,2x-y-1=0x-y+2=0,2 x-y-1=0$x-y+2=0,2x-y-1=0$, where $x$$x$xx$x$ and $y$$y$yy$y$ are the numerator and denominator of the fraction; $\frac{3}{5}$$\frac{3}{5}$(3)/(5)\frac{3}{5}$\frac{3}{5}$.

EXERCISE 4.1

1. (i) Yes
   (ii) Yes
   (iii) No
   (iv) Yes
   (v) Yes
   (vi) No
   (vii) No
   (viii) Yes
2. (i) $2x^2+x-528=0$$2x^2+x-528=0$2x^(2)+x-528=02 x^{2}+x-528=0$2x^2+x-528=0$, where $x$$x$xx$x$ is breadth (in metres) of the plot.

EXERCISE 4.2

1. (i) $-2,5$$-2,5$-2,5-2,5$-2,5$

2. (i) 9,36

3. Numbers are 13 and 14 .
4. Positive integers are 13 and 14 .
5. $5\mathrm{c}\mathrm{m}$$5\mathrm{c}\mathrm{m}$5cm5 \mathrm{~cm}$5\mathrm{c}\mathrm{m}$ and $12\mathrm{c}\mathrm{m}$$12\mathrm{c}\mathrm{m}$12cm12 \mathrm{~cm}$12\mathrm{c}\mathrm{m}$
6. Number of articles $=6$$=6$=6=6$=6$, Cost of each article $=₹15$$=₹15$=₹15=₹ 15$=₹15$

EXERCISE 4.3

2. (i) $k=\pm 2\sqrt{6}$$k=\pm 2\sqrt{6}$k=+-2sqrt6k= \pm 2 \sqrt{6}$k=\pm 2\sqrt{6}$

3. Yes. $40\mathrm{m},20\mathrm{m}$$40\mathrm{m},20\mathrm{m}$40m,20m40 \mathrm{~m}, 20 \mathrm{~m}$40\mathrm{m},20\mathrm{m}$
4. No
5. Yes. $20\mathrm{m},20\mathrm{m}$$20\mathrm{m},20\mathrm{m}$20m,20m20 \mathrm{~m}, 20 \mathrm{~m}$20\mathrm{m},20\mathrm{m}$

EXERCISE 5.1

1. (i) Yes. $15,23,31,\dots$$15,23,31,\dots$15,23,31,dots15,23,31, \ldots$15,23,31,\dots$ forms an $\mathrm{A}\mathrm{P}$$\mathrm{A}\mathrm{P}$AP\mathrm{AP}$\mathrm{A}\mathrm{P}$ as each succeeding term is obtained by adding 8 in its preceding term.

3. (i) $a=3,d=-2$$a=3,d=-2$a=3,d=-2a=3, d=-2$a=3,d=-2$

4. (i) No

EXERCISE 5.2

1. (i) $a_n=28$$a_n=28$a_(n)=28a_{n}=28$a_n=28$
   (ii) $d=2$$d=2$d=2d=2$d=2$
   (iii) $a=46$$a=46$a=46a=46$a=46$
   (iv) $n=10$$n=10$n=10n=10$n=10$
   (v) $a_n=3.5$$a_n=3.5$a_(n)=3.5a_{n}=3.5$a_n=3.5$
2. (i) $\mathrm{C}$$\mathrm{C}$C\mathrm{C}$\mathrm{C}$

5. (i) 34

6. No
7. 178
8. 64
9. 5th term
10. 1
11. 65 th term
12. 100
13. 128
14. 60
15. 13
16. $4,10,16,22,\dots$$4,10,16,22,\dots$4,10,16,22,dots4,10,16,22, \ldots$4,10,16,22,\dots$
17. 20th term from the last term is 158 .
18. $-13,-8,-3$$-13,-8,-3$-13,-8,-3-13,-8,-3$-13,-8,-3$
19. 11th year
20. 10

EXERCISE 5.3

1. (i) 245
   (ii) -180
   (iii) 5505
   (iv) $\frac{33}{20}$$\frac{33}{20}$(33)/(20)\frac{33}{20}$\frac{33}{20}$
2. (i) $1046\frac{1}{2}$$1046\frac{1}{2}$1046(1)/(2)1046 \frac{1}{2}$1046\frac{1}{2}$

3. (i) $n=16,{\mathrm{S}}_n=440$$n=16,{\mathrm{S}}_n=440$n=16,S_(n)=440n=16, \mathrm{~S}_{n}=440$n=16,{\mathrm{S}}_n=440$

4. 12. By putting $a=9,d=8,\mathrm{S}=636$$a=9,d=8,\mathrm{S}=636$a=9,d=8,S=636a=9, d=8, \mathrm{~S}=636$a=9,d=8,\mathrm{S}=636$ in the formula $\mathrm{S}=\frac{n}{2}[2a+(n-1)d]$$\mathrm{S}=\frac{n}{2}[2a+(n-1)d]$S=(n)/(2)[2a+(n-1)d]\mathrm{S}=\frac{n}{2}[2 a+(n-1) d]$\mathrm{S}=\frac{n}{2}[2a+(n-1)d]$, we get a quadratic equation $4n^2+5n-636=0$$4n^2+5n-636=0$4n^(2)+5n-636=04 n^{2}+5 n-636=0$4n^2+5n-636=0$. On solving, we get $n=-\frac{53}{4},12$$n=-\frac{53}{4},12$n=-(53)/(4),12n=-\frac{53}{4}, 12$n=-\frac{53}{4},12$. Out of these two roots only one root 12 is admissible.
5. $n=16,d=\frac{8}{3}$$n=16,d=\frac{8}{3}$n=16,d=(8)/(3)n=16, d=\frac{8}{3}$n=16,d=\frac{8}{3}$
6. $n=38,\mathrm{S}=6973$$n=38,\mathrm{S}=6973$n=38,S=6973n=38, \mathrm{~S}=6973$n=38,\mathrm{S}=6973$
7. Sum $=1661$$=1661$=1661=1661$=1661$
8. ${\mathrm{S}}_{51}=5610$${\mathrm{S}}_{51}=5610$S_(51)=5610\mathrm{S}_{51}=5610${\mathrm{S}}_{51}=5610$
9. $n^2$$n^2$n^(2)n^{2}$n^2$
10. (i) $S_{15}=525$$S_{15}=525$S_(15)=525S_{15}=525$S_{15}=525$

11. ${\mathrm{S}}_1=3,{\mathrm{S}}_2=4;a_2={\mathrm{S}}_2-{\mathrm{S}}_1=1;{\mathrm{S}}_3=3,a_3={\mathrm{S}}_3-{\mathrm{S}}_2=-1$${\mathrm{S}}_1=3,{\mathrm{S}}_2=4;a_2={\mathrm{S}}_2-{\mathrm{S}}_1=1;{\mathrm{S}}_3=3,a_3={\mathrm{S}}_3-{\mathrm{S}}_2=-1$S_(1)=3,S_(2)=4;a_(2)=S_(2)-S_(1)=1;S_(3)=3,a_(3)=S_(3)-S_(2)=-1\mathrm{S}_{1}=3, \mathrm{~S}_{2}=4 ; a_{2}=\mathrm{S}_{2}-\mathrm{S}_{1}=1 ; \mathrm{S}_{3}=3, a_{3}=\mathrm{S}_{3}-\mathrm{S}_{2}=-1${\mathrm{S}}_1=3,{\mathrm{S}}_2=4;a_2={\mathrm{S}}_2-{\mathrm{S}}_1=1;{\mathrm{S}}_3=3,a_3={\mathrm{S}}_3-{\mathrm{S}}_2=-1$, $a_{10}={\mathrm{S}}_{10}-{\mathrm{S}}_9=-15;a_n={\mathrm{S}}_n-{\mathrm{S}}_{n-1}=5-2n$$a_{10}={\mathrm{S}}_{10}-{\mathrm{S}}_9=-15;a_n={\mathrm{S}}_n-{\mathrm{S}}_{n-1}=5-2n$a_(10)=S_(10)-S_(9)=-15;a_(n)=S_(n)-S_(n-1)=5-2na_{10}=\mathrm{S}_{10}-\mathrm{S}_{9}=-15 ; a_{n}=\mathrm{S}_{n}-\mathrm{S}_{n-1}=5-2 n$a_{10}={\mathrm{S}}_{10}-{\mathrm{S}}_9=-15;a_n={\mathrm{S}}_n-{\mathrm{S}}_{n-1}=5-2n$.
12. 4920
13. 960
14. 625
15. ₹ 27750
16. Values of the prizes (in ₹) are $160,140,120,100,80,60,40$$160,140,120,100,80,60,40$160,140,120,100,80,60,40160,140,120,100,80,60,40$160,140,120,100,80,60,40$.
17. 234
18. $143\mathrm{c}\mathrm{m}$$143\mathrm{c}\mathrm{m}$143cm143 \mathrm{~cm}$143\mathrm{c}\mathrm{m}$
19. 16 rows, 5 logs are placed in the top row. By putting $S=200,a=20,d=-1$$S=200,a=20,d=-1$S=200,a=20,d=-1S=200, a=20, d=-1$S=200,a=20,d=-1$ in the formula $\mathrm{S}=\frac{n}{2}[2a+(n-1)d]$$\mathrm{S}=\frac{n}{2}[2a+(n-1)d]$S=(n)/(2)[2a+(n-1)d]\mathrm{S}=\frac{n}{2}[2 a+(n-1) d]$\mathrm{S}=\frac{n}{2}[2a+(n-1)d]$, we get, $41n-n^2=400$$41n-n^2=400$41 n-n^(2)=40041 n-n^{2}=400$41n-n^2=400$. On solving, $n=16,25$$n=16,25$n=16,25n=16,25$n=16,25$. Therefore, the number of rows is either 16 or $25.a_{25}=a+24d=-4$$25.a_{25}=a+24d=-4$25.a_(25)=a+24 d=-425 . a_{25}=a+24 d=-4$25.a_{25}=a+24d=-4$

20. $370\mathrm{m}$$370\mathrm{m}$370m370 \mathrm{~m}$370\mathrm{m}$

EXERCISE 5.4 (Optional)

1. 32 nd term
2. ${\mathrm{S}}_{16}=20,76$${\mathrm{S}}_{16}=20,76$S_(16)=20,76\mathrm{S}_{16}=20,76${\mathrm{S}}_{16}=20,76$
3. $385\mathrm{c}\mathrm{m}$$385\mathrm{c}\mathrm{m}$385cm385 \mathrm{~cm}$385\mathrm{c}\mathrm{m}$
4. 35
5. $750{\mathrm{m}}^3$$750{\mathrm{m}}^3$750m^(3)750 \mathrm{~m}^{3}$750{\mathrm{m}}^3$

EXERCISE 6.1

1. (i) Similar
   (ii) Similar
   (iii) Equilateral
   (iv) Equal, Proportional
2. No

1. (i) $2\mathrm{c}\mathrm{m}$$2\mathrm{c}\mathrm{m}$2cm2 \mathrm{~cm}$2\mathrm{c}\mathrm{m}$

2. (i) No

9. Through $\mathrm{O}$$\mathrm{O}$O\mathrm{O}$\mathrm{O}$, draw a line parallel to $\mathrm{D}\mathrm{C}$$\mathrm{D}\mathrm{C}$DC\mathrm{DC}$\mathrm{D}\mathrm{C}$, intersecting $\mathrm{A}\mathrm{D}$$\mathrm{A}\mathrm{D}$AD\mathrm{AD}$\mathrm{A}\mathrm{D}$ and $\mathrm{B}\mathrm{C}$$\mathrm{B}\mathrm{C}$BC\mathrm{BC}$\mathrm{B}\mathrm{C}$ at $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$ and $\mathrm{F}$$\mathrm{F}$F\mathrm{F}$\mathrm{F}$ respectively.

EXERCISE 6.3

1. (i) Yes. $\mathrm{A}\mathrm{A}\mathrm{A},\mathrm{△}\mathrm{A}\mathrm{B}\mathrm{C}\sim \mathrm{\Delta }\mathrm{P}\mathrm{Q}\mathrm{R}$$\mathrm{A}\mathrm{A}\mathrm{A},\mathrm{△}\mathrm{A}\mathrm{B}\mathrm{C}\sim \mathrm{\Delta }\mathrm{P}\mathrm{Q}\mathrm{R}$AAA,/_\ABC∼DeltaPQR\mathrm{AAA}, \triangle \mathrm{ABC} \sim \Delta \mathrm{PQR}$\mathrm{A}\mathrm{A}\mathrm{A},\mathrm{△}\mathrm{A}\mathrm{B}\mathrm{C}\sim \mathrm{\Delta }\mathrm{P}\mathrm{Q}\mathrm{R}$

14. Produce $\mathrm{A}\mathrm{D}$$\mathrm{A}\mathrm{D}$AD\mathrm{AD}$\mathrm{A}\mathrm{D}$ to a point $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$ such that $\mathrm{A}\mathrm{D}=\mathrm{D}\mathrm{E}$$\mathrm{A}\mathrm{D}=\mathrm{D}\mathrm{E}$AD=DE\mathrm{AD}=\mathrm{DE}$\mathrm{A}\mathrm{D}=\mathrm{D}\mathrm{E}$ and produce $\mathrm{P}\mathrm{M}$$\mathrm{P}\mathrm{M}$PM\mathrm{PM}$\mathrm{P}\mathrm{M}$ to a point $\mathrm{N}$$\mathrm{N}$N\mathrm{N}$\mathrm{N}$ such that $\mathrm{P}\mathrm{M}=\mathrm{M}\mathrm{N}$$\mathrm{P}\mathrm{M}=\mathrm{M}\mathrm{N}$PM=MN\mathrm{PM}=\mathrm{MN}$\mathrm{P}\mathrm{M}=\mathrm{M}\mathrm{N}$. Join $\mathrm{E}\mathrm{C}$$\mathrm{E}\mathrm{C}$EC\mathrm{EC}$\mathrm{E}\mathrm{C}$ and NR.
15. $42\mathrm{m}$$42\mathrm{m}$42m42 \mathrm{~m}$42\mathrm{m}$

EXERCISE 7.1

1. (i) $2\sqrt{2}$$2\sqrt{2}$2sqrt22 \sqrt{2}$2\sqrt{2}$
   (ii) $4\sqrt{2}$$4\sqrt{2}$4sqrt24 \sqrt{2}$4\sqrt{2}$
   (iii) $2\sqrt{a^2+b^2}$$2\sqrt{a^2+b^2}$2sqrt(a^(2)+b^(2))2 \sqrt{a^{2}+b^{2}}$2\sqrt{a^2+b^2}$
2. $39;39\mathrm{k}\mathrm{m}$$39;39\mathrm{k}\mathrm{m}$39;39km39 ; 39 \mathrm{~km}$39;39\mathrm{k}\mathrm{m}$
3. No
4. Yes
5. Champa is correct.
6. (i) Square

10. $3x+y-5=0$$3x+y-5=0$3x+y-5=03 x+y-5=0$3x+y-5=0$

EXERCISE 7.2

1. $(1,3)$$(1,3)$(1,3)(1,3)$(1,3)$
2. $(2,-\frac{5}{3});(0,-\frac{7}{3})$$\left(2,-\frac{5}{3}\right);\left(0,-\frac{7}{3}\right)$(2,-(5)/(3));(0,-(7)/(3))\left(2,-\frac{5}{3}\right) ;\left(0,-\frac{7}{3}\right)$(2,-\frac{5}{3});(0,-\frac{7}{3})$
3. $\sqrt{61}\mathrm{m};5$$\sqrt{61}\mathrm{m};5$sqrt61m;5\sqrt{61} \mathrm{~m} ; 5$\sqrt{61}\mathrm{m};5$ th line at a distance of $22.5\mathrm{m}$$22.5\mathrm{m}$22.5m22.5 \mathrm{~m}$22.5\mathrm{m}$
4. $2:7$$2:7$2:72: 7$2:7$
5. $1:1;(-\frac{3}{2},0)$$1:1;\left(-\frac{3}{2},0\right)$1:1;(-(3)/(2),0)1: 1 ;\left(-\frac{3}{2}, 0\right)$1:1;(-\frac{3}{2},0)$
6. $x=6,y=3$$x=6,y=3$x=6,y=3x=6, y=3$x=6,y=3$
7. $(3,-10)$$(3,-10)$(3,-10)(3,-10)$(3,-10)$
8. $(-\frac{2}{7},-\frac{20}{7})$$\left(-\frac{2}{7},-\frac{20}{7}\right)$(-(2)/(7),-(20)/(7))\left(-\frac{2}{7},-\frac{20}{7}\right)$(-\frac{2}{7},-\frac{20}{7})$
9. $(-1,\frac{7}{2}),(0,5),(1,\frac{13}{2})$$\left(-1,\frac{7}{2}\right),(0,5),\left(1,\frac{13}{2}\right)$(-1,(7)/(2)),(0,5),(1,(13)/(2))\left(-1, \frac{7}{2}\right),(0,5),\left(1, \frac{13}{2}\right)$(-1,\frac{7}{2}),(0,5),(1,\frac{13}{2})$
10. 24 sq. units

EXERCISE 8.1

1. (i) $\sin \mathrm{A}=\frac{7}{25},\cos \mathrm{A}=\frac{24}{25}$$\sin \mathrm{A}=\frac{7}{25},\cos \mathrm{A}=\frac{24}{25}$sin A=(7)/(25),cos A=(24)/(25)\sin \mathrm{A}=\frac{7}{25}, \cos \mathrm{A}=\frac{24}{25}$\sin \mathrm{A}=\frac{7}{25},\cos \mathrm{A}=\frac{24}{25}$
   (ii) $\sin \mathrm{C}=\frac{24}{25},\cos \mathrm{C}=\frac{7}{25}$$\sin \mathrm{C}=\frac{24}{25},\cos \mathrm{C}=\frac{7}{25}$sin C=(24)/(25),cos C=(7)/(25)\sin \mathrm{C}=\frac{24}{25}, \cos \mathrm{C}=\frac{7}{25}$\sin \mathrm{C}=\frac{24}{25},\cos \mathrm{C}=\frac{7}{25}$
2. 0
3. $\cos \mathrm{A}=\frac{\sqrt{7}}{4},\tan \mathrm{A}=\frac{3}{\sqrt{7}}$$\cos \mathrm{A}=\frac{\sqrt{7}}{4},\tan \mathrm{A}=\frac{3}{\sqrt{7}}$cos A=(sqrt7)/(4),tan A=(3)/(sqrt7)\cos \mathrm{A}=\frac{\sqrt{7}}{4}, \tan \mathrm{A}=\frac{3}{\sqrt{7}}$\cos \mathrm{A}=\frac{\sqrt{7}}{4},\tan \mathrm{A}=\frac{3}{\sqrt{7}}$
4. $\sin \mathrm{A}=\frac{15}{17},\sec \mathrm{A}=\frac{17}{8}$$\sin \mathrm{A}=\frac{15}{17},\sec \mathrm{A}=\frac{17}{8}$sin A=(15)/(17),sec A=(17)/(8)\sin \mathrm{A}=\frac{15}{17}, \sec \mathrm{A}=\frac{17}{8}$\sin \mathrm{A}=\frac{15}{17},\sec \mathrm{A}=\frac{17}{8}$
5. $\sin \theta =\frac{5}{13},\cos \theta =\frac{12}{13},\tan \theta =\frac{5}{12},\cot \theta =\frac{12}{5},\mathrm{cosec}\theta =\frac{13}{5}$$\sin \theta =\frac{5}{13},\cos \theta =\frac{12}{13},\tan \theta =\frac{5}{12},\cot \theta =\frac{12}{5},\mathrm{cosec}\theta =\frac{13}{5}$sin theta=(5)/(13),cos theta=(12)/(13),tan theta=(5)/(12),cot theta=(12)/(5),cosec theta=(13)/(5)\sin \theta=\frac{5}{13}, \cos \theta=\frac{12}{13}, \tan \theta=\frac{5}{12}, \cot \theta=\frac{12}{5}, \operatorname{cosec} \theta=\frac{13}{5}$\sin \theta =\frac{5}{13},\cos \theta =\frac{12}{13},\tan \theta =\frac{5}{12},\cot \theta =\frac{12}{5},\mathrm{cosec}\theta =\frac{13}{5}$
6. (i) $\frac{49}{64}$$\frac{49}{64}$(49)/(64)\frac{49}{64}$\frac{49}{64}$

8. Yes
9. (i) 1

11. (i) False

EXERCISE 8.2

1. (i) 1
   (ii) 2
   (iii) $\frac{3\sqrt{2}-\sqrt{6}}{8}$$\frac{3\sqrt{2}-\sqrt{6}}{8}$(3sqrt2-sqrt6)/(8)\frac{3 \sqrt{2}-\sqrt{6}}{8}$\frac{3\sqrt{2}-\sqrt{6}}{8}$
   (iv) $\frac{43-24\sqrt{3}}{11}$$\frac{43-24\sqrt{3}}{11}$(43-24sqrt3)/(11)\frac{43-24 \sqrt{3}}{11}$\frac{43-24\sqrt{3}}{11}$
   (v) $\frac{67}{12}$$\frac{67}{12}$(67)/(12)\frac{67}{12}$\frac{67}{12}$
2. (i) $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$
   (ii) $\mathrm{D}$$\mathrm{D}$D\mathrm{D}$\mathrm{D}$
   (iii) $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$
   (iv) $\mathrm{C}$$\mathrm{C}$C\mathrm{C}$\mathrm{C}$
3. $\mathrm{\angle }\mathrm{A}={45}^{\circ },\mathrm{\angle }\mathrm{B}={15}^{\circ }$$\mathrm{\angle }\mathrm{A}={45}^{\circ },\mathrm{\angle }\mathrm{B}={15}^{\circ }$/_A=45^(@),/_B=15^(@)\angle \mathrm{A}=45^{\circ}, \angle \mathrm{B}=15^{\circ}$\mathrm{\angle }\mathrm{A}={45}^{\circ },\mathrm{\angle }\mathrm{B}={15}^{\circ }$
4. (i) False
   (ii) True
   (iii) False
   (iv) False
   (v) True

1. $\sin \mathrm{A}=\frac{1}{\sqrt{1+{\cot }^2\mathrm{A}}},\tan \mathrm{A}=\frac{1}{\cot \mathrm{A}},\sec \mathrm{A}=\frac{\sqrt{1+{\cot }^2\mathrm{A}}}{\cot \mathrm{A}}$$\sin \mathrm{A}=\frac{1}{\sqrt{1+{\cot }^2\mathrm{A}}},\tan \mathrm{A}=\frac{1}{\cot \mathrm{A}},\sec \mathrm{A}=\frac{\sqrt{1+{\cot }^2\mathrm{A}}}{\cot \mathrm{A}}$sin A=(1)/(sqrt(1+cot^(2)A)),tan A=(1)/(cot A),sec A=(sqrt(1+cot^(2)A))/(cot A)\sin \mathrm{A}=\frac{1}{\sqrt{1+\cot ^{2} \mathrm{~A}}}, \tan \mathrm{A}=\frac{1}{\cot \mathrm{A}}, \sec \mathrm{A}=\frac{\sqrt{1+\cot ^{2} \mathrm{~A}}}{\cot \mathrm{A}}$\sin \mathrm{A}=\frac{1}{\sqrt{1+{\cot }^2\mathrm{A}}},\tan \mathrm{A}=\frac{1}{\cot \mathrm{A}},\sec \mathrm{A}=\frac{\sqrt{1+{\cot }^2\mathrm{A}}}{\cot \mathrm{A}}$
2. $\sin \mathrm{A}=\frac{\sqrt{{\sec }^2\mathrm{A}-1}}{\sec \mathrm{A}},\cos \mathrm{A}=\frac{1}{\sec \mathrm{A}},\tan \mathrm{A}=\sqrt{{\sec }^2\mathrm{A}-1}$$\sin \mathrm{A}=\frac{\sqrt{{\sec }^2\mathrm{A}-1}}{\sec \mathrm{A}},\cos \mathrm{A}=\frac{1}{\sec \mathrm{A}},\tan \mathrm{A}=\sqrt{{\sec }^2\mathrm{A}-1}$sin A=(sqrt(sec^(2)A-1))/(sec A),cos A=(1)/(sec A),tan A=sqrt(sec^(2)A-1)\sin \mathrm{A}=\frac{\sqrt{\sec ^{2} \mathrm{~A}-1}}{\sec \mathrm{A}}, \cos \mathrm{A}=\frac{1}{\sec \mathrm{A}}, \tan \mathrm{A}=\sqrt{\sec ^{2} \mathrm{~A}-1}$\sin \mathrm{A}=\frac{\sqrt{{\sec }^2\mathrm{A}-1}}{\sec \mathrm{A}},\cos \mathrm{A}=\frac{1}{\sec \mathrm{A}},\tan \mathrm{A}=\sqrt{{\sec }^2\mathrm{A}-1}$

3. (i) $\mathrm{B}$$\mathrm{B}$B\mathrm{B}$\mathrm{B}$
   (ii) $\mathrm{C}$$\mathrm{C}$C\mathrm{C}$\mathrm{C}$
   (iii) $\mathrm{D}$$\mathrm{D}$D\mathrm{D}$\mathrm{D}$
   (iv) $\mathrm{D}$$\mathrm{D}$D\mathrm{D}$\mathrm{D}$

EXERCISE 9.1

1. $10\mathrm{m}$$10\mathrm{m}$10m10 \mathrm{~m}$10\mathrm{m}$
2. $8\sqrt{3}\mathrm{m}$$8\sqrt{3}\mathrm{m}$8sqrt3m8 \sqrt{3} \mathrm{~m}$8\sqrt{3}\mathrm{m}$
3. $3\mathrm{m},2\sqrt{3}\mathrm{m}$$3\mathrm{m},2\sqrt{3}\mathrm{m}$3m,2sqrt3m3 \mathrm{~m}, 2 \sqrt{3} \mathrm{~m}$3\mathrm{m},2\sqrt{3}\mathrm{m}$
4. $10\sqrt{3}\mathrm{m}$$10\sqrt{3}\mathrm{m}$10sqrt3m10 \sqrt{3} \mathrm{~m}$10\sqrt{3}\mathrm{m}$
5. $40\sqrt{3}\mathrm{m}$$40\sqrt{3}\mathrm{m}$40sqrt3m40 \sqrt{3} \mathrm{~m}$40\sqrt{3}\mathrm{m}$
6. $19\sqrt{3}\mathrm{m}$$19\sqrt{3}\mathrm{m}$19sqrt3m19 \sqrt{3} \mathrm{~m}$19\sqrt{3}\mathrm{m}$
7. $20(\sqrt{3}-1)\mathrm{m}$$20(\sqrt{3}-1)\mathrm{m}$20(sqrt3-1)m20(\sqrt{3}-1) \mathrm{m}$20(\sqrt{3}-1)\mathrm{m}$
8. $0.8(\sqrt{3}+1)\mathrm{m}$$0.8(\sqrt{3}+1)\mathrm{m}$0.8(sqrt3+1)m0.8(\sqrt{3}+1) \mathrm{m}$0.8(\sqrt{3}+1)\mathrm{m}$
9. $16\frac{2}{3}m$$16\frac{2}{3}m$16(2)/(3)m16 \frac{2}{3} m$16\frac{2}{3}m$
10. $20\sqrt{3}\mathrm{m},20\mathrm{m},60\mathrm{m}$$20\sqrt{3}\mathrm{m},20\mathrm{m},60\mathrm{m}$20sqrt3m,20m,60m20 \sqrt{3} \mathrm{~m}, 20 \mathrm{~m}, 60 \mathrm{~m}$20\sqrt{3}\mathrm{m},20\mathrm{m},60\mathrm{m}$
11. $10\sqrt{3}\mathrm{m},10\mathrm{m}$$10\sqrt{3}\mathrm{m},10\mathrm{m}$10sqrt3m,10m10 \sqrt{3} \mathrm{~m}, 10 \mathrm{~m}$10\sqrt{3}\mathrm{m},10\mathrm{m}$
12. $7(\sqrt{3}+1)\mathrm{m}$$7(\sqrt{3}+1)\mathrm{m}$7(sqrt3+1)m7(\sqrt{3}+1) \mathrm{m}$7(\sqrt{3}+1)\mathrm{m}$
13. $75(\sqrt{3}-1)\mathrm{m}$$75(\sqrt{3}-1)\mathrm{m}$75(sqrt3-1)m75(\sqrt{3}-1) \mathrm{m}$75(\sqrt{3}-1)\mathrm{m}$
14. $58\sqrt{3}\mathrm{m}$$58\sqrt{3}\mathrm{m}$58sqrt3m58 \sqrt{3} \mathrm{~m}$58\sqrt{3}\mathrm{m}$
15. 3 seconds

EXERCISE 10.1

1. Infinitely many
2. (i) One
   (ii) Secant
   (iii) Two
   (iv) Point of contact
   EXERCISE 10.2
3. A
4. $\mathrm{B}$$\mathrm{B}$B\mathrm{B}$\mathrm{B}$
5. $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$
6. $3\mathrm{c}\mathrm{m}$$3\mathrm{c}\mathrm{m}$3cm3 \mathrm{~cm}$3\mathrm{c}\mathrm{m}$
7. $8\mathrm{c}\mathrm{m}$$8\mathrm{c}\mathrm{m}$8cm8 \mathrm{~cm}$8\mathrm{c}\mathrm{m}$
8. $\mathrm{A}\mathrm{B}=15\mathrm{c}\mathrm{m},\mathrm{A}\mathrm{C}=13\mathrm{c}\mathrm{m}$$\mathrm{A}\mathrm{B}=15\mathrm{c}\mathrm{m},\mathrm{A}\mathrm{C}=13\mathrm{c}\mathrm{m}$AB=15cm,AC=13cm\mathrm{AB}=15 \mathrm{~cm}, \mathrm{AC}=13 \mathrm{~cm}$\mathrm{A}\mathrm{B}=15\mathrm{c}\mathrm{m},\mathrm{A}\mathrm{C}=13\mathrm{c}\mathrm{m}$
9. $\mathrm{D}$$\mathrm{D}$D\mathrm{D}$\mathrm{D}$

EXERCISE 11.1

1. $\frac{132}{7}{\mathrm{c}\mathrm{m}}^2$$\frac{132}{7}{\mathrm{c}\mathrm{m}}^2$(132)/(7)cm^(2)\frac{132}{7} \mathrm{~cm}^{2}$\frac{132}{7}{\mathrm{c}\mathrm{m}}^2$
2. $\frac{77}{8}{\mathrm{c}\mathrm{m}}^2$$\frac{77}{8}{\mathrm{c}\mathrm{m}}^2$(77)/(8)cm^(2)\frac{77}{8} \mathrm{~cm}^{2}$\frac{77}{8}{\mathrm{c}\mathrm{m}}^2$
3. $\frac{154}{3}{\mathrm{c}\mathrm{m}}^2$$\frac{154}{3}{\mathrm{c}\mathrm{m}}^2$(154)/(3)cm^(2)\frac{154}{3} \mathrm{~cm}^{2}$\frac{154}{3}{\mathrm{c}\mathrm{m}}^2$
4. (i) $28.5{\mathrm{c}\mathrm{m}}^2$$28.5{\mathrm{c}\mathrm{m}}^2$28.5cm^(2)28.5 \mathrm{~cm}^{2}$28.5{\mathrm{c}\mathrm{m}}^2$
   (ii) $235.5{\mathrm{c}\mathrm{m}}^2$$235.5{\mathrm{c}\mathrm{m}}^2$235.5cm^(2)235.5 \mathrm{~cm}^{2}$235.5{\mathrm{c}\mathrm{m}}^2$
5. (i) $22\mathrm{c}\mathrm{m}$$22\mathrm{c}\mathrm{m}$22cm22 \mathrm{~cm}$22\mathrm{c}\mathrm{m}$
   (ii) $231{\mathrm{c}\mathrm{m}}^2$$231{\mathrm{c}\mathrm{m}}^2$231cm^(2)231 \mathrm{~cm}^{2}$231{\mathrm{c}\mathrm{m}}^2$
   (iii) $(231-\frac{441\sqrt{3}}{4}){\mathrm{c}\mathrm{m}}^2$$\left(231-\frac{441\sqrt{3}}{4}\right){\mathrm{c}\mathrm{m}}^2$(231-(441sqrt3)/(4))cm^(2)\left(231-\frac{441 \sqrt{3}}{4}\right) \mathrm{cm}^{2}$(231-\frac{441\sqrt{3}}{4}){\mathrm{c}\mathrm{m}}^2$
6. $20.4375{\mathrm{c}\mathrm{m}}^2;686.0625{\mathrm{c}\mathrm{m}}^2$$20.4375{\mathrm{c}\mathrm{m}}^2;686.0625{\mathrm{c}\mathrm{m}}^2$20.4375cm^(2);686.0625cm^(2)20.4375 \mathrm{~cm}^{2} ; 686.0625 \mathrm{~cm}^{2}$20.4375{\mathrm{c}\mathrm{m}}^2;686.0625{\mathrm{c}\mathrm{m}}^2$
7. $88.44{\mathrm{c}\mathrm{m}}^2$$88.44{\mathrm{c}\mathrm{m}}^2$88.44cm^(2)88.44 \mathrm{~cm}^{2}$88.44{\mathrm{c}\mathrm{m}}^2$
8. (i) $19.625{\mathrm{m}}^2$$19.625{\mathrm{m}}^2$19.625m^(2)19.625 \mathrm{~m}^{2}$19.625{\mathrm{m}}^2$
   (ii) $58.875{\mathrm{c}\mathrm{m}}^2$$58.875{\mathrm{c}\mathrm{m}}^2$58.875cm^(2)58.875 \mathrm{~cm}^{2}$58.875{\mathrm{c}\mathrm{m}}^2$
9. (i) $285\mathrm{m}\mathrm{m}$$285\mathrm{m}\mathrm{m}$285mm285 \mathrm{~mm}$285\mathrm{m}\mathrm{m}$
   (ii) $\frac{385}{4}{\mathrm{m}\mathrm{m}}^2$$\frac{385}{4}{\mathrm{m}\mathrm{m}}^2$(385)/(4)mm^(2)\frac{385}{4} \mathrm{~mm}^{2}$\frac{385}{4}{\mathrm{m}\mathrm{m}}^2$
10. $\frac{22275}{28}{\mathrm{c}\mathrm{m}}^2$$\frac{22275}{28}{\mathrm{c}\mathrm{m}}^2$(22275)/(28)cm^(2)\frac{22275}{28} \mathrm{~cm}^{2}$\frac{22275}{28}{\mathrm{c}\mathrm{m}}^2$
11. $\frac{158125}{126}{\mathrm{c}\mathrm{m}}^2$$\frac{158125}{126}{\mathrm{c}\mathrm{m}}^2$(158125)/(126)cm^(2)\frac{158125}{126} \mathrm{~cm}^{2}$\frac{158125}{126}{\mathrm{c}\mathrm{m}}^2$
12. $189.97{\mathrm{k}\mathrm{m}}^2$$189.97{\mathrm{k}\mathrm{m}}^2$189.97km^(2)189.97 \mathrm{~km}^{2}$189.97{\mathrm{k}\mathrm{m}}^2$
13. ₹ 162.68
14. $\mathrm{D}$$\mathrm{D}$D\mathrm{D}$\mathrm{D}$

EXERCISE 12.1

1. $160{\mathrm{c}\mathrm{m}}^2$$160{\mathrm{c}\mathrm{m}}^2$160cm^(2)160 \mathrm{~cm}^{2}$160{\mathrm{c}\mathrm{m}}^2$
2. $572{\mathrm{c}\mathrm{m}}^2$$572{\mathrm{c}\mathrm{m}}^2$572cm^(2)572 \mathrm{~cm}^{2}$572{\mathrm{c}\mathrm{m}}^2$
3. $214.5{\mathrm{c}\mathrm{m}}^2$$214.5{\mathrm{c}\mathrm{m}}^2$214.5cm^(2)214.5 \mathrm{~cm}^{2}$214.5{\mathrm{c}\mathrm{m}}^2$
4. Greatest diameter $=7\mathrm{c}\mathrm{m}$$=7\mathrm{c}\mathrm{m}$=7cm=7 \mathrm{~cm}$=7\mathrm{c}\mathrm{m}$,surface area $=332.5{\mathrm{c}\mathrm{m}}^2$$=332.5{\mathrm{c}\mathrm{m}}^2$=332.5cm^(2)=332.5 \mathrm{~cm}^{2}$=332.5{\mathrm{c}\mathrm{m}}^2$
5. $\frac{1}{4}{l}^2(\pi +24)$$\frac{1}{4}{l}^2(\pi +24)$(1)/(4)l^(2)(pi+24)\frac{1}{4} l^{2}(\pi+24)$\frac{1}{4}{l}^2(\pi +24)$
6. $220{\mathrm{m}\mathrm{m}}^2$$220{\mathrm{m}\mathrm{m}}^2$220mm^(2)220 \mathrm{~mm}^{2}$220{\mathrm{m}\mathrm{m}}^2$
7. $44{\mathrm{m}}^2$$44{\mathrm{m}}^2$44m^(2)44 \mathrm{~m}^{2}$44{\mathrm{m}}^2$, ₹ 22000
8. $18{\mathrm{c}\mathrm{m}}^2$$18{\mathrm{c}\mathrm{m}}^2$18cm^(2)18 \mathrm{~cm}^{2}$18{\mathrm{c}\mathrm{m}}^2$
9. $374{\mathrm{c}\mathrm{m}}^2$$374{\mathrm{c}\mathrm{m}}^2$374cm^(2)374 \mathrm{~cm}^{2}$374{\mathrm{c}\mathrm{m}}^2$

EXERCISE 12.2

1. $\pi {\mathrm{c}\mathrm{m}}^3$$\pi {\mathrm{c}\mathrm{m}}^3$picm^(3)\pi \mathrm{cm}^{3}$\pi {\mathrm{c}\mathrm{m}}^3$
2. $66{\mathrm{c}\mathrm{m}}^3$$66{\mathrm{c}\mathrm{m}}^3$66cm^(3)66 \mathrm{~cm}^{3}$66{\mathrm{c}\mathrm{m}}^3$. Volume of the air inside the model $=$$=$==$=$ Volume of air inside $($$($(($($ cone + cylinder + cone $)$$)$))$)$ $=(\frac{1}{3}\pi r^2{h}_1+\pi r^2{h}_2+\frac{1}{3}\pi r^2{h}_1)$$=\left(\frac{1}{3}\pi r^2{h}_1+\pi r^2{h}_2+\frac{1}{3}\pi r^2{h}_1\right)$=((1)/(3)pir^(2)h_(1)+pir^(2)h_(2)+(1)/(3)pir^(2)h_(1))=\left(\frac{1}{3} \pi r^{2} h_{1}+\pi r^{2} h_{2}+\frac{1}{3} \pi r^{2} h_{1}\right)$=(\frac{1}{3}\pi r^2{h}_1+\pi r^2{h}_2+\frac{1}{3}\pi r^2{h}_1)$, where $r$$r$rr$r$ is the radius of the cone and the cylinder, $h_1$$h_1$h_(1)h_{1}$h_1$ is the height (length) of the cone and $h_2$$h_2$h_(2)h_{2}$h_2$ is the height (length) of the cylinder.

EXERCISE 13.1

1. 8.1 plants. We have used direct method because numerical values of $x_i$$x_i$x_(i)x_{i}$x_i$ and $f_i$$f_i$f_(i)f_{i}$f_i$ are small.
2. ₹ 545.20
3. $f=20$$f=20$f=20f=20$f=20$
4. 75.9
5. 57.19
6. ₹ 211
7. $0.099\mathrm{p}\mathrm{p}\mathrm{m}$$0.099\mathrm{p}\mathrm{p}\mathrm{m}$0.099ppm0.099 \mathrm{ppm}$0.099\mathrm{p}\mathrm{p}\mathrm{m}$
8. 12.48 days
9. $69.43\mathrm{\%}$$69.43\mathrm{\%}$69.43%69.43 \%$69.43\mathrm{\%}$

1. Mode $=36.8$$=36.8$=36.8=36.8$=36.8$ years, Mean $=35.37$$=35.37$=35.37=35.37$=35.37$ years. Maximum number of patients admitted in the hospital are of the age 36.8 years (approx.), while on an average the age of a patient admitted to the hospital is 35.37 years.
2. 65.625 hours
3. Modal monthly expenditure $=₹1847.83$$=₹1847.83$=₹1847.83=₹ 1847.83$=₹1847.83$, Mean monthly expenditure $=₹2662.5$$=₹2662.5$=₹2662.5=₹ 2662.5$=₹2662.5$.
4. Mode:30.6, Mean=29.2. Most states/U.T. have a student teacher ratio of 30.6 and on an average, this ratio is 29.2 .
5. Mode $=4608.7$$=4608.7$=4608.7=4608.7$=4608.7$ runs
6. Mode $=44.7$$=44.7$=44.7=44.7$=44.7$ cars

1. Median $=137$$=137$=137=137$=137$ units, Mean $=137.05$$=137.05$=137.05=137.05$=137.05$ units, Mode $=135.76$$=135.76$=135.76=135.76$=135.76$ units.

4. Median length $=146.75\mathrm{m}\mathrm{m}$$=146.75\mathrm{m}\mathrm{m}$=146.75mm=146.75 \mathrm{~mm}$=146.75\mathrm{m}\mathrm{m}$
5. Median life $=3406.98$$=3406.98$=3406.98=3406.98$=3406.98$ hours
6. Median $=8.05,$$=8.05,$=8.05,quad=8.05, \quad$=8.05,$ Mean $=8.32,$$=8.32,$=8.32,quad=8.32, \quad$=8.32,$ Modal size $=7.88$$=7.88$=7.88=7.88$=7.88$
7. Median weight $=56.67\mathrm{k}\mathrm{g}$$=56.67\mathrm{k}\mathrm{g}$=56.67kg=56.67 \mathrm{~kg}$=56.67\mathrm{k}\mathrm{g}$

EXERCISE 14.1

1. (i) 1

2. The experiments (iii) and (iv) have equally likely outcomes.
3. When we toss a coin, the outcomes head and tail are equally likely. So, the result of an individual coin toss is completely unpredictable.
4. $\mathrm{B}$$\mathrm{B}$B\mathrm{B}$\mathrm{B}$
5. 0.95
6. (i) 0
   (ii) 1
7. 0.008
8. (i) $\frac{3}{8}$$\frac{3}{8}$(3)/(8)\frac{3}{8}$\frac{3}{8}$

9. (i) $\frac{5}{17}$$\frac{5}{17}$(5)/(17)\frac{5}{17}$\frac{5}{17}$

10. (i) $\frac{5}{9}$$\frac{5}{9}$(5)/(9)\frac{5}{9}$\frac{5}{9}$

13. (i) $\frac{1}{2}$$\frac{1}{2}$(1)/(2)\frac{1}{2}$\frac{1}{2}$

14. (i) $\frac{1}{26}$$\frac{1}{26}$(1)/(26)\frac{1}{26}$\frac{1}{26}$

15. (i) $\frac{1}{5}$$\frac{1}{5}$(1)/(5)\frac{1}{5}$\frac{1}{5}$

17. (i) $\frac{1}{5}$$\frac{1}{5}$(1)/(5)\frac{1}{5}$\frac{1}{5}$

18. (i) $\frac{9}{10}$$\frac{9}{10}$(9)/(10)\frac{9}{10}$\frac{9}{10}$

19. (i) $\frac{1}{3}$$\frac{1}{3}$(1)/(3)\frac{1}{3}$\frac{1}{3}$

21. (i) $\frac{31}{36}$$\frac{31}{36}$(31)/(36)\frac{31}{36}$\frac{31}{36}$

22. (i)

23. $\frac{3}{4}$$\frac{3}{4}$(3)/(4)\frac{3}{4}$\frac{3}{4}$; Possible outcomes are : HHH,TTT, HHT, HTH, HTT, THH, THT, TTH. Here, THH means tail in the first toss, head on the second toss and head on the third toss and so on.
24. (i) $\frac{25}{36}$$\frac{25}{36}$(25)/(36)quad\frac{25}{36} \quad$\frac{25}{36}$ (ii) $\frac{11}{36}$$\frac{11}{36}$(11)/(36)\frac{11}{36}$\frac{11}{36}$
25. (i) Incorrect. We can classify the outcomes like this but they are not then 'equally likely'. Reason is that 'one of each' can result in two ways - from a head on first coin and tail on the second coin or from a tail on the first coin and head on the second coin. This makes it twicely as likely as two heads (or two tails).

EXERCISE A1.1

1. (i) Ambiguous
   (ii) True
   (iii) True
   (iv) Ambiguous
   (v) Ambiguous
2. (i) True

3. Only (ii) is true.
4. (i) If $a>0$$a>0$a > 0a>0$a>0$ and $a^2>b^2$$a^2>b^2$a^(2) > b^(2)a^{2}>b^{2}$a^2>b^2$, then $a>b$$a>b$a > ba>b$a>b$.

EXERCISE A1.2

1. A is mortal
2. $ab$$ab$aba b$ab$ is rational
3. Decimal expansion of $\sqrt{17}$$\sqrt{17}$sqrt17\sqrt{17}$\sqrt{17}$ is non-terminating non-recurring.
4. $y=7$$y=7$y=7y=7$y=7$
5. $\mathrm{\angle }\mathrm{A}={100}^{\circ },\mathrm{\angle }\mathrm{C}={100}^{\circ },\mathrm{\angle }\mathrm{D}={180}^{\circ }$$\mathrm{\angle }\mathrm{A}={100}^{\circ },\mathrm{\angle }\mathrm{C}={100}^{\circ },\mathrm{\angle }\mathrm{D}={180}^{\circ }$/_A=100^(@),/_C=100^(@),/_D=180^(@)\angle \mathrm{A}=100^{\circ}, \angle \mathrm{C}=100^{\circ}, \angle \mathrm{D}=180^{\circ}$\mathrm{\angle }\mathrm{A}={100}^{\circ },\mathrm{\angle }\mathrm{C}={100}^{\circ },\mathrm{\angle }\mathrm{D}={180}^{\circ }$
6. $\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$$\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$PQRS\mathrm{PQRS}$\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$ is a rectangle.
7. Yes, because of the premise. No, because $\sqrt{3721}=61$$\sqrt{3721}=61$sqrt3721=61\sqrt{3721}=61$\sqrt{3721}=61$ which is not irrational. Since the premise was wrong, the conclusion is false.

EXERCISE A1.3

1. Take two consecutive odd numbers as $2n+1$$2n+1$2n+12 n+1$2n+1$ and $2n+3$$2n+3$2n+32 n+3$2n+3$ for some integer $n$$n$nn$n$.

EXERCISE A1.4

1. (i) Man is not mortal.

EXERCISE A1.5

1. (i) If Sharan sweats a lot, then it is hot in Tokyo.

2. (i) If the base angles of triangle $\mathrm{A}\mathrm{B}\mathrm{C}$$\mathrm{A}\mathrm{B}\mathrm{C}$ABC\mathrm{ABC}$\mathrm{A}\mathrm{B}\mathrm{C}$ are equal, then it is isosceles. True.

EXERCISE A1.6

1. Suppose to the contrary $b\le d$$b\le d$b <= db \leq d$b\le d$.
2. See Example 10 of Chapter 1.
3. See Theorem 5.1 of Class IX Mathematics Textbook.

EXERCISE A2.2

1. (i) $\frac{1}{5}$$\frac{1}{5}$(1)/(5)quad\frac{1}{5} \quad$\frac{1}{5}$ (ii) 160
2. Take $1{\mathrm{c}\mathrm{m}}^2$$1{\mathrm{c}\mathrm{m}}^2$1cm^(2)1 \mathrm{~cm}^{2}$1{\mathrm{c}\mathrm{m}}^2$ area and count the number of dots in it. Total number of trees will be the product of this number and the area (in ${\mathrm{c}\mathrm{m}}^2$${\mathrm{c}\mathrm{m}}^2$cm^(2)\mathrm{cm}^{2}${\mathrm{c}\mathrm{m}}^2$ ).
3. Rate of interest in instalment scheme is $17.74\mathrm{\%}$$17.74\mathrm{\%}$17.74%17.74 \%$17.74\mathrm{\%}$, which is less than $18\mathrm{\%}$$18\mathrm{\%}$18%18 \%$18\mathrm{\%}$.

1. Students find their own answers.

Proofs in Mathematics

A1.1 Introduction

A1.2 Mathematical Statements Revisited

• always true
• always false
• ambiguous

Solution :

Solution :

Solution :

Solution :

EXERCISE A1.1

1. State whether the following statements are always true, always false or ambiguous. Justify your answers.

2. State whether the following statements are true or false. Justify your answers.
   (i) All hexagons are polygons.
   (ii) Some polygons are pentagons.
   (iii) Not all even numbers are divisible by 2.
   (iv) Some real numbers are irrational.
   (v) Not all real numbers are rational.
3. Let $a$$a$aa$a$ and $b$$b$bb$b$ be real numbers such that $ab\ne 0$$ab\ne 0$ab!=0a b \neq 0$ab\ne 0$. Then which of the following statements are true? Justify your answers.
   (i) Both $a$$a$aa$a$ and $b$$b$bb$b$ must be zero.
   (ii) Both $a$$a$aa$a$ and $b$$b$bb$b$ must be non-zero.
   (iii) Either $a$$a$aa$a$ or $b$$b$bb$b$ must be non-zero.
4. Restate the following statements with appropriate conditions, so that they become true.
   (i) If $a^2>b^2$$a^2>b^2$a^(2) > b^(2)a^{2}>b^{2}$a^2>b^2$, then $a>b$$a>b$a > ba>b$a>b$.
   (ii) If $x^2=y^2$$x^2=y^2$x^(2)=y^(2)x^{2}=y^{2}$x^2=y^2$, then $x=y$$x=y$x=yx=y$x=y$.
   (iii) If $(x+y{)}^2=x^2+y^2$$(x+y{)}^2=x^2+y^2$(x+y)^(2)=x^(2)+y^(2)(x+y)^{2}=x^{2}+y^{2}$(x+y{)}^2=x^2+y^2$, then $x=0$$x=0$x=0x=0$x=0$.
   (iv) The diagonals of a quadrilateral bisect each other.

A1.3 Deductive Reasoning

EXERCISEA1.2

1. Given that all women are mortal, and suppose that $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$ is a woman, what can we conclude about A?
2. Given that the product of two rational numbers is rational, and suppose $a$$a$aa$a$ and $b$$b$bb$b$ are rationals, what can you conclude about $ab$$ab$aba b$ab$ ?
3. Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and $\sqrt{17}$$\sqrt{17}$sqrt17\sqrt{17}$\sqrt{17}$ is irrational, what can we conclude about the decimal expansion of $\sqrt{17}$$\sqrt{17}$sqrt17\sqrt{17}$\sqrt{17}$ ?
4. Given that $y=x^2+6$$y=x^2+6$y=x^(2)+6y=x^{2}+6$y=x^2+6$ and $x=-1$$x=-1$x=-1x=-1$x=-1$, what can we conclude about the value of $y$$y$yy$y$ ?
5. Given that $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ABCD\mathrm{ABCD}$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram and $\mathrm{\angle }\mathrm{B}={80}^{\circ }$$\mathrm{\angle }\mathrm{B}={80}^{\circ }$/_B=80^(@)\angle \mathrm{B}=80^{\circ}$\mathrm{\angle }\mathrm{B}={80}^{\circ }$. What can you conclude about the other angles of the parallelogram?
6. Given that $\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$$\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$PQRS\mathrm{PQRS}$\mathrm{P}\mathrm{Q}\mathrm{R}\mathrm{S}$ is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?
7. Given that $\sqrt{p}$$\sqrt{p}$sqrtp\sqrt{p}$\sqrt{p}$ is irrational for all primes $p$$p$pp$p$ and also suppose that 3721 is a prime. Can you conclude that $\sqrt{3721}$$\sqrt{3721}$sqrt3721\sqrt{3721}$\sqrt{3721}$ is an irrational number? Is your conclusion correct? Why or why not?

A1.4 Conjectures, Theorems, Proofs and Mathematical Reasoning

Solution :

Solution :

Theorem A1.1 (Converse of the

Pythagoras Theorem) : If in a triangle the square of the length of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Proof :

EXERCISE A1.3

1. Prove that the sum of two consecutive odd numbers is divisible by 4 .
2. Take two consecutive odd numbers. Find the sum of their squares, and then add 6 to the result. Prove that the new number is always divisible by 8 .
3. If $p\ge 5$$p\ge 5$p >= 5p \geq 5$p\ge 5$ is a prime number, show that $p^2+2$$p^2+2$p^(2)+2p^{2}+2$p^2+2$ is divisible by 3 .

4. Let $x$$x$xx$x$ and $y$$y$yy$y$ be rational numbers. Show that $xy$$xy$xyx y$xy$ is a rational number.
5. If $a$$a$aa$a$ and $b$$b$bb$b$ are positive integers, then you know that $a=bq+r,0\le r<b$$a=bq+r,0\le r<b$a=bq+r,0 <= r < ba=b q+r, 0 \leq r<b$a=bq+r,0\le r<b$, where $q$$q$qq$q$ is a whole number. Prove that $\mathrm{HCF}(a,b)=\mathrm{HCF}(b,r)$$\mathrm{HCF}(a,b)=\mathrm{HCF}(b,r)$HCF(a,b)=HCF(b,r)\operatorname{HCF}(a, b)=\operatorname{HCF}(b, r)$\mathrm{HCF}(a,b)=\mathrm{HCF}(b,r)$.

6. A line parallel to side $\mathrm{B}\mathrm{C}$$\mathrm{B}\mathrm{C}$BC\mathrm{BC}$\mathrm{B}\mathrm{C}$ of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}$$\mathrm{A}\mathrm{B}\mathrm{C}$ABC\mathrm{ABC}$\mathrm{A}\mathrm{B}\mathrm{C}$, intersects $\mathrm{A}\mathrm{B}$$\mathrm{A}\mathrm{B}$AB\mathrm{AB}$\mathrm{A}\mathrm{B}$ and $\mathrm{A}\mathrm{C}$$\mathrm{A}\mathrm{C}$AC\mathrm{AC}$\mathrm{A}\mathrm{C}$ at $\mathrm{D}$$\mathrm{D}$D\mathrm{D}$\mathrm{D}$ and $\mathrm{E}$$\mathrm{E}$E\mathrm{E}$\mathrm{E}$ respectively.

A1.5 Negation of a Statement

Solution :

EXERCISE A1.4

1. State the negations for the following statements :
   (i) Man is mortal.
   (ii) Line $l$$l$ll$l$ is parallel to line $m$$m$mm$m$.
   (iii) This chapter has many exercises.
   (iv) All integers are rational numbers.
   (v) Some prime numbers are odd.
   (vi) No student is lazy.
   (vii) Some cats are not black.
   (viii) There is no real number $x$$x$xx$x$, such that $\sqrt{x}=-1$$\sqrt{x}=-1$sqrtx=-1\sqrt{x}=-1$\sqrt{x}=-1$.

2. In each of the following questions, there are two statements. State if the second is the negation of the first or not.
   (i) Mumtaz is hungry.
   (ii) Some cats are black.
   Mumtaz is not hungry.
   Some cats are brown.
   (iii) All elephants are huge.
   (iv) All fire engines are red.
   One elephant is not huge.
   All fire engines are not red.

A1.6 Converse of a Statement

Solution :

EXERCISE A1.5

1. Write the converses of the following statements.

2. Write the converses of the following statements. Also, decide in each case whether the converse is true or false.

A1.7 Proof by Contradiction

• Let us assume that we want to establish the truth of a statement $p$$p$pp$p$ (here we want to show that $p:$$p:$p:p:$p:$ 'A is mortal' is true).
• So, we begin by assuming that the statement is not true, that is, we assume that the negation of $p$$p$pp$p$ is true (i.e., A is not mortal).
• We then proceed to carry out a series of logical deductions based on the truth of the negation of $p$$p$pp$p$. (Since $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$ is not mortal, we have a counter-example to the statement 'All women are mortal.' Hence, it is false that all women are mortal.)
• If this leads to a contradiction, then the contradiction arises because of our faulty assumption that $p$$p$pp$p$ is not true. (We have a contradiction, since we have shown that the statement 'All women are mortal' and its negation, 'Not all women are mortal' is true at the same time. This contradiction arose, because we assumed that A is not mortal.)
• Therefore, our assumption is wrong, i.e., $p$$p$pp$p$ has to be true. (So, $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$ is mortal.) Let us now look at examples from mathematics.

Solution :

EXERCISEA1.6

1. Suppose $a+b=c+d$$a+b=c+d$a+b=c+da+b=c+d$a+b=c+d$, and $a<c$$a<c$a < ca<c$a<c$. Use proof by contradiction to show $b>d$$b>d$b > db>d$b>d$.
2. Let $r$$r$rr$r$ be a rational number and $x$$x$xx$x$ be an irrational number. Use proof by contradiction to show that $r+x$$r+x$r+xr+x$r+x$ is an irrational number.
3. Use proof by contradiction to prove that if for an integer $a,a^2$$a,a^2$a,a^(2)a, a^{2}$a,a^2$ is even, then so is $a$$a$aa$a$. [Hint : Assume $a$$a$aa$a$ is not even, that is, it is of the form $2n+1$$2n+1$2n+12 n+1$2n+1$, for some integer $n$$n$nn$n$, and then proceed.]
4. Use proof by contradiction to prove that if for an integer $a,a^2$$a,a^2$a,a^(2)a, a^{2}$a,a^2$ is divisible by 3 , then $a$$a$aa$a$ is divisible by 3 .
5. Use proof by contradiction to show that there is no value of $n$$n$nn$n$ for which $6^n$$6^n$6^(n)6^{n}$6^n$ ends with the digit zero.
6. Prove by contradiction that two distinct lines in a plane cannot intersect in more than one point.

A1.8 Summary

1. Different ingredients of a proof and other related concepts learnt in Class IX.
2. The negation of a statement.
3. The converse of a statement.
4. Proof by contradiction.

MathematicalModeLling

A2.1 Introduction

• An adult human body contains approximately $1,50,000\mathrm{k}\mathrm{m}$$1,50,000\mathrm{k}\mathrm{m}$1,50,000km1,50,000 \mathrm{~km}$1,50,000\mathrm{k}\mathrm{m}$ of arteries and veins that carry blood.
• The human heart pumps 5 to 6 litres of blood in the body every 60 seconds.
• The temperature at the surface of the Sun is about $6,{000}^{\circ }\mathrm{C}$$6,{000}^{\circ }\mathrm{C}$6,000^(@)C6,000^{\circ} \mathrm{C}$6,{000}^{\circ }\mathrm{C}$.

A2.2 Stages in Mathematical Modelling

• define variables
• write equations or inequalities
• gather data and organise into tables
• make graphs
• calculate probabilities

EXERCISE A2.1

1. Consider the following situation.

A2.3 Some Illustrations

Solution :

Solution :

EXERCISE A2.2

1. An ornithologist wants to estimate the number of parrots in a large field. She uses a net to catch some, and catches 32 parrots, which she rings and sets free. The following week she manages to net 40 parrots, of which 8 are ringed.

2. Suppose the adjoining figure represents an aerial photograph of a forest with each dot representing a tree. Your purpose is to find the number of trees there are on this tract of land as part of an environmental census.

3. A T.V. can be purchased for ₹ 24000 cash or for ₹ 8000 cashdown payment and six monthly instalments of ₹ 2800 each. Ali goes to market to buy a T.V., and he has ₹ 8000 with him. He has now two options. One is to buy TV under instalment scheme or to make cash payment by taking loan from some financial society. The society charges simple interest at the rate of $18\mathrm{\%}$$18\mathrm{\%}$18%18 \%$18\mathrm{\%}$ per annum simple interest. Which option is better for Ali?

A2.4 Why is Mathematical Modelling Important?

• To gain understanding. If we have a mathematical model which reflects the essential behaviour of a real-world system of interest, we can understand that system better through an analysis of the model. Furthermore, in the process of building the model we find out which factors are most important in the system, and how the different aspects of the system are related.
• To predict, or forecast, or simulate. Very often, we wish to know what a realworld system will do in the future, but it is expensive, impractical or impossible to experiment directly with the system. For example, in weather prediction, to study drug efficacy in humans, finding an optimum design of a nuclear reactor, and so on.

• To estimate. Often, we need to estimate large values. You've seen examples of the trees in a forest, fish in a lake, etc. For another example, before elections, the contesting parties want to predict the probability of their party winning the elections. In particular, they want to estimate how many people in their constituency would vote for their party. Based on their predictions, they may want to decide on the campaign strategy. Exit polls have been used widely to predict the number of seats, a party is expected to get in elections.

EXERCISE A2.3

1. Based upon the data of the past five years, try and forecast the average percentage of marks in Mathematics that your school would obtain in the Class $\mathrm{X}$$\mathrm{X}$X\mathrm{X}$\mathrm{X}$ board examination at the end of the year.

A2.5 Summary

1. A mathematical model is a mathematical description of a real-life situation. Mathematical modelling is the process of creating a mathematical model, solving it and using it to understand the real-life problem.
2. The various steps involved in modelling are : understanding the problem, formulating the mathematical model, solving it, interpreting it in the real-life situation, and, most importantly, validating the model.
3. Developed some mathematical models.
4. The importance of mathematical modelling.

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1. Note: The Article 51A containing Fundamental Duties was inserted by the Constitution (42nd Amendment) Act, 1976 (with effect from 3 January 1977).
   *(k) was inserted by the Constitution (86th Amendment) Act, 2002 (with effect from 1 April 2010).
2. • Not from the examination point of view.
3. • Plotting of graphs of quadratic or cubic polynomials is not meant to be done by the students, nor is to be evaluated.
4. • $\alpha ,\beta$$\alpha ,\beta$alpha,beta\alpha, \beta$\alpha ,\beta$ are Greek letters pronounced as 'alpha' and 'beta' respectively. We will use later one more letter ' $\gamma$$\gamma$gamma\gamma$\gamma$ ' pronounced as 'gamma'.
5. • Not from the examination point of view.
6. • These exercises are not from the examination point of view.
7. *The word 'tangent' comes from the Latin word 'tangere', which means to touch and was introduced by the Danish mathematician Thomas Fineke in 1583.
8. • Not from the examination point of view.
9. • Not from the examination point of view.
10. • Not from the examination point of view.