O Math 10 2075

= Value of 68 th
2
item

= Value of 34th item

Cumulative frequency just greater than 34 is 50 and the median lies in the corresponding class i.e.
30 – 40. The median of the given data is

N – cf 34 – 30
2 f 20
Md =  + × h = 30 + × 10 = 30 + 2 = 32

Now mean deviation from median,

M.D. = ∑f |x – Md| = 824 = 12.118
N 68

Co-efficient of M.D. = M.D. = 12.118 = 0.379
Md 32

Hence, mean deviation from median is 12.118 and its co-efficient is 0.379.

Exercise 8.2

1. Find the quartile deviation and its co-efficient of the following data:

a. Marks 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30 30 – 35 35 – 40 40 – 45

No. of students 5 6 15 10 5 4 2 2

b.
Daily wages (Rs.) 300 – 400 400 – 500 500 – 600 600 – 700 700 – 800 800 – 900

No. of person 2 5 8 4 3 2

c. Class 100 – 120 120 – 140 140 – 160 160 – 180 180 – 200 200 – 220
Frequency
15 20 24 25 12 4

d. Class 60 – 65 60 – 70 60 – 75 60 – 80 60 – 85 60 – 90
Friequency 7 12 20 24 27 30

2. Find the mean deviation from mean and its co-efficient from the following data.

a. Marks 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
3
No .of students 4 5 3 6 Statistics

b. Age (yrs.) 5 – 15 15 – 25 25 – 35 35 – 45 45 – 55 55 – 65
4 5 6 3 3
No .of students 7

Partition Values / 297

c. Height (cm.) 10 – 20 10 – 30 10 – 40 10 – 50 10 – 60 10 – 70
No. of plants 8 14 26 31 33 40

d. Marks 5 ≤ x < 10 10 ≤ x < 15 15 ≤ x < 20 20 ≤ x < 25 25 ≤ x < 30

No .of students 7 4 5 6 3

e. Marks 10 ≤ x < 20 20 ≤ x < 30 30 ≤ x < 40 40 ≤ x < 50 50 ≤ x < 60

No .of students 8 12 15 9 6

3. Find the mean deviation from median and its co-efficient from the following data.

a. Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
2
No .of students 6 8 11 18 5

b. Class 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80
Frequency 4 6 10 20 10 6 4

c. Age (yrs.) 10 – 20 10 – 30 10 – 40 10 – 50 10 – 60 10 – 70

No .of passengers 4 10 19 33 50 70

d. Marks 0 ≤ x < 10 10 ≤ x < 20 20 ≤ x < 30 30 ≤ x < 40 40 ≤ x < 50

No .of students 5 2 9 2 2

4. Calculate the mean deviation from mean and median and its co-efficient from the following data.

a. Marks 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80

No .of students 5 7 8 6 4 2

b. Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70

No.of Students 7 12 18 28 16 14 8



298 / Optional Mathematics Class 10

8.3 Standard Deviation

The absolute value of the deviation of the items from the average is used in the caluclation of
the mean deviation. The standard deviation remove the drawback of the mean deviation. The standard
deviation is denoted by s and defined as the positive square root of the mean of the squares of the deviation
of the items taken from their arithmetic mean.

Direct Method

For the given frequency distribution of continuous series, the standard deviation can be calculated

by the following formula.

s= ∑f(x – ‒x )2 or ∑fx2 – ∑fx 2
N N N

Short Cut (Deviation) Method

When the values of the observations and the frequencies are large in case of discrete or continuous
series, the calculation of standard deviation can be made easy by using the following formula.

s= ∑fd2 ∑fd 2
N N
–

Where, d = x – A and A is the assumed mean.

Step Deviation Method

When the values of variables or mid-values have some common factor, the calculation of the
standard deviation can be made still easier by taking the deviation as follows.

s = ∑fd'2 – ∑fd' 2 h
N N
×

where d' = x – A where A is the assumed mean and h is the common factor from all x – A.
h
Co-efficient of Standard Deviation

The relative measure or the co-efficient of standard deviation is given by co-efficient of

S.D. = Standard Deviation i.e. Co-efficient of S.D = s
Mean x‒

Note: a. The square of the standard deviation is called variance.

∴ Variance = s2 = ∑fx2 – ∑fx 2
N N

b. The co-efficient of variance is C.V. = s × 100% Statistics
x‒

Example 1: Find the standard deviation and its co-efficient from the following frequency distribution.

Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60

No. of students 8 7 10 12 6 3

Standard Deviation / 299

Solution:

Marks No. of students (f) Mid-value (x) fx fx2
0 – 10 85 40 200

10 – 20 7 15 105 1575

20 – 30 10 25 250 6250

30 – 40 12 35 420 14700

40 – 50 6 45 270 12150

50 – 60 3 55 165 9075
Total N = 46
∑fx = 1250 ∑fx2 = 43950

Here, N = 46, ∑fx = 1250, ∑fx2 = 43950

Standard deviation (σ) = ∑fx2 – ∑fx 2
= N N

43950 – 1250 2
46 46

= 955.43 – 738.42 Arithmetic mean ( ‒x ) = ∑fx = 1250 = 27.17
= 217.01 = 14.73 N 46
Hence, the standard deviation (σ) = 14.73
Co-efficient of S.D. = s = 14.73 = 0.542.
‒x 27.17

Example 2: Following are the marks obtained by students in a test exam.

Marks 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90

No. of students 4 6 10 17 11 9 3

Calculate the standard deviation and its co-efficient by

a. Direct method b. Shortcut method

c. Step deviation method

Solution:

a. Direct method

Marks No. of students (f) Mid-value (x) fx fx2
2500
20 – 30 4 25 100 7350
20250
30 – 40 6 35 210 51425
46475
40 – 50 10 45 450 50625
21675
50 – 60 17 55 935
∑fx2 = 200300
60 – 70 11 65 715

70 – 80 9 75 675

80 – 90 3 85 255

Total N = 60 ∑fx = 3340

Here, N = ∑f = 60, ∑fx = 3340 and ∑fx2 = 200300.

300 / Optional Mathematics Class 10

Standard deviation (σ) = ∑fx2 – ∑fx 2
N N

= 200300 – 3340 2
60 60

= 3338.33 – 3098.78 = 239.55 = 15.48

Arithmetic mean ( x‒ ) = ∑fx = 3340 = 55.67
N 60

Co-efficient of S.D. = s = 15.48 = 0.279.
x‒ 55.67

b. Shortcut method

Marks Frequency (f) Mid-value (x) d = x – 45 fd fd2
3600
20 – 30 4 25 – 30 – 120 2400
1000
30 – 40 6 35 – 20 – 120
0
40 – 50 10 45 – 10 – 100 1100
3600
50 – 60 17 55 0 0 2700
∑fd'2 = 14400
60 – 70 11 65 10 110

70 – 80 9 75 20 180

80 – 90 3 85 30 90

Total 60 ∑fd' = 40

Here, N = 60, ∑fd = 40, ∑fd2 = 14400

∑fd2 ∑fd 2
N N
Standard deviation (σ) = –

= 14400 – 40 2
60 60

= 240 – 0.44 = 239.56

= 15.48

Hence, the standard deviation (σ) = 15.48

Arithmetic mean ( x‒ ) = A + ∑fd = 55 + 40
N 60

= 55 + 0.67 = 55.67

Co-efficient of S.D. = s = 5155..4687 = 0.279. Statistics
x‒

Standard Deviation / 301

c. Step deviation method

x – 55
10
Marks Frequency Mid-value d' = fd' fd'2

20 – 30 4 25 –3 – 12 36
30 – 40 6 35 – 12 24
40 – 50 10 45 –2 – 10 10
50 – 60 17 55 0
60 – 70 11 65 –1 0 11
70 – 80 9 75 11 36
80 – 90 3 85 0 18 27
Total N = 60 9 ∑fd'2 = 144
1 ∑fd' = 4

2

3

Here, N = 60, ∑fd' = 4, ∑fd'2 = 144, h = 10

∑fd'2 ∑fd' 2
N N
Standard deviation (σ) = – × h

= 144 – 4 2 10
60 60
×

= 2.4 – 0.0044 × 10

= 1.548 × 10 = 15.48

Hence, the standard deviation (σ) = 15.48

Arithmetic mean ( ‒x ) = A + h × ∑fd' = 55 + 10 × 4 = 55.67
N 60

Co-efficient of S.D. = s = 1555..4687 = 0.279.
x‒

Exercise 8.3

1. Find the standard deviation of the following data.

a. Class interval 5 – 15 15 – 25 25 – 35 35 – 45 45 – 55
6 4 5
Frequency 7 3

b. Class 25 – 35 35 – 45 45 – 55 55 – 65 65 – 75
Frequency 5 4 6 7 3

c. Class 0–4 4–8 8 – 12 12 – 16 16 – 20 20 – 24
Frequency 7 7 10 15 7 6

302 / Optional Mathematics Class 10

2. Calculate the standard deviation and its co-efficient from the following data.

a. Scores 25 – 35 35 – 45 45 – 55 55 – 65 65 – 75
4
No. of players 7 8 6 5
50 – 60
b. Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 6

No .of students 3 7 8 2 4 55 – 65
15 – 25 25 – 35 35 – 45 45 – 55 3
c. Age (yrs.) 5 – 15
50 – 60
No .of students 7 4 5 6 3 6

3. Calculate the standard deviation and its co-efficient from the following data by

i. Short cut method ii. Step deviation method

a. Marks 10 – 20 20 – 30 30 – 40 40 – 50

No .of students 4 10 12 8

b. Marks 800 – 1000 1000 – 1200 1200 – 1400 1400 – 1600 1600 – 1800

No .of students 4 8 12 5 2

4. Calculate the standard deviation and its co-efficient from the following data. Calculate the variance and the
co-efficient of variation also.

a. Class 0–8 8 – 16 16 – 24 24 – 32 32 – 40

Frequency 6 7 10 8 9

b. Marks 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50 50 – 55

No. of students 3 4 8 10 3 1

c. Class 0–4 4–8 8 – 12 12 – 16 16 – 20 20 – 24
Frequency 7 7 10 15 7 2

d. Wages (Rs.) 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
12 20 30 18 10
No. of workers 10

e. Wages (Rs.) 1200 – 1250 1250 – 1300 1300 – 1350 1350 – 1400 1400 – 1450

No. of persons 20 26 32 21 15

f. Class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 Statistics
Frequency 4 6 10 20 10 6 4



Standard Deviation / 303

Specification Grid (Optional Mathematics Class 10)

Contents Knowledge Understanding Application Higher ability Total Total
Quesitons Marks
Algebra Each of 1 mark Each of 2 marks Each of 4 marks Each of 5 marks
Continuity 8 21
Matrices 2321 2 5
Co-ordinate 4 9
Trigonometry 1–1– 6 15
8 20
Vector 121– 4 10
Transformation 3 10
2211 3 10
Statistics 100
Total 233– 38

12–1

1–11

–12–

10 13 11 4

MODEL QUESTION (SEE Examination) Full Marks : 100
Optional Mathematics 10×1 = 10

Time : 3 hrs
Group ‘A’

1. a. If f = {(1, 2), (3, 5), (5, 7)}, find the inverse function f–1.

b. Find the geometric mean between a and b.

2. a. Write the condition of a function f for being continuous at x = a.

b. What is the inverse matrix of A = 1 0 ?
0 1

3. a. What is the slope of a line parallel to ax + by + c = 0.

b. Find the angle between the straight lines represented by the equation ax2 + 2hxy + by2 = 0.

4. a. If sin θ = 45, then find cos 2θ. P'

b. Solve : 2 sin θ = 1 [0° ≤ θ ≤ 180°]. P
O rA
5. a. If →a =  1  and →b =  0   , then find the angle between →a and →b.
0 1

b. Write the relation between OP, OP' and r in the given inversion circle.

Group ‘B’ 13×2 = 26

6. a. If f = {(1, 2), (2, 4), (3, 6)} and g = {(2, 4), (4, 6), (6, 8)}, then find g⸰f.

b. If x – 2 is a factor of ax3 – 6x + 2 2, then find the value of a.

c. If a + 4, a – 2 and a – 6 are three succesive terms in GP, then find the value of a.

7. a. If  x  (3) =  6  , find the value of x and y.
y9
2x + y = 4
b. Solve the following equations using Cramer's rule: x – y = 5

8. a. If the lines represented by 2x2 – 4xy + 3my2 = 0 are orthogonal, then find the value of m.

b. Find the centre and radius of circle 2x2 + 2y2 = 50.

9. a. If cos 30° = 23, then show that cos 15° = 3 +21.
2
cos 35° – sin 35° 1
b. Prove that: cos 35° + sin 35° = cot 10°

304 / Optional Mathematics Class 10

c. If A + B + C = πc then prove that: tan A + tan B + tan C = tan A tan B tan C.

10. a. In the given ∆PQR, if → . → = 10, then find Q O
PQ PR

the value of q. →→ 4 →a →b

b. In the given figure is 4AB = 3 BP then find the q R AB
→
value of OP in the terms of →a and →b. P5 Q.N.10b P

Q.N.10a

c. If Q1 = 35 and Q3 = 55, find quartile deviation and its co-efficient.

2x + 1 3x – 1 Group ‘C’ 11×4 = 44
11. 3 2 f–1(x) × g(x) = g–1(x)
If f(x) = and g(x) = and find the value of x2.

12. Solve : x3 + 2x2 – x – 2 = 0

13. If f(x) = x + 2 1≤x<2
3x – 2 x≥2

a. Find f(x) if x = 1.99 b. Find f(x) if x = 2.01

c. Is lim f(x) = lim f(x)? d. Is f(x) continuous at x = 2?
x→2– x→2+

14. Solve by matrix method: 4 + 3 – 7 = 0 and 3 = 4 – 2
x y x y

15. Find the equation of a circle with centre (3, – 2) and passing through the centre of the circle

x2 + y2 – 6x + 8y + 9 = 0. πc – A πc – B πc – C
2 2 2
16. If A + B + C = 180° then prove that: cos A + cos B + cos C = 1 + 4 cos cos cos

17. Solve : tan2A + 3 = ( 3 + 1) tan A (0° ≤ A ≤ 360°)

18. The angle of elevation of the top of the tower from a point was observed to be 45° on walking 30m

away from that point it was found to be 30°, find the height of the tower. 0 3 4 1
0 3 4 1
19. Find the transformation matrix which transforms the unit square to the parallelogram .

20. Find the mean deviation from mean and its co-efficient.

Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
No. of students 2 3 6 5 4

21. Find the standard deviation and its co-efficient.

CI 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49 50 – 54
1
frequency 2 3 4 7 2 4×5 = 20

Group ‘D’

22. Solve by graphical method and substitution method: y = x2 – x – 3 and y = x.

23. Prove that angle between the line represented by the equation

(x2 + y2) sin2α – (x cos θ – y sin θ)2 = 0 is 2α.

24. Vectorically prove that the middle point of hypotenuse of right angled triangle is equidistant from its vertices. Statistics

25. ∆PQR is reflected in the line x = 2 and the enlarged by the scale factor 2 with the centre at origin so

that the final image are P"(4, 6), Q"(0, 10) and R"(– 4, 4). Find the co-ordinates of ∆PQR and also

show on graph paper.

THE END

Standard Deviation / 305

ANSWERS

Exercise 1.1A

2. a. Composite function 2. b. Constant function

2. c. Quadratic function 2. d. Linear function
3. a.
5. a. – 3, – 2 3. b. – 11, – 8, – 23 4. a. 3 + 1, 2 4. b. 3, 0
2

{(– 3, 3), (– 1, 7), (– 2, 0), (2, 3), (0, 0)} 5. b. {(4, 9), (3, 4), (2, 1), (1, 0)}

5. c. {(2, 1), (3, 2), (4, 3)}, {(2, 1), (3, 2), (4, 3)} 6. a. i. 5, 2, 5 6. a. ii. 3, 1, 3

6. b. i. 7 6. b. ii. 8 6. b. iii. 5

7. a. g = {(– 1, 7), (– 5, 0), (– 2, 4)} 7. b. f = {(0, 2), (1, 3), (2, 4)}

8. a. 2x + 1, 2(x + 1) 8. b. i. x2 – 2 8. b. ii. (x – 2)2 8. b. iii. x4

8. b. iv. x – 4 8. b. v. 10 8. b. vi. 16 11. a. 5

11. b. – 11 12. a. – 1 12. b. 3 13. a. 9
13. b. 4 x + 2, 14. b. 2x – 3, 3 14. c. 4x + 5
14. a. 3 3 15. b. i. 3x – 2 5. b. ii. 3x – 2

15. a. 4(3x – 4), 4(3x – 10)

15. b. iii. 3 15. b. iv. 24

Exercise 1.1B

1. a. {(2, 1), (4, 3), (6, 5)} 1. b. {(10, 9), (12, 11), (14, 13)}

1. c. {(b, a), (d, c), (f, e)} 1. d. {(3, 1), (7, 5), (11, 9)}

2. a. x – 3, 0 2. b. 5x––24x, – 1 2. c. x – 4, 23 2. d. x – 1, – 3
2 3 3

43.. aa.. 2xx+4–57, ,4x 3. b. 2x – 3, – 4 3. c. x + 2, a + 2 3. d. x – 1, 1–x
5. a. {(4, 1), (6, 2), (5, 3)} 2 2x
+ 7 4. b. 2(xx––13), x – 26
2 x –

5. b. {(4, 3), (6, 5), (8, 7)} 5. c. {(2, 5), (3, 6), (4, 7)}, {(2, 5), (3, 6), (4, 7)}

Exercise 1.1C

1. a. x – 6, x – 2 1. b. 5 1. c. 3x + 4, 2(x – 4) 2. a. 0, 2
2. b. 0, 1 3. a. – 1 6

3. b. 7 3. c. 5

4. a. 1 4. b. 1 7. a. 1 7. b. 1, 9

8. a. – 1 8. b. 83 8. c. 5x––22x, 1 9. a. – 7
9. b. – 12 10. a. 1
10. b. 4
3
Exercise 1.2

1. a. 2x2 + 5x – 12 1. b. x4 + 6x3 + 12x2 + 9x + 2
2. b. x2 + x + 4 + x2–04
2. a. x3 – x2 + 4x – 18 + x7+54 3. b. x2 + x – 3, – 15

3. a. 6x + 5, 22

3. c. x2 – 3x – 2, – 11 3. d. 2x2 + 4x – 2, – 6

3. e. x2 – 2x + 15, 0 3. f. 3x3 – 6x2 + 3x – 6, 8

4. a. x2 – 4x + 3, 0 4. b. 6x2 – 8x + 8, 0 4. c. 6x2 – 7x + 4, – 4 4. d. x3 – 4x2 + 2x – 1, 6
x3 – 9x2 + 24x – 20
5. a. x3 + x2 – x – 1 5. b. x3 + 4x2 + 3x + 3 5. c.

306 / Optional Mathematics Class 10

Exercise 1.3A

1. a. 54 1. b. – 5 1. c. 20 1. d. 6

2. a. – 16 2. b. – 3 2. c. 148 2. d. 5
2. e. 4 2. f. 3 3. a. 20 3. b. 3
3. c. 5 4. a. 4 4. b. – 5 4. c. 28
4. d. – 2 4. e. 4 5. a. 2 5. b. x – 2
5. c. 5x + 2 8. a. 3 8. b. 3 9. a. 3
9. b. 6
Exercise 1.3B

1. a. – 6, – 12, 0, 0, 0, 5276; (x + 2)(x – 3)(2x – 1) 1. b. 0, 20, 0, 0, 60; (x – 1)(x + 2)(x + 3)

2. a. 7, 14 2. b. 3, – 27; 1 3. a. 2, – 5 3. b. – 17, 27

4. a. m = 3, n = – 6 4. b. 1, 2 4. c. 3, – 5 7. a. 1, – 1, – 21
7. d. – 2, 3, 12 7. d. 1, 12, – 32
7. b. 2, – 3, – 12 7. c. 2, ± 72
7. e. 1, 21, – 23 7. f. – 2, 12, – 23 8. a. (x + 1)(x – 2)(x + 3); – 1, 2, – 3

8. b. (x – 2)(x – 3)(x – 4); 2, 3, 4 8. c. (x + 2)(x – 1)(2x – 1); – 2, 1, 1
2

8. d. (x – 3)(x – 1)(2x – 3); 3, 1, 23 8. e. (x – 1)(x – 2)(x – 3) (x + 2); 1, 2, 3, – 2

8. f. (x – a)(x + a)(x – 2a); a, – a, 2a

Exercise 1.4A

2. a. Arithmetic sequence; 5 2. b. Arithmetic sequence; 7
2

2. c. Non-Arithmetic sequence 2. d. Arithmetic sequence; b –23a

3. a. 30; –3; 18, 15; 33 – 3n 3. b. 85; 25; 156, 158; 2n + 6
5

3. c. 22; – 25; 12, 912; 49 – 5n 3. d. –3a – b; a + b; 5a + 7b, 7a + 9b; –5a – 3b + an + bn
2 2 2 2 2

4. a. a 5. a. 18, 1321, 9; –18 5. b. 25, 2221, 20; 10

5. c. 1013, 11, 1123; 1631 6. a. 23 6. b. 8
x = 4, 5, 9, 13
6. c. – 14 7. a. 7. b. m = 125; 64, 43, 22

7. c. k = 27, 15, 21, 27 8. a. Yes 8. b. 13

8. c. 15 9. a. 40 9. b. 15 10. a. 45
10. b. 99 11. a. 11; 67
11. b. 21; – 123
Exercise 1.4B

1. a. d = 3; 97 1. b. 7; 28 1. c. 30 2. a. 25; – 2 Answers
3. a. 18; 34 3. b. 18, 1834, 1912, 2041
2. b. 25, 23, 21, 19 2. c. – 15 4. b. 80 4. c. 42

3. c. 30 4. a. –39 7. a. Rs.165 7. b. Rs.370
5. a. 8 5. b. 21
7. c. Rs.60000

Answers / 307

1. a. a + b 2. a. 24 Exercise 1.5A 2. c. 3k + 2
2 3. a. 5 2. b. 15152 2
3. b. 9
2. d. a2 + b2 3. c. – 1, – 5

3. d. – 12, – 7, – 2 3. e. 35, 25, 15 4. a. 12 4. b. 46

4. c. 7 4. d. 29 4. e. 22, 18 5. a. 5

5. b. 11 6. a. 4, 20 6. b. 100, 60 6. c. 10, 20

1. a. 8, 4, 0 Exercise 1.5B
2. a. 12, 14
1. b. 3, 412, 6, 721 1. c. 1456, 1223, 1012, 813, 661

2. b. 3, 4 3. a. – 2 3. b. 15

4. a. 5 4. b. 5 5. a. a = – 1, b = 39, m1 = 7, m3 = 23
5. b. 3, 16 21 6. a. 6
6. d. 8 6. b. 9 6. c. 6

1. a. n(n2+ 1) Exercise 1.6A
2. a. 187
3. a. 6 1. b. n(n + 1) 1. c. n2 1. d. n(n + 1)(2n + 1)
4. c. 207 6
6. a. 2n (5n + 1)
8. a. 950 2. b. – 12 2. c. 290 2. d. 115
3. b. 24 21 4. a. 36
4. d. 500 5. a. 6 4. b. 189
6. b. n2 (5n + 21) 7. a. 1829
8. b. 1260 8. c. 5796 5. b. 1

7. b. 910

9. a. 13, 17, 21

9. b. 7, 13 10. a. 2n 10. b. n 10. c. 2n – 1
Exercise 1.6B

1. a. 345 1. b. 380 1. c. 5623 1. d. 108

2. a. 5, 20 2. b. 4, 9 2. c. 4, 8 3. a. 85

3. b. –1696 3. c. 2950 4. a. 16 4. b. 900

5. a. 50 5. b. a = d = 5 5. c. 100 5. d. 483

6. a. 16, 8 6. b. –1, 7, 15 6. c. n = 25, d = 4 7. a. 2, 26

7. b. 740 8. a. Rs.63 8. b. 3 or 30 9. a. 4, 11, 18

9. b. 4, 6, 8, 10 10. a. n(n + 2); n6(n + 1) (2n + 7)
10. b. (4n2 – 1); n3(4n2 + 6n – 1) 10. c. n(n + 1)2, 1n2(n + 1) (n + 2) (3n + 5)

308 / Optional Mathematics Class 10

Exercise 1.7A

1. a. arn – 1 2. a. GS, 15 2. b. NGS 2. c. NGS
2. d. GS, 1b 3. a. 2; 48, 96
3. d. an, am + 3n, am + 4n 4. a. 32, – 1, 32; – 8161 3. b. – 12; 4, – 2 3. c. 12; 212, 1
4. b. 54, 18, 6; 227 4

5. b. 2 4. c. 5, – 15, 45; 405

4. d. 32, 48, 72; 108 5. a. 9 5. c. 5

6. a. 6 6. b. 9 6. c. 6 6. d. 4
7. c. 3, 48
7. a. 45 7. b. 14 8. b. 1 7. d. 6 or 2
5

7. e. 12cm, 16cm 8. a. 2048 8. c. 3, 6, 12, ...

9. a. No. 9. b. 9th

Exercise 1.7B

1. a. – 2187 1. b. 128 2. a. 21 2. b. 614
2. d. 332 3. a. 243, 81, 27, … …
2. c. 4, – 8, 16, – 32, … or – 43, – 83, – 136, – 332, …
4. b. 5 5. a. 8, 16, 32 or 32, 16, 8
3. b. 1218 4. a. 32, 48

5. b. 3, 5, 7

Exercise 1.8A

1. a. ab 1. b. N = n + 2 2. a. 9 2. b. 1
3
2. c. 12 2. d. a2 – b2 3. a. – 16, 214
3. b. 54, 36, 24
4. a. 4, – 6 4. b. 16 or 0 4. c. 36
4. d. ± 4

5. a. 1, – 27 5. b. 6, 54 5. c. 32, 6, 24
Exercise 1.8B

1. a. 10, 20, 40 1. b. 31, 1, 3 or – 31, 1, – 3 1. c. 18, 12, 8, 16
3

1. d. 70, 140, 280, 560, 1120 2. a. 3, 40 2. b. 6
3. b. 128
2. c. 5 3. a. 1 4. c. 5 3. c. 160
4. a. 4 4 6. a. 2, 8

4. b. 6 5. a. 4, 16

5. b. 9, 81 5. c. 116, 1 6. b. 3, 27

6. c. 81, 9 6. d. 20, 80 7. a. 16, 4 7. b. 2, 4, 8, 16

Exercise 1.9A Answers

1. a. 728 1. b. 1562 1. c. 44 1. d. 73 89
2. b. 2 2801 2. c. 62 362125 2. d. 127 2 – 126
2. a. 3577 3. b. 3 4. a. 1089
4. d. 336 4. b. 2184
3. a. 7
4. c. 373

Answers / 309

Exercise 1.9B

1. a. 5 1. b. 8 2. a. 2 2. b. 3
4. a. 3280
3. a. 729 31 3. b. 511 12 3. c. 364 49 5. b. 190 12
6. c. 4, 6, 9 or 9, 6, 4
4. b. 6 11078 4. c. 7 225565 5. a. 14
5. d. x + y ≤ 0
6. a. 25, 1, 5 or 52, 1, 52 6. b. 18, 12, 8 or 8, 12, 18 5. h. 3x + 2y < – 4
2

7. a. 4 yrs 7. b. 4 yrs

Exercise 1.10A

5. a. x < – 1 5. b. y ≤ 2 5. c. x – y ≤ 0
3. e. 3x + 2y > 4 5. f. 2x – y > – 4 5. g. x + y ≤ 3
5. i. x – 2y ≥ 1

Exercise 1.10B

1. a. Max. 52 at (5, 6), Min. 21 at (7, 1) 1. b. Max. 35 at (0, 5), Min. 6 at (3, 0)
2. a. Max. 21 at (3, 6), Min. 0 at (0, 0)
1. c. Max. 39 at (2, 5), Min. 13 at (3, 1) 2. c. Max. 28 at (6, 5), Min. 0 at (0, 0)
4. a. 11 at (5, 1) 4. b. 11 at (1, 3)
2 b. Max. 29 at (5, 7), Min. 14 at (4, 1) 5. b. Max. 30 at (0, 6), Min. 0 at (0, 0)
5. 3. 96 at (6, 3) 7. a. 0 at (0, 0)
3. a. 30 at (0, 6) 3. b. 52 at (5, 1) 8. a. x  – 3, x – y  1, 2x + 3y  12; Min – 42 at (– 3, 6)

5. a. Max. 22 at (6, 2), Min. – 2 at (0, 2)

6. a. 22 at (4, 2) 6. b. 24 at 3,  23

7. b. 9 at (3, 0) 7. c. 24 at (0, 2)

8. b. A(4, 2); x  – 2, x – 2y  0; Max. 14 at (– 2, 8) 8. c. x – 2y  1, x ≥ 0, y ≥ 0; 20 at (0, 4)

9. a. 4 units of A and 3 units of B 9. b. Max. Rs.2000; Bus = 10, Car = 50
Exercise 1.11A

1. a. (0, – 2) 1. b. (2, 2) 1. c. (1, 6) 2. a. (0, 0)

2. b. (0, 0) 2. c. (0, 0) 2. d. (0, – 2) 2. e. (0, 4)

2. f. (0, – 1) 4. a. (–1, 0) 4. b. (1, – 6) 4. d. (– 2, 5)
4. e. (– 1, 5)
6. c. (1, – 11), x = 1 4. f. (1, 2) 6. a. – 21,  141 , x = – 12 6. b. (1, 9), x = 1

6. d. 23,  141 , x = 23 6. e. 34,  487 , x = 43 6. f. (1, 7), x = 1

7. a. y = 2x2 7. b. y = x2 – 6x + 9 7. c. y = – x2 + 2x + 2

Exercise 1.11B

1. a. 2, –1 1. b. 1, – 3 1. c. 5, – 3 1. d. 23, – 2
1. e. 1, – 23 1. f. 12, 5
2. a. (1, 1), (– 2, 4) 2. b. (– 2, – 1), (4, 11)

2. c. (2, 5), (4, 9) 2. d. (4, 11), (– 3,– 3) 3. a. (4, 0), (0, 4) 3. b. (– 3, 4), (4, – 3)

4. a. (1, – 2), (– 3, 6) 4. b. (5, 1), (2, 4)

310 / Optional Mathematics Class 10

Exercise 2

2. a. 1 2. b. 3 2. c. 0 2. d. 4

3. a. 1 3. b. 6 3. c. 2 4. a. 5

4. b. 1 4. c. 9 5. a. 8 5. b. 1

5. c. 5 5. d. 31 5. e. 3 5. f. 0

5. g. 16 5. h. 14 6. a. Discontinuous 6. b. Continuous

6. c. Discontinuous 6. d. Continuous 6. e. Continuous 6. f. Disontinuous

6. g. Continuous 6. h. Continuous 6. i. Discontinuous 6. j. Discontinuous

8. a. Discontinuous 8. b. Continuous 8. c. Discontinuous 8. d. Continuous

8. e. Continuous 8. f. Continuous 8. g. Discontinuous 8. h. Discontinuous

8. i. Discontinuous 8. j. Continuous
Exercise 3.1A

1. a. 10 1. b. 15 1. c. – 6 1. d. – 32

3. a. 4 3. b. – 2 3. c. 2 3. d. 4

4. a. 2 4. b. 4 4. c. 6 4. d. 4, – 145

5. a. 2 5. b. 8 5. c. 4 5. d. – 6, 3

6. a. 120 6. b. 298 6. c. 172

7. a. 5 – 3 7. b. – 5 6 7. c. 1 0 7. d. 5 2
– 3 2 6 – 7 0 1 – 8 – 3

8. a. 8 3 8. b. 2 – 3
5 2 – 3 5

Exercise 3.1B

1. a. – 7 1. b. 31 4. a. x = 2, y = – 1 4. b. m = 2, n = – 7

6. a. 1 6. b. 3 6. c. 2 6. d. 1
2 4 – 1 0

Exercise 3.2

1. a. iii. 2, 1 1. b. iii. 9, – 3 2. a. – 3, 5 2. b. 3, 3

2. c. – 2, 5 2. d. 12, 15 2. e. 1, 1 2. f. 2, – 1

2. g. 3, 4 2. h. 1, 0 2. i. 2, 1 2. j. 4, – 5

2. k. 3, 23 2. . 32 , 2 2. m. 32 , 32 2. n. 13 , 5
2

2. o. 2, 5 2. p. 15, 7 3. a. Rs.72, Rs.32 3. b. Rs.2400, Rs.10000

Exercise 3.3

1. a. – 7, – 7, – 21 1. b. Yes 2. a. 0, 75, – 45 2. b. No solution. Answers

3. a. 0, 0, 0 3. b. No, they have infinite solutions.

4. a. – 1, 2 4. b. 4, – 1 4. c. 7, 2 4. d. – 6, – 5
4. f. 2, 7 5. a. 2, 1 5. b. 21 , 3
4. e. 1, 0 5. d. 4, 1 5. e. 3, 2
5. c. 13, 1

Answers / 311

Exercise 4.1

1. a. tan–1 1 1. b. 30° 1. c. 45° 1. d. tan–1 1
3 8
2. b. tan–1 – 121
2. a. 135° 2. c. 120° 2. d. 120°

3. a. 30° or 150° 3. b. 90° 3. c. 30° or 150° 3. d. tan–1 ± p22p–qq2

4. a. 45°, 135° 4. b. 90° 7. a. 6 7. b. – 2

7. c. 78 7. d. 3 7. e. 125 10. a. – 20

10. b. 15 11. a. 920° 11. b. tan–1 (± 2)

Exercise 4.2

1. a. 3x – 4y + 2 = 0 1. b. 3x – 2y + 9 = 0 1. c. 4x – y = 0 1. d. 2x – 3y + 13 = 0
e. x – 9y + 25 = 0
2. a. 12x – 5y = 99 3. b. 4x + 3y = 11
c. x + y = 12
2. b. 3x – 2y – 23 = 0 2. c. x – 2y + 10 = 0 2. d. 8x – 5y + 58 = 0 2.
b. 3x – 2y + 30 = 0
3. a. i. x – 3y – 4 = 0 3. a. ii. 3x – 4y – 2 = 0

3. c. 6x – 7y – 5 = 0 4. a. 2x – 4y + 9 = 0 4. b. x + 4y – 9 = 0 4.

4. d. 4x + 3y = 1 5. a. x – 5y = 21, 5x + y = 1

5. b. 2x – y – 1 = 0, x + 2y – 8 = 0 6. a. x – 3y = 0, x = 0

6. b. y = 0, 3x – y = 3 7. a. 3x + y = 11

7. b. 12x + 3y = 4 8. a. 352 , 11 8. b. (3, 2)
5

9. a. (2 – 3)x – y – 5 + 2 3 = 0, (2 + 3)x – y – 5 – 2 3 = 0

9. b. ( 3 – 2)x + y – 2 3 + 1 = 0, (2 + 3)x – y – 2 3 – 1 = 0

10. a. (0, – 1), (0, 3) 10. b. (2, – 2), (6, 4) 11. a. 2x + 3y = 12 11.
Exercise 4.3A

1. a. x2 – y2 = 0 1. b. x2 + 3xy + 2y2 = 0
1. c. 4x2 – 4xy – 3y2 = 0 1. d. 33x2 + 71xy – 14y2 = 0
1. e. abx2 + (a2 – b2)xy – aby2 = 0 1. f. (a2 – b2)(x2 + y2) + 2(a2 + b2) xy = 0
2. a. x + 2y = 0, 3x + y = 0 2. b. x – 3y = 0, x – y = 0
2. c. x – y = 0, x + 2y = 0 2. d. x – y = 0, x + 3y = 0
2. e. x + by = 0, ax + y = 0 2. f. px + y = 0, px – qy = 0
2. g. ax – by = 0, bx + ay = 0 2. h. ax – by = 0, bx – ay = 0
3. a. x – 2y = 0, x – 2y – 1 = 0 3. b. x – y = 0, x – y + 2 = 0
3. c. x + y + 3 = 0, x + y + 2 = 0 3. d. x – y – 2 = 0, x – y – 1 = 0
Exercise 4.3B

1. a. 2x2 + 5xy + 3y2 + 8x + 10y + 8 = 0 1. b. 2x2 + 5xy + 2y2 + 10x + 8y + 8 = 0

1. c. 2x2 + 3xy + y2 + 5x + 2y – 3 = 0 1. d. 6x2 – 7xy + 2y2 + 32x – 20y + 32 = 0

1. e. x2 + 2 xy sec θ + y2 = 0 3. a. x + 3y + 5 = 0, x + 3y – 1 = 0

3. b. x – y + 2 = 0, 2x + 3y – 4 = 0 3. c. x + y + 3 = 0, 2x + y – 1 = 0

3. d. x – 4y + 10 = 0, x – 2y + 4 = 0 4. a. x + y(cosec q + cot q) = 0, x + y(cosec q – cot q) = 0

4. b. x – y(sec q + tan q) = 0, x – y(sec q – tan q) = 0 4. c. y + x(cot q – cosec q) = 0, y + x(cot q + cosec q) = 0
4. d. x cos q + y sin q = 0, x sin q +y cos q = 0 5. c. 247 sq. units

5. d. 36x2 – 25y2 – 252x + 350y – 784 = 0 5. e. 58

312 / Optional Mathematics Class 10

Exercise 4.4A

3. a. – 5 3. b. 6 3. c. 12 3. d. 9
2

3. e. ±  6

Exercise 4.4B

1. a. 45°, 135° 1. b. 60°, 120° 1. c. tan–1 ± 17 1. d. tan–1 ±  57
4
cos 2q
1. e. tan–1 ±  sin q 1. f. α

2. a. x + y(cosec a + cot a) = 0, x + y(cosec a – cot a) = 0; 90° + a, 90° – a

2. b. x sin q – y cos q = 0, x cos q – y sin q = 0; 90 + 2q, 90° – 2q

3. a. x + 2y + 1 = 0, 3x – y + 6 = 0, tan1(±7) 3. b. x – y + 1 = 0, x + y – 2 = 0
4. b. 6x2 – 13xy + 6y2 = 0
3. c. x – 5y + 3 = 0, 2x – 3y – 1 = 0, tan–1 ± 177
4. a. 3x2 + 2xy – y2 = 0

4. c. 3x2 – 8xy + 4y2 = 0 4. d. 3x2 – 7xy – 6y2 = 0

6. a. 5x2 + 8xy + 3y2 – 44x – 34y + 95 = 0 6. b. 4x2 + 3xy – y2 – 25x + 25 = 0

6. c. 6x2 – 7xy + 2y2 – 3x + 2y = 0 6. d. 5x2 – xy – 4y2 – 17x + 26y – 22 = 0

7. a. x2 – 6xy + 8y2 + 16x – 42y + 55 = 0 7. b. 4x2 + 8xy – 5y2 – 32x + 22y – 17 = 0

7. c. 3x2 + 2xy – 8y2 – 12x + 46y – 63 = 0 7. d. 4x2 + 5xy – 6y2 – 23x + 31y – 35 = 0

8. a. 2x2 + 5xy – 3y2 + 7x – 28y – 49 = 0 8. b. x2 – 5xy + 4y2 + 3x – 3y = 0
9. a. (– 2, – 1), 90°
9. b. (2, 3); tan–1 ± 92
Exercise 4.5A

1. a. x2 + y2 = 36 1. b. x2 + y2 = 9 1. c. x2 + y2 = 25 1. d. x2 + y2 = 20

1. e. x2 + y2 = 2a 1. f. x2 + y2 = 3a 2. a. x2 + y2 + 2x + 6y + 1 = 0

2. b. x2 + y2 – 8x + 4y – 5 = 0 2. c. 9x2 + 9y2 + 54x – 90y + 185 = 0

2. d. x2 + y2 – 6x – 8y + 20 = 0 2. e. x2 + y2 – 2ax – 2ay = 0
2. f. x2 + y2 – 2px – 2qy + 2q2 = 0 3. a. x2 + y2 – 8x – 9 = 0

3. b. x2 + y2 – 4x – 6y – 12 = 0 3. c. x2 + y2 + 2x – 8y – 83 = 0
3. d. x2 + y2 – 2ax – 2by + b2 = 0 4. a x2 + y2 + x – 15 = 0
4. b. x2 + y2 – 10x – 4y + 9 = 0 4. c. x2 + y2 – 2x – 8y + 9 = 0

4. d. x2 + y2 – 5x + 3y = 22 4. e. x2 + y2 = 2c2 5. c. (– 2, 3), 5 2
4. f. x2 + y2 = a2 5. a. (0, 0), 6 5. b. (4, 0), 7

5. d. (m, m), 2 m 5. e. (0, – 3), 34 5. f. (3, – 1), 3 6. c. x2 + y2 + 4x – 4y – 2 = 0
6. a. x2 + y2 = 18 6. b. x2 + y2 – 6y – 4 = 0

Exercise 4.5B Answers

1. a. 32 , 1 , 3 1. b. – 1 , 1 , 23 1. c. (5, – 2), 4 1. d. (1, – 3), 2
2 2

1. e. – 32p , 3p  , 74p 1. f. (a cos θ , a sin θ), a 2. a. 4, 6
4
12, – 32 5 12, 3
2. b. 6 2. c. ± 4 2. d. , 2. e. ± 2
2

Answers / 313

3. a. x2 + y2 – 6x + 8y = 0 3. b. x2 + y2 – 4x – 4y – 42 = 0
3. c. x2 + y2 – 2x – 6y – 15 = 0
4. a. x2 + y2 – 8x – 12y + 27 = 0 3. d. x2 + y2 – 2x – 4y – 20 = 0
4. c. x2 + y2 – 8x + 4y + 10 = 0
4. e. x2 + y2 – 6x – 10y + 25 = 0 4. b. x2 + y2 – 4x – 6y – 12 = 0
6. a. x2 + y2 – 16x – 10y + 64 = 0
6. c. x2 + y2 – 2x + 6y = 40 4. d. x2 + y2 – 6x + 2y – 35 = 0
7. a. x2 + y2 – 5y = 0 or x2 + y2 – 11y + 24 = 0
5. b. (1, 2) 5. c. (4, 1)

6. b. x2 + y2 – 2x – 4y = 20

6. d. x2 + y2 – 8x – 6y + 16 = 0

7. b. 4

Exercise 4.6A

1. a. x2 + y2 – 6x – 10y + 9 = 0 1. b. x2 + y2 + 10x – 6y + 25 = 0
1. c. x2 + y2 + 8x + 12y + 16 = 0 1. d. x2 + y2 – 8x – 6y + 9 = 0 or x2 + y2 – 8x + 6y + 9 = 0
2. a. x2 + y2 – 8x – 12y + 36 = 0 2. b. x2 + y2 + 6x – 12y + 36 = 0
2. c. x2 + y2 – 6x + 10y + 25 = 0 2. d. x2 + y2 – 12x + 10y + 25 = 0 or x2 + y2 + 12x + 10y + 25 = 0
3. a. x2 + y2 + 6x – 6y + 9 = 0 3. b. x2 + y2 – 10x + 10y + 25 = 0
4. a. x2 + y2 + 8x – 8y + 16 = 0 4. b. x2 + y2 – 10x + 10y + 25 = 0

Exercise 4.6B

1. a. (3, –1), x2 + y2 – 6x + 2y + 6 = 0 1. b. (3, 2), x2 + y2 – 6x – 4y + 9 = 0

1. c. (4, – 1), x2 + y2 – 8x + 2y – 12 = 0 1. d. (2, 3), x2 + y2 – 4x – 6y – 12 = 0

2. a. x2 + y2 – 6x – 8y + 15 = 0 2. b. x2 + y2 – 2x – 2y – 8 = 0

2. c. x2 + y2 + 3x + 12y + 2 = 0 3. a. x2 + y2 – 6x – 8y + 21= 0

3. b. x2 + y2 – 6x – 6y + 17 = 0 5. a. x2 + y2 + 2x – 6y + 5 = 0

5. b. x2 + y2 – 9x – 3y + 10 = 0 6. a. x – 2y + 10 = 0

6. b. 9x – 14y – 37 = 0 6. c. x ± 3 y – 4 = 0

7. a. 2x2 + 2y2 – 15x – 8y + 28 = 0 7. b. x2 + y2 + 4x + 6y – 12 = 0

8. a. x2 + y2 – 6x – 8y + 20 = 0 8. b. x2 + y2 + 6x – 5y – 15 = 0

9. a. x2 + y2 – 6x – 4y + 4 = 0 9. b. x – y – 1 = 0
9. c. x + y – 2 = 0, 90°
9. d. 29 sq. units.
Exercise 5.1A

1. a. 2254 , 275 , 274 1. b. 112609 , – 111699 , – 111290  1. c. 2254  , –2 57 , – 24
7

1. d. 5622511 , 122275 , 5612711 1. e. 275 , 2254 , 274 1. f. 2254 , 275 , 24
7

2. a. 14245 , 111275 , 14147  2. b. 0, – 1, 0 2. c. 111275 , –1 2454 , – 41417 

Exercise 5.1B

1. a. 1 1. b. – 1

Exercise 5.2A

1. a. 2254 , 275 , 274 1. b. 23 , – 21 , –  3 1. c. 112690 , 111699 , 111290  4. a. 121

4. b. 1
3

314 / Optional Mathematics Class 10

Exercise 5.2B

1. a. 3 – 1 , 3 + 1  1. b. 3 – 1 , 3 + 1 6. a. 10 – 2 5 6. b. 3
22 22 22 22 4 2

Exercise 5.3A

1. a. sin 68° – sin 28° 1. b. 12 cos 10° 1. c. 3+ 2 1. d. cos 4θ – cos 10θ
4

1. f. 1 1. e. 21 (sin 6θ + sin 2θ)
4

Exercise 5.4A

1. a. – 2 sin 55° sin 15° 1. b. 2 sin 25° 1. c. 3
2

1. d. – 2 sin 3θ sin θ 1. e. 2 cos 4θ sin 3θ 1. f. 2 sin 9b cos 5b
2 2

2. a. 1 2. b. 3 2. c. 3 2. d. 1

Exercise 5.6

1. a. 60° 1. b. 45°, 135° 1. c. 15°, 165° 1. d. 30°, 150°

2. a. 30° 2. b. 9°, 45°,81° 2. c. 18°, 90° 2. d. 15°, 45°

3. a. 0°, 90°, 180° 3. b. 0°, 90° 3. c. 30°, 150° 3. d. 60°, 180°

4. a. 60°, 120° 4. b. 45°, 135° 4. c. 120° 4. d. 0°, 180°

4. e. 30°, 150° 4. f. 0°, 60° 4. g. 30°, 90°, 150° 4. h. 0°, 60°, 180°

5. a. 45° 5. b. 105°, 345° 5. c. 0°, 90°, 360° 5. d. 60°

5. e. 0°, 60°, 360° 5. f. 75°, 165° 5. g. 255°, 345°, 5. h. 53°

5. i. 0°, 67°, 360° 5. j. 0°, 120°, 360° 6. a. 60°, 180°, 300° 6. b. 0°, 30°, 150°, 180°, 360°

6. c. 60°, 300° 6. d. 0°, 120°, 240°, 360° 6. e. 120°, 240°

6. f. 60°, 135°, 240°, 315° 6. g. 45°, 225°

6. h. 120°, 150°, 300°, 330° 7. a. 0°, 30°, 150°, 180°, 210°, 330°, 360°

7. b. 45°, 120°, 225°, 240° 7. c. 0°, 90°, 120°, 180°, 240°, 270°, 360°

7. d. 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360°
1
8. a. 20°, 30° 8. b. 0°, 60°, 120°, 180°, tan–1 ±  8. c. 0°, 45°, 180°, 225°, 360°

2

8. d. 60°, 120° 8. e. 30°, 150° 8. f. 90°
Exercise 5.7

1. a. 170m 1. b. 21.65m 1. c. 20.78m 1. d. 11.55m

2. a. 5.57m 2. b. 15m, 30m 2. c. 34.64ft 3. a. 138.56m

3. b. 50.72m 3. c. 73.2m 4. a. 173.2m 4. b. 40.98m

4. c. 2.366km 4. d. 42.26m 5. a. 11.83m 5. b. 106.47m, 61.47m

5. c. 43.92m 5. d. 5.07km 6. a. 30m 6. b. 132m

6. c. 24m 7. a. 5m 7. b. 23.66m, 13.66m, 13.66m Answers

7. c. 20.49m, 70.98m 7. d. 30 2 m 8. a. 8.87m 8. b. 165.62m

8. c. 54.64m, 54.64m 9. a. 20m 9. b. 86.60m, 50m 9. c. 13.66m, 23.66m; 13.66m

9. d. 32m, 13.86m 10. a. 73.20m 10. b. 12.68m 10. c. 12.68m

10. d. 42.16 m 10. e. 178.89 m 11. a. 36.74m, 21.21m 12. a. 146.4m/s

12. b. 45m, 1485.71m

Answers / 315

Exercise 6.1A

1. a. →a .→b = – 2 1. b. m→ .→n = 6 1. c. →a .→b = 0 1. d. →u.→v = – 13
1. e. →a .→b = 6 1. f. →p.→q = 0 2. a. 25 2. b. 1
2. f. 13
2. c. – 24 2. d. 85 2. e. 29 3. d. 31
6. d. 1
3. a. – 8 3. b. 42 3. c. 41 8. d. 2
6. c. 73 9. b. 32
6. a. 5 6. b. 2 8. c. – 2 11. c. 150°
9. a. 14
8. a. – 8 8. b. 12 11. b. 30° 2. d. 121°
3. b. 30°
8. e. – 32 8. f. – 1 Exercise 6.1B 4 a. 60°
8. b. 45°
9. c. 4 2 11. a. 60°

11. d. 53°

2. a. 127° 2. b. 61° 2. c. 90°
2. e. 63.43° 2. f. 76° 3. a. 60°
3. a. 60° 3. b. 114° 3. c. 106.60°
4. b. 60° 4. c. 75.52° 8. a. 90°
8. c. 40°

Exercise 6.2A

1. a. 4→i – 2→j 1. b. 6→i – →j 1. c. – 4→i + 5→j 1. d. 11→i + 18→j
2. a. 5→i + 2→j
2. b. – 54 →i – 156 →j 2. c. 5 →i – 9→j 3. b. ii. – 5→b
4. d. 2(2→a + →b)
3. a. i. 3 3. a. ii. 9 3. b. i. 12→a – 13→b
0 3 5

4. a. 23 →v + 1 →u 4. b. 3 →b – 2→a 4. c. 4→b –  3→a
3

4. e. →p + 4→q 5. a. 3 5. b. 8 5. c. 3 →i + 4→j
2 –6
5. c. 10→i + 4→j

Exercise 6.2B

1. a. 3 →i – 5→j , – 3→i + 5→j 1. b. 6 →i + 10→j , – 4→i – 2→j

1. c. – →i + 3→j , →i + 2→j 1. d. – 23 →i + 6→j , 32→i – 6→j

2. a. →c – →→ac  , →a 2– →b 2. b. →r + 3→p – 4→q , →r – →p
4. a. →a 2 4. b. 34 →a , 12 4

+ 12 →b, 12 →b + 38 →a , 38 7. b. →a – →b + →c
4

Exercise 7.1A

1. a. –52 1. b. i. –2 1. b. ii. 4 1. b. iii. 0
1. c. 91 –1 0 9

3. a. (7, 2) 1. d. 0 2. a. A'(3, 7) 2. b. (8, 3)
6

3. b. (6, 3) 3. c. (– 2, – 6) 3. d. (0, 9)

3. e. (3, – 2) 3. f. (a + 4, b – 7) 3. g. (12, – 10) 3. h. (– 2, 10)
3. i. (1, – 6)
4. a. a = 1, b = – 5 4. b. 8
–3

316 / Optional Mathematics Class 10

Exercise 7.1B

1. a. A'(0, 7), B'(1, 8), C'(– 2, 6) 1. b. A'(1, – 1), B'(6, – 6), C'(5, 0)

1. c. A'(1, 5), B'(– 1, 3), C'(0, 7); A"(4, 1), B"(2, – 1), C"(3, 3) 2. a. B'(6, – 2), C'(6, 1)

2. b. P'(0, 1), R'(– 2, – 5), S'(– 6, – 3) 3. a. A'(– 1, 0), B'(3, 4), C'(9, 0), D'(3, – 4)

3. b. O'(1, 1), A'(3, 1), B'(4, 2), C'(2, 2); O"(– 1, 0), A"(1, 1), B"(2, 2), C"(0, 2)
Exercise 7.2A

1. a. (– 3, 2) 1. b. (– 3, 2) 1. c. (11, – 2) 1. d. (0, 3)
2. a. (9, 7) 2. b. A'(3, 2) 2. c. (– 4, – 1) 2. d. P'(8, 6)
3. a. (– 7, 2) 3. b. M'(– 7, – 2) 3. c. A(– 2, – 7) 4. a. (2, – 4)
4. b. (– 1, – 5) 4. c. (7, 4) 4. d. (3, 4) 4. e. (– 6, 3)
4. f. (1, – 2) 5. a. 10 5. b. 5 5. c. 1
6. a. 4, 2 6. b. 1, – 2 6. c. 5, 2
Exercise 7.2B

1. a. A'(10, 1), B'(7, 3), C'(9, 5) – 14

1. b. A'(9, 2), B'(6, –1), C'(8, 5); A"(– 13, – 6), B"(– 10, – 1), C"(– 12, 5), Translation by 0
0
1. c. N'(2, – 7), V'(4, – 1), M'(1, – 4); N"(2, 11), V"(4, 5), M"(1, 8), Translation by 6

2. a. (10, 3) 2. b. (– 4, – 2) 2. c. (8, 5)

3. a. P'(2, – 4), Q'(5, – 2); P"(4, – 2), Q"(2, – 5) 3. b. P'(1, 3), Q'(2, 6), R'(5, 4); P"(1, – 7), Q"(2, – 10), R"(4, – 8)

4. a. Rotation through 180° about origin; A"(– 2, – 3), B"(– 3, 4), C"(– 1, 2)

4. b. N"(– 3, 2), V"(– 6, 5), M"(– 4, – 1), Rotation through 180° about (– 1, 2)

4. c. A"(– 3, – 1), B"(– 4, – 3), C"(– 6, – 2) 4. d. P'(– 3, 10), Q'(– 9, 10), R'(– 8, 12)
Exercise 7.3A

1. a. (3, – 4) 1. b. (– 7, 2) 1. c. (1, 2) 2. a. (4, – 1)
2. d. (– 3, 2) 2. e. (1, 3)
2. b. (2, 2) 2. c. (3, 4) 3. b. 3, – 4 4. a. 3, 2

2. f. (q + 5, 1 – p) 3. a. 2, 3 Exercise 7.3B

4. b. 5, 5

1. a. A'(– 1, – 7), B'(– 2, – 2), C'(3, – 5) 1. b. P'(– 2, 3), Q'(– 1, 7), R'(– 5, 5)

1. c. A'(2, 5), B'(4, 4), C'(1, 2); A"(– 2, – 5), B"(– 4, – 4), C"(– 1, – 2)
2. a. P'(2, – 6), Q'(1, – 1), R'(– 3, – 4); P"(3, 5), Q"(– 2, 4), R"(1, 0)
2. b. D'(– 3, 2), E'(– 4, 6), F'(0, 4); D"(– 3, – 4), E"(– 2, – 8), F"(– 6, – 6)

3. a. A'(– 5, 1), B'(– 2, 3); A"(– 3, 1), B"(– 1, – 2)
3. b. P'(1, 3), Q'(3, 4), R'(1, 6); P"(– 3, 1), Q"(– 5, 0), R"(– 3, – 2)
4. a. P'(– 1, 1), Q'(– 1, 5), R'(– 3, 4); P"(– 1, – 1), Q"(– 1, – 5), R"(1, – 4); 270°, (– 1, 1)

4. b. A'(0, 3), B'(3, 2), C'(2, 4); A"(–2, –1), B"(–5, 0), C"(– 4, – 2); 270º, (0, 2)
Exercise 7.4A

1. a. (– 30, – 36) 1. b. A'(– 15, 12) 1. c. (18, – 30), (– 18, 24) Answers

2. a. (0, – 4) 2. b. (2, 0) 2. c. (3, – 2) 2. d. (– 2, 4)
2. f. (p – 1, q + 1) 3. a. 1, 6 3. b. 3, 8
2. e. 1 – 2a , 2 – b 3. d. 2, 18
2

3. c. 2, – 5

Answers / 317

Exercise 7.4B

1. a. A'(– 4, – 1), B'(– 1, – 2), C'(– 3, – 3); A"(8, 2), B"(2, 4), C"(6, 6)

1. b. P'(2, 1), Q'(1, 3), R'(3, 2); P"(4, 2), Q"(2, 6), R"(6, 4)

1. c. A1(– 3, 0), B1(– 1, – 2), C1(– 3, – 3) 2. a. i. A'(2, 4), B'(6, 8), C'(6, 4)
2. a. ii. A"(3, 6), B"(9, 12), C"(9, 6) 2. b. i. A'(– 1, – 1), B'(– 4, – 1), C'(– 2, – 3)

2. b. ii. A"(– 1, 5), B"(5, 5), C"(1, 9) 2. b. iii. (3, – 3), 2

3. a. A'(6, 2), B'(4, 6); A"(5, 6), B"(– 4, 6) 3. b. P'(8, 3), Q'(5, 6), R'(6, 3); P"(7, 0), Q"(1, 6), R"(3, 0)

3. c. P'(8, 2), Q'(0, 6), R'(4, 8); P"(– 4, 4), Q"(4, 0), R"(0, – 2)

4. a. A1(2, 2), B1(8, 2), C1(8, 6), D1(2, 6); A'(– 2, – 2), B'(– 8, – 2), C'(– 8, – 6), D'(– 2, – 6)

4. b. P'(5, – 3), Q'(1, – 4), R'(0, – 6), S'(4, – 5); P"(– 10, 6), Q"(2, – 8), R"(0, – 12), S"(8, – 10)
Exercise 7.5A

1. a. (– 6, – 11) 1. b. (– 15, 9) 1. c. (3, – 2) 1. d. (4, 3)
1. e. (– 7, – 3) 2. c. (6, – 8)
2. d. (– 1, – 1) 2. a. (– 2, 10) 2. b. (6, – 6) 3. c. (11, 2)
3. d. (5, – 7) 4. a. (– 6, 4)
4. b. (3, 2) 3. a. (– 2, 3) 3. b. (6, – 4) 5. a. – 4, 5
5. b. 2, 1
3. e. (4, – 12) 3. f. (5, 0)

4. c. (1, 6) 4. d. (4, – 8)

5. c. 3, – 3
Exercise 7.5B

1. a. A'(– 3, 1), B'(– 6, 1), C'(– 5, 3); A"(3, 1), B"(6, 1), C"(5, 3)

1. b. A'(4, – 2), B'(5, 3), C'(3, 5); A"(2 ,4), B"(– 3, 5), C"(– 5, 3)

1. c. A"(1, – 3), B"(2, 2), C"(3, 0)

2. a. X'(3, 3), Y'(1, – 1), Z(7, 1); X"(3, – 3), Y"(1, 1), Z"(7, – 1)

2. b. A'(2, 2), B'(3, 3), C'(4, 1); A"(2, 2), B"(1, 3), C"(0, 1)

3. a. P'(– 2, 1), Q'(– 5, 3), R'(– 7, 1); P"(– 4, 2), Q"(– 10, 6), R"(– 14, 2)

3. b. A'(0, 2), B'(1, 3), C'(1, 1); A"(3, 8), B"(5, 10), C"(5, 6)

4. a. K'(5, – 2), L'(3, 1), M'(1, – 4); K"(10, – 4), L"(6, 2), M"(2, – 8)

4. b. A'(– 1, – 1), B'(– 3, – 1), C'(– 2, – 3); A"(– 2, – 2), B"(– 6, – 2), C"(– 4, – 6)

5. a. A'(– 5, 2), B'(– 3, – 1), C'(– 1, 4); A"(9, 2), B"(7, – 1), C"(5, 4)

5. b. A'(– 9, 5), B'(– 13, 2), C'(– 11, 8); A"(5, 9), B"(2, 13), C"(8, 11)

6. a. O'(0,2), A'(2, 2), B'(3, 4), C'(1, 4); O"(6, 2), A"(4, 2), B"(3, 4), C"(5, 4)

6. b. A'(6, – 2), B'(2, – 2), C'(2, – 6), D'(6, – 6); A"(– 2, 6), B"(– 2, 2), C"(– 6, 2), D"(– 6, 6)

6. c. A'(– 2, – 3), B'(– 5, – 4), C'(– 7, – 1), D'(– 3, 0); A"(– 3, 2), B"(– 4, 5), C"(– 1, 7), D"(0, 3)

6. d. A'(– 2, 2), B'(– 4, 0), C'(– 8, 4), D'(– 4, 6); A"(1, – 1), B"(2, 0), C"(4, – 2), D"(2, – 3)
Exercise 7.6

1. a. A'(0, 2), B'  21, 0 , C'  12, 1 , D'(1, – 1), E'(2, 1), F'(– 4, 2)
2

2. a. 2cm 2. b. 1cm 2. c. 2 2cm 3. a. (1, 0)

3. b. (0, – 1) 3. c. (0, – 2) 3. d. 21, 0 3. e. 0, 1
3. f. (– 2, 0) 2

3. g. 21, 1 3. h. 14, 1 4. a. (2, 0)
2 4

318 / Optional Mathematics Class 10

4. b. (0, – 2) 4. c. (0, – 4) 4. d. (1, 0) 4. e. (0, 1)

4. f. (– 2, 0) 4. g. (1, 1) 4. h. (2, 2) 5. a. 21, – 12

5. b. 14, – 14 5. c. 15, 2 5. d. 52, 1 5. e. 110, 3
5 5 10

5. f. 130, 1 5. g. 133, – 123 5. h. 334, 5 6. a. 43, 0
10 34

6. b. (0, – 1) 6. c. 0, – 45 6. d. 0, 4 6. e. 0, – 43
3

6. f. – 183, 12 6. g. 2410, 16 6. h. 2152, 16 7. a. (4, 3)
13 41 25

7. b. 130, 3 7. c. (2, 4) 7. d. 154, 23 7. e. 152, 21
5 5

7. f. (6, 3) 7. g. 158, 19 6. h. 154, 17 8. a. 2
5 5

8. b. 3 8. c. 3 9. a. 2 9. b. 2 5

9. c. 13 10. a. 163, 9 10. b. 1110, 17
13 10
Exercise 7.7A

1. a. A'(2, 2) 1. b. M'(5, – 9) 2. a. 4 2. b. –8
–3 2

2. c. –8 3. a. A'(3, 12) 3. b. B'(17, 21) 4. a. Reflection in y = x
5

4. b. Rotation through 270° about O. 4. c. Rotation through 180° about O.

4. d. Enlargement about O with scale factor k. 5. a. 0 1 5. b. –1 0
1 0 0 1

5. c. 0 –1 5. d. 2 0 6. a. 3 , (9, 4) 6. b. –3 , (7, 2)
1 0 0  2 –6 –6

7. a. (2, –4) 7. b. 2, –3 7. c. 5, 4 7. d. –1, 4

8. a. A'(8, 9), B'(7, 3) 8. b. A'(8, 7), B'(18, 17) 9. a. A'(2, 3), B'(5, 5), C'(3, 7)

9. b. A'(2, – 3), B'(–1, –2), C'(0, –4), D'(3, – 6) 9. c. A'(– 1, – 1), B'(– 1, – 3), C'(– 4, – 2)
10. a. Reflection in Y-axis, (3, – 1) 10. b. Reflection in y = – x, (– 2, 3)

10. c. Rotation through 90° about O; (– b, a) 11. a. B'(12, 29) 11. b. A'(15, 12)

12. a. A'(7, 7), B'(4, – 9) 12. b. P'(5, 0), Q'(4, 7), R'(11, 13)

13. a. 3, 2, – 5, – 2 13. b. 1, – 2, 3, – 1 13. a. – 2, 3, 2
Exercise 7.7B

1. a. A'(1, – 1), B' (2, – 4), C'(5, – 4), Rotation through – 90º about O(0, 0).

1. b. P' (–2, 1), Q'(–3, –3), R'(–5, –2). Half turn about O(0, 0).

1. c. A'(– 4, 9), B'(– 8, – 3), C'(– 6, 15) 1. d. P'(11, 7), Q'(7, – 1), R'(9, 8)

2. a. –1 0 , P'(–1, 2), Q'(– 3, – 1), R'(– 5, 3) 2. b. 0 1 , A'(1, –1), B'(2, 3), C'(6, 2)
0 1 –1 0

2. c. 2 0 , A'(2, 0), B' (6, 4), C'(4, 6) 3. a. 0 1 3. b. –1 0
0 2 1 0 0 1

3. c. Reflection in y = – x, 0 – 1 4. a. 0 – 1 ; Rotation through +90° about O
– 1 0 1 0

4. b. Rotation through 270° about O; 0 1 4. c. –1 0 5. a. 2 0 Answers
– 1 0 0 –1 0  2

5. b. 1/2 0 6. a. 1 2 6. b. 1 0 6. c. 1 2
0 1/2 1 –2 2 1 1 – 2

6. d. 2 0 7. a. 3 2 7. b. 3 1 8. a. 6 2
0 – 2 1 1 1 2 2 4

8. b. 4 2 9. a. 4, 1, 6, 2 9. b. 3, 7, 2, 4 10. a. 1, 2, – 5, 2
1 2

Answers / 319

10. b. 3, – 3, 2, 5 11. a. O'(0, 0), A'(0, 2), B'(4, 2), C'(4, 0); O"(– 3, 2), A"(– 3, 4), B"(1, 4), C"(1, 2)
11. b.
12. a. O'(0, 0), A'(4, 0), B'(4, 2), C'(0, 2); O"(2, 4), A"(6, 4), B"(6, 6), C"(2, 6)
12. b.
A'(3, 7), B'(5, 13), C'(9, 21), D'(7, 15); A"(– 1, – 2), B"(– 3, – 2), C"(– 3, – 6), D"(– 1, – 6)

P'(3, – 3), Q'(– 1, – 3), R'(3, – 9), S'(7, – 9); P"(– 3, –3), Q"(– 7, – 11), R"(– 15, – 21), S"(– 11, – 13)

Exercise 8.1

1. a. 11.83 1. b. 42 1. c. 97.2 1. d. 34
2. a. 5
3. a. 27.5 2. b. 10 2. c. 10 2. d. 4
3. e. 77.92
5. a. 20.31 3. b. 24 3. c. 10.75 3. d. 137.5
6. c. 75, 88.21
4. a. 61.43 4. b. 45.36 4. c. 90.75

5. b. 12.92 6. a. 47.5, 59.23 6. b. 30.8, 42.71

7. a. 8.16, 14.04 7. b. 16.14, 29.29
Exercise 8.2

1. a. Q1 = 15.42, Q3= 25.75; 5.165, 0.251 1. b. Q1 = 480, Q3= 675; 97.5, 0.169

1. c. Q1 = 130, Q3= 172.8; 21.4, 0.141 1. d. Q1 = 65.5, Q3= 78.125; 6.31, 0.088

2. a. 11.97, 0.347 2. b. 14.08, 0.453 2. c. 13.6, 0.368 2. d. 6.14, 0.377

2. e. 10.08, 0.3 3. a. 10.8, 0.36 3. b. 11.33, 0.252 3. c. 12.498, 0.244

3. d. 9.501, 0.407 4. a. 11.798, 0.257; 11.563, 0.257

4. b. 12.867, 0.362; 12.766, 0.363

Exercise 8.3

1. a. 14.78 1. b. 13.11 1. c. 6.05 2. a. 13.454, 0.286

2. b. 16.683, 0.556 2. c. 16.549, 0.533 3. a. i. 12.031, 0.339

3. a. ii. 12.031, 0.034 3. b. i. 213.78, 0.17 3. b. ii. 213.78, 0.17

4. a. 10.865, 0.508, 118.048, 50.8% 4. b. 6.173, 0.158, 38.106, 15.8%

4. c. 5.535, 0.496, 30.636, 49.6% 4. d. 13.154, 0.419, 173.028, 41.9%

4. e. 63.699, 0.0483, 4057.56, 4.83% 4. f. 15.275, 0.436, 233.33, 43.6%
Model Question

1. a. {(2, 1), (5, 3), (7, 5)} 1. b. ab 2. a. lim f(x) = f(a)
x→a

2. b. 1 0 3. a. – ba 3. b. tan–1 ± h2 – ab 4. a. – 275
0 1 a+b

4. b. 30°, 150° 5. a. 90° 5. b. OP×OP' = r2 6. a. {(1, 4), (2, 6), (3, 8)}

6. b. 2 6. c. 14 7. a. 2, 3 7. b. 3, – 2

8. a. – 32 8. b. (0, 0), 5 10. a. 60° 10. b. 7b – 4a
10. c. 10, 2.22 11. 1, – 12 3

12. 1, – 1, – 2 13. a. 3.99

13. b. 4.03 13. c. Yes 13. d. Yes 14. – 21, 15

15. x2 + y2 – 6x + 4y + 9 = 0 17. 45°, 60°, 225°, 240° 18. 40.98m

19. 3 1 20. 10.4, 0.52 21. 7.12, 0.186 22. (3, 3), (– 1, – 1)
3 1

25. P(2, 3), Q(4, 5), R(2, 2)

320 / Optional Mathematics Class 10