Final PDF file of OPT Class 9 2077

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

a. Age (months) 8 10 13 15 17 18

No. of children 5 12 15 16 20 7

b. Wages (Rs.) 700 750 780 850 900

No. of workers 5 8 12 13 10

10. Calculate the mean deviation from mean and median and their co-efficient of the following data.

a. Marks obtained 20 25 30 35 40 45

No. of students 12 15 20 22 18 10

b. Heights (inch) 55 56 57 59 62 65 66

No. of workers 10 15 20 22 17 8 3

11. a. In a discrete series, if ∑fx = 1200, N = 100 and ∑f|x – –x | = 224, then find the mean deviation from
mean and its co-efficient.

b. From a discrete series, if Md = 24, N = 100 and ∑f|x – Md| = 448, then find the mean deviation
from median and its co-efficient.

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

a. 12, 25, 29, 37, 41, 45, 49 b. 12, 22, 15, 18, 24, 38, 25, 44, 29
13. a.
From an individual series, if ∑x = 114, –x  = 119 and ∑(x – x– )2 = 232, then find the standard

deviation and its co-efficient.

b. From a discrete series, if N = 48, ∑fx = 576 and ∑f(x – x– )2 = 48, then find the standard and its
co-efficient.

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

a. Size 2 4 6 8 10 12 14

Frequency 1235 321

b. Score (x) 10 12 17 21 26

Frequency (f) 24 8 5 1

15. Calculate the variance its and co-efficient of the following data.

a. Height 5 10 15 20 25 30

Frequency 10 8 3 7 5 2 Statistics

b. Weight 4 6 8 10 12 14 16

Frequency 7 11 12 10 4 6 3



Dispersion ~ 297

EXERCISE – 1.1

1. b 2. a. x = 2, y = 5 2. b. x = 2, y = 10

2. c. x = 6, y = 2 2. d. x = 7, y = – 1

3. a. A × B = {(a, p), (a, q), (a, r), (b, p), (b, q), (b, r)},

B × A = {(p, a), (p, b), (q, a), (q, b), (r, a), (r, b)}

3. b. P × Q = {(– 2, 2), (– 2, 4), (– 2, 5), (0, 2), (0, 4), (0, 5), (3, 2), (3, 4), (3, 5)}

Q × P = {(2, – 2), (2, 0), (2, 3), (4, – 2), (4, 0), (4, 3), (5, – 2), (5, 0), (5, 3)}

3. c. A = {– 1, 2}, B = {3, 5} 3. e. i. {(2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}

3. e. ii. {(4, 3), (5, 3)} 3. e. iii. {(2, 3), (2, 6), (3, 3), (3, 6), (4, 3), (4, 6), (5, 3), (5, 6)}

3. e. iv. {(2, 4), (2, 5), (3, 4), (3, 5), (4, 3), (4, 6), (5, 3), (5, 6)}

4. a. {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}

4. b. {(m, l), (m, o), (m, v), (m, e), (y, l), (y, o), (y, v), (y, e)}

5. a. {(8, 8), (8, 9), (9, 8), (9, 9)}

5. b. {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}

6. a. {(1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1), (2, 2), (2, 3), (3, 0), (3, 1), (3, 2), (3, 3)}

{(0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

6. b. {(3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}

EXERCISE – 1.2

1. a. i. {(2, 7), (2, 9), (3, 7), (3, 9)} 1. a. ii. {(2, 7), (2, 9), (3, 2), (3, 7), (3, 9)}

1. b. i. {(2, 2), (4, 4)} 1. b. ii. {(2, 1), (4, 1), (4, 2), (5, 1), (5, 2), (5, 4)}

1. b. iii. {(2, 2), (2, 4), (4, 4)} 1. b. iv. {(2, 1), (2, 2), (4, 1), (4, 2), (4, 4), (5,1)}

1. c. {(1, 2), (2, 4)} 1. d. i. {(1, 1), (4, 2), (4, – 2)}

1. d. ii. {(1, – 2), (4, 1), (4, 2), (4, – 2), (8, 1), (8, 2), (8, – 2), (8, 5)}

1. e. i. {(5, 2), (6, 2), (6, 5)}, D = {5, 6}, R = {2, 5}, R– 1 = {(2, 5), (2, 6), (5, 6)}

1. e. ii. {(2, 5), (2, 6), (5, 2), (5, 6), (6, 2), (6, 5)}, D = {2, 5, 6}, R = (2, 5, 6},

R– 1 = {(5, 2), (6, 2), (2, 5), (6, 5), (2, 6), (5, 6)}

2. a. i. D = {1, 2, 3, 5}, R = {5, 8, 10} 2. a. ii. D = {2, 3, 4}, R = {1, 3, 5}

2. b. R = {1, 5, 7, 9, 11} 2. c. i. R = {0, 6, 9, 12}

2. c. ii. R = {3, 2, 3, 6, 11} 2. c. iii. R = {– 2, 1, 4, 7, 10}

3. a. onto 3. b. into 3. c. onto 3. d. into

4. a. one to one onto 4. b. one to one into 4. c. one to many onto 4. d. many to one into

298 ~ Perfect Optional Mathematics Class 9

4. e. one to many into 4. e. many to one into 5. a. R = {(2, 1), (4, 3), (5, 4)}, R– 1 = {(1, 2), (3, 4), (4, 5)}

5. b. i. R = {(3, 2), (4, 2), (4, 3), (5, 2), (5, 3), (5, 4)} 5. b. ii. D = {3, 4, 5}, R = {2, 3, 4}
5. b. iii. R– 1 = {(2, 3), (2, 4), (3, 4), (2, 5), (3, 5), (4, 5)} 5. b. iv. D = {2, 3, 4}, R = {3, 4, 5}
5. c. R = {(1, 1), (3, 9), (5, 25), (7, 49)}, R– 1 = {(1, 1), (9, 3), (25, 5), (49, 7)}; D = {1, 3, 5, 7}, R = {1, 9, 25, 49}

6. a. i. {(2, 4), (1, 2), (4, 8), (3, 6), (– 1, – 2)} 6. a. ii. {(x, y): x∈A, y∈B and y = 2x }
6. a. iii. {(4, 2), (2, 1), (8, 4), (6, 3), (– 2, – 1)} 6. b. i. {(x, y): x∈A, y∈B, x = y3}
6. b. ii. {(x, y), x ≤ y}
EXERCISE – 1.3

1. a. Yes. 1. b. No. 1. c. Yes. 1. d. Yes.

2. a. Yes. 2. b. Yes. 2. c. No. 2. d. Yes.

3. a. Yes. 3. b. Yes. 3. c. No. 3. d. No.

4. a. yes. 4. b. No. 4. c. Yes. 4. d. Yes.

5. a. D = {2, 4, 8, 9}, C = {a, b, c, d}, R = {a, b, c} 5. b. D = {– 2, 4, 5, 6, 8}, C = {p, q, r}, R = {p, q, r}

5. c. D = {a, b, c, d, e}, C = {b, d, q, r}, R = {b, d, q, r} 5. d. D = {1, 2, 3, 5, 7}, C = {a, b, c, d}, R = {a, b, c, d}

6. a. D = {a, b, c, d}, C = {4, 5, 6, 7}, R = {4, 5, 6} 6. b. D = {2, 4, 7, 8}, C = {a, b, c}, R = {a, b, c}

6. c. D = {a, b, c, d}, C = {2, 4, 5, 7}, R = {2, 4, 5, 7} 6. d. D = {– 2, – 1, 2, 3, 4}, C = {a, b, c}, R = {a, b, c}

7. a. Range = {– 3, 5, 15, 47} 7. b. 3, 15 7. c. 25

7. d. 6 7. e. 4 7. f. – 3 8. b and d are functions

9. a. Many one into 9. b. Many one onto 9. c. One one onto 9. d. One one into

10. a. i. {(2, 5), (3, 7), (4, 9)} 10. a. iii. x 2 3 4
y 5 7 2

10. a. iv. y = 2x + 1 10. b. ii. {(1, 4), (2, 5), (3, 6), (5, 8)}

EXERCISE – 1.4

1. a. f(0) = 3, f(– 2) = – 1, f 1 = 2 + 3, f –7   = – 35 , f(1 + h) = 2h + 5, f(1 + h) – f(1) = 2
x x 3 x

1. b. i. – 10 1. b. ii. 81 1. b. iii. – 213 1. b. iv. 82

1. c. i. 8 1. c. ii. 55 1. c. iii. No. 1. c. iv. No
2. a. i. 13 2. a. ii. – 10 2. a. iii. 427
2. b. ii. 150 2. b. iii. 1207 2. a. iv. 5
4. b. 2, 1 4. c. 5, 3
2. b. i. – 11 5. b. 4 5. c. 0, 19, 1 2. b. iv. 127
4. d. 3, 2 9

4. a. 19, 1 Answers
2

5. a. 5, – 3, 24 6. a. {– 3, – 2, – 1, 0, 1, 2, 3}

6. b. {– 9, – 7, – 5, – 3, 5, 9, 13} 6. c. – 5, 9, 10 7. 0, 2 + 1

8. – 1, 0 9. a. 5, 4, 3, 1 9. b. 1, 0, 1, 3 9. c. 4, 0, 1, 4

Answers ~ 299

EXERCISE – 1.5

1. a. 2, 5 1. b. 3, 3 1. c. 5, 1 2. a. 4, y2

2. b. 3, y3 2. c. 4, yz 3. a, b, d and f are polynomials

4. a. 2 4. b. 5 4. c. 50 4. d. 4

5. a. x5 + 3x3 – 8 5. b. 3 + 2x – x3 + x4 5. c. 7 – 2y + y3 + 2y4 – 3y5

5. d. – x + 3 5. e. 4x5 + 2x4 + 7x2 + 21x 6. a. Yes

6. b. No. 6. c. Yes. 6. d. No. 6. e. No.

7. a. m = 4 and a = 2 7. b. 2, 2 8. a. 7x2 + 6x – 1, – x2 + 8x – 5

8. b. 3x3 + 4x2 – 7x – 5, – x3 – 4x2 – 7x + 9 8. c. 2x4 + x3 – 4x2 + 5x + 2, 2x4 – x3 – 2x2 + x – 2

8. d. 5x5 + 4x3 – 7x2 – x + 1, x5 + 4x3 + x2 + x – 7 9. a. – x3 + 4x2 + 3x

9. b. 8 – 3x + 7x2 – x4 9. c. 3 + 7x – 4x2 + 3x3

9. d. – 4x4 + 3 x3 – 2x – 5 10. a. 2x3 – 2x2 + 3x – 3

10. b. 6x3 – 5x2 + 9x – 4 10. c. x4 – 2x3 – 2x2 + 5x – 2 10. d. 2 x3 – (2 + 2)x2 + 5x – 3
11. a. x2 + 5x + 5; 12
13. a. 0 11. b. x2 – xy + y2, 0 12. a. a = 85, b = 1, c = 4 12. b. a = 4, b = 8, c = 6
14. b. 6x2y + 2y3
13. b. 32 x3 – 12 x2 – 35 x – 9 14. a. – 23 x3 + 12 x2 + 98 x – 11
1. a. 31, 4n – 9
EXERCISE – 1.6

1. b. – 37, 43 – 8n 2. a. 5, 8, 11, 14; 38 2. b. – 1, 5, 15, 29; 285

2. c. 1, 3, 6, 10; 78 2. d. – 2, 34, –9 4, 156; 13 2. e. 32, 32, 152, 130; 78
2. f. 3, 7, 13, 21; 157 3. a. 144 3. b. 7

7, 9, 11, 13, 15; 55 3, 35, 75, 79, 191; 8318151 3. c. – 21, 23, –4 3, 54, –6 5; – 37
60

3. d. 2, – 5, 10, – 17, 26; 16 3. e. 0, 3, 10, 21, 36; 70 3. f. 1, 3, 7, 13, 21; 45

4. a. 9, – 2 4. b. 25, – 18 4. c. n3 – 2n, 980 5. a. 3, 9, 21, 45, 93
5. d. 3, 8, 13, 18, 23
5. b. 21, 25, 123, 229, 621 5. c. – 2, 12, 74, 189, 4163 6. b. 23, – 12 5. e. 76, 25, 8, 73, 4
7. d. 7 21570230 9

5. f. –1 761, – 831, – 411, – 12, 4 6. a. 2, 1 7. a. 54

7. b. 40 7. c. 325 7. e. 72

7. f. x6 – 1 8. a. 3n 8. b. 5n 8. c. 4n + 1

8. d. 4n – 3 8. e. n + 2 8. f. 12 n(n + 1) 8. g. 3n + 1

10 3 n – 1 13 3 n – 1
2 2
9. a. 5(5 – n); ∑  5(5 – n) 9. b. 32  ; ∑ 32 
n =1
r =1

9. c. (– 1)n nn + 24; n∑ 5=1 (– 1)n nn + 24 9. d. 31 (7n + 29);  r∑1 =51 13 (7n + 29)
+ +

5 9

9. e. n2 + n; ∑ n(n + 1) 9. f. (n + 1) (n – 2); ∑ (n + 1)(n – 2)
n=1 n =1

10. a. 56

300 ~ Perfect Optional Mathematics Class 9

, 72 10. b. 16 10. c. 2n 11. a. 285, 385

11. b. 100 11. c. n2 12. a. 1296, 2025, 3025 12. b. 729, 1000

12. c. n3 13. a. 5n – 2 13. b. 10n + 3 13. c. 7n – 5
13. d. – 2n + 20 14. b. 2n2 – 3n + 3 14. c. n2 + 7n – 3
14. d. – n2 + 13n – 4 14. a. 21 (n2 – n + 20)

1. a. 8, 9 EXERCISE – 2.1
3. a. 16, 15; – ∞
1. b. ∞ 2. a. 18, 21 2. b. ∞

3. b. 60, 50; – ∞ 4. a. 2003000, 20030000; 0 4. b. 614, 1218; 0

5. a. 2.000001, 2.0000001, 2.00000001; 2 5. b. 5.299999, 5.2999999, 5.29999999; 5.3

6. a. – 10, 10; no limit 6. b. 4, – 4; no limit 7. a. 0, 5, 10, 15, 20; ∞ 7. b. 2, 8, 18, 32, 50; ∞

8. a. 21, 31, 41, 51, 61; 0 8. b. 21, 41, 61, 81, 110; 0 9. a. 73, 163, 199, 285, 1315; 2 9. b. 23, 34, 54, 65, 76; 1
1. a. 16cm, 8cm
EXERCISE – 2.2

1. b. 0 1. c. 0 1. d. 0

2. a. 5 2cm, 5cm 2. b. 0 2. c. 0 2. d. 0

3. b. 0 3. c. 0 4. b. 0 4. c. 0

4. d. infinite 5. b. infinite 5. c. 0 5. d. 0

6. b. infinite 6. b. 0 7. b. infinite 7. c. ∞
3. c. 2 4. c. 128 5. b. 51
1. a. x → 3 1. b. x → a EXERCISE – 2.3 1. d. x → – a

5. a. 13

EXERCISE – 2.4

1. c. x → – 2

2. a. x approaches a 2. b. x approaches – a 2. c. x approaches 8 2. d. x approaches – 2

2. e. limit as x approaches a from left 2. f. limit as x approaches a from right

2. g. limit of f(x) as x approaches a from left 2. h. limit of f(x) as x approaches a from right

4. a. 10 4. b. 7 4. c. 4 4. d. 31

5. a. 3 5. b. 6 5. c. 6 5. d. – 5

5. e. 3 5. f. 5 6. a. 2 6. b. 8

6. c. – 2 6. d. 1 6. e. – 2 6. f. 4 Answers

7. a. 1 7. b. 4 7. c. 4 7. d. 4

9. a. DNE, 2, 2; 2 9. b. 3, 5, 5; 5 9. c. 1, 1, 4; DNE 9. d. 4, 4, 4; 4

9. e. DNE, 1, 5; DNE 9. f. 3, 4, 1; DNE 10. a. 1 10. b. – 2

Answers ~ 301

10. c. DNE 10. d. 2 10. e. DNE 10. f. – 4
11. a. 4 11. b. 1 11. c. – 2 11. d. 0
11. e. 4 11. f. DNE 11. g. 1 11. h. 0
11. i. 1 12. a. DNE 12. b. 3 12. c. 5
12. d. DNE 12. e. 3 12. f. DNE
1. d. – 25
1. a. 4 1. b. – 4 EXERCISE – 2.5 1. h. – 23
1. e. 1 1. f. 2 2. d. 3b2
2. a. 2 1. c. 45 2. h. – 21
2. b. 14  1. g. 0 3. d. 32
2. e. 32a  2. c. – 8 3. h. – 1
2. f. – 12 2. g. 27
3. a. 14 3. b. 81 3. c. 6 4 2a
3. e. 2 6 3. f. 1 3. g. – 35
3. i. 0
4a a EXERCISE – 3.1
3. j. 0

2. a. Scalar matrix 2. b. Diagonal matrix 3. a. 2×3

3. b. 2×3, Rectangula matrix 4. a. 3 4. b. 3

5. a. 9 5. b. 3, 4 6. a. 1×8, 8×1, 4×2, 2×4

6. b. 10 7. a. 2 × 3 7. b. 3 × 2 7. c. 4 × 3

A B C Apples Mangoes Oranges
270 184 203
8. a. Boys 209 201 143 8. b. A 210 167 78
Girls B 198 200 103

9. a. i. 3 × 4 9. a. ii. 4, – 4, – 7 9. a. iii. – 1, 15 9. b. i. 3 × 3, 2 × 3

9. b. ii. 8, 6, 4, – 3 9. b. iii. – 1, 0 10. a. Zero or Null matrix 10. b. Column matrix

10. c. Row matrix 10. d. Diagonal matrix 10. e. Scalar matrix 10. f. Symmetric matrix

10. g. Lower triangular 10. h. Unit or Identity 10. i. Rectangular matrix 10. j. Diagonal matrix

10. k. Symmetric matrix 10. l. Upper triangular 10. m. Lower triangular 10. n. Upper triangular

10. o. Diagonal matrix 10. p. Symmetric matrix

11. A = E, B = F 12. a. 3 5 7 12. b. 1 2 3 12. c. 1 – 1 – 3
4 6 8 1 2 3 4 2 0
5 8 11
13. a. 7 10 13 2 5 8 1 – 2 – 5
13. b. 1 4 7 13. c. 5 2 – 1
9 12 15
14. a. x = – 3, y = 2 0 3 6 9 6 3

14. b. a = 5, b = 4 14. c. x = 4, y = – 2

302 ~ Perfect Optional Mathematics Class 9

14. d. a = 2, b = 3, c – 3, d = 2 14. e. p = 2, q = 6, r = 4, s = 3

EXERCISE – 3.2A

1. a. a = – 4, b = 2, c = – 3 1. b. a = 3, b = 5, c = – 2

1. c. a = 1, b = – 5, c = 6, d = 9 1. d. a = 3, b = 1, c = 6, d = 4

2. a. – 1 4 2. b. 0 0 5 2 1 2 4 – 6
–   11 10 4 8 1 2. c. 4 – 3
2. d. 8 – 10 12
5 3 – 14 16 18

3. a. 9 3. b. – 2 1 3 3. c. a 2b – 2 6
3 0 – 2 0 –  5b – 3a 3. d. 6 – 1

0 – 7

4. a. 6 7 4. b. –  10 – 1 4. c. – 4 6 4. d. – 4 – 4
14 – 6 – 6 6 8 0 – 5 3

5. a. – 4 5 5. b. – 3 5 5. c. 2925 ––  412 5. d. – 3 4
4 11 5 10 3 9

0 13 16 19 3 4 – 1 – 3
5. e. 2 6 3 12 – 4 1
21 5. f. 92 5 6. a. 6. b.
9
2

6. c. x = 3, y = 10 6. d. x = 3, y = 4 7. a. – 2 1 2 – 1
4 – 5 7. b. – 3 – 4

– 6 7

7. c. – 2 – 3 – 4 – 1 4 0 8. a. 7 – 1 8. b. – 1 8
– 1 – 4 3 7. d. – 2 5 – 7 – 2 14 1 3

– 3 6 – 9

8. c. 11 0 8. d. – 2 16 9. a. P= 0 2 ,Q= 3 1
– 3 23 2 6 2 3 0 2

9. b. A = 1 2 ,B= 2 0 10. a. – 12, 3 10. b. 1, – 21, 2
3 1 4 1 11. d. 121 7 – 23

11. a. 0 3 – 8 11. b. 7 15 – 2 11. c. – 16 2 – 167 5 8 6
7 – 2 11 17 26 25 – 32 2 7

12. a. 1 – 3 12. b. 2 0 13. a. a = 13, b = 4, c = 5, d = 1
– 6 – 5 2 – 6

13. b. x = 2, z = 3, y = 5 13. c. x = – 1, y = 2, w = 141, z = – 1 14. a. 2 3
4 7

1 1 14. c. 1 2
14. b. 2 3 4
4 0

EXERCISE – 3.2B

1. a. (23) 1. b. (47) 1. c. 10 1. d. 2 0 Answers
9 0 3

2. a. 11 – 3 , 2 0 3 2. b. a = 2, b = 1 3. a. x = 8
9 16 8 16 6
5 – 4 9

3. b. x = 2 3. c. x = 1, y = – 1 3. d. x = 1, y = 2 3. e. a = 2, b = 10

Answers ~ 303

3. f. x = 1, y = 2 4. b. 2, – 7 5. a. 1 1 5. b. 1 0 3
3 4 4 – 2 5

5. c. 13 – 6 18 5. d. 5 – 2 8 13 17 5. f. 5 – 2 8
17 – 8 23 19 – 8 29 5. e. – 6 – 8 19 – 8 29

18 23

7. a. AB = 9 – 6 , BA = 2 – 4 12 , AB ≠ BA 9. a. 1 9. b. – 1
1 26 9 –12 – 15 – 2 2
– 8 – 14 21

10. a. – 7 12 10. b. 170 12. a. 14 10
11 –16 – 79 – 5 – 1

EXERCISE – 4.1A

1. a. (0, 4) 1. b. 2 , 1 2. a. (– 5, 4) 2. b. (9, 4)
3 4. b. 5:2
5. c. (– 2, 1)
3. b. (– 6, 11) 3. c. (– 6, 2) 4. a. 2:1
2. b. a = 5
4. c. 3:1 5. a. 4, 14 5. b. 5, 0 2. f. (0, 3)
3 4. c. (– 3, – 4), (– 6, 10)

6. a. – 4 6. b. 4

1. a. 45, 19 , (8, 11) EXERCISE – 4.1B
5
1. b. (0, – 1), (– 8, 15) 2. a. a = – 5

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

3. a. 10 units 3. b. 69 units 4. b. (0, 0)

4. d. (– 4, 2), (– 2, 8) 4. e. a = 8, b = 7 5. a. (15, 5), (5, 0) 5. b. (6, 4), (7, 5)

5. c. (– 2, 5), (0, 2), (2, – 1) 5. d. 5 units, (11, 10)

6. b. 52 units 6. c. (4, 5), (0, 3), (1, 4) 8. a. 1:2 8. b. P(1, 0), Q(6, 5)
4. a. 10x + 12y – 7 = 0
8. d. 130, 2
3

1. b, f 2. a, c, f EXERCISE – 4.2

3. a. 5

4. b. bx – ay = 0 4. c. 3x2 – y2 – 8x – 6y – 25 = 0

4. d. x2 – 3y2 + 2x – 2y + 2 = 0 4. e. 7x2 + 7y2 + 50ax + 7a2 = 0

5. a. x – 3 = 0 5. b. 3x – y = 0 5. c. x2 + y2 = 36 5. d. x2 + y2 – 4x + 8y = 5

5. e. 4x + 6y = 13 6. a. x2 + y2 – 5x + y + 6 = 0 6. b. 3x2 + 3y2 – 14x + 4y + 13 = 0

6. c. 4x2 + 4y2 – 15x + 9y + 18 = 0 7. a. x2 + y2 = 4 7. b. i. x2 + y2 = a2

7. b. ii. (m2 – n2)(x2 + y2 + a2) – 2ax(m2 + n2) = 0 8. b. 1:2

9. a. x + y = 4 9. b. 3x + 6y = 20

EXERCISE – 4.3

1. a. y = 6 1. b. y = – 3 1. c. x = 2 1. b. x = – 6
3. a. y = 0 3. b. y = – 2 4. a. x = – 2 4. b. x = 3

304 ~ Perfect Optional Mathematics Class 9

5. a. 1 5. b. 1 5. c. 3 5. d. – 3
3 6. b. 60° 6. c. 90° 6. d. 150°

6. a. 45°

7. a. 0 7. b. ∞ 7. c. 0 7. d. ∞
8. a. 1 8. b. – 2 8. c. 170 9. a. 6

9. b. 2 11. a. 8 11. b. 6

EXERCISE – 4.4

1. a. x – 3y + 12 = 0 1. b. x + 2y = 12 2. a. y = x – 2 2. b. y = –  3x + 1
2
3. a. y = 1  x + 2
3 3. b. 3x + y + 2 = 0 3. c. y = –  1  x + 3 4. a. y = – 3x
3
4. b. y = –  1  x 5. a. y = 3 x – 2 6. a. y = x + 5
3 7. a. 3x – y + 17 = 0 5. b. y = 12 x + 5 7. c. 3 x + y = 4
7. b. 6x – y = 34
6. b. 7x – y = 3

8. a. y = 3 x + 5 8. b. y – x + 5 = 0 or x + y + 5 = 0 9. a. x – y + 1 = 0 and x + y = 9

9. b. x – y = – 1 and x + y = 5 10. a. x – 3 y + 4 3 – 3 = 0

10. b. x – y + 1 = 0 11. a. m = 1 , c = – 3 11. b. m = – ba , c = – c
3 b

EXERCISE – 4.5

1. a. 2x + 3y = 6 1. b. 5x + 6y = 60 2. a. a = 3, b = 2; 2x + 3y = 6

2. b. a = 6, b = 6 3; 3x + y = 6 3 2. c. a = 8, b = 6; 3x + 4y = 24

3. a. x – y = 3 3. b. 9x + 8y = 6 4. a. 3x + 2y = 6 4. b. 4x – 3y + 12 = 0

5. a. x + y = 2 5. b. i. x + y = 7 5. b. ii. x – y = 1 5. c. x + 2y = 7

6. a. 3x – 2y = 12 6. b. 2x – y = 6 6. c. x + y = 4 6. d. 9x – 20y + 96 = 0

7. a. 2x + y = 6, x + 2y = 6 7. b. 6y – x = 6, 2x + y = 14

7. c. x + y = 2, 2x + y = 3 7. d. x + 2y = 4 8. a. 4x + 5y = 40 8. b. 2x + 3y = 6

9. a. 2 13 units 9. b. 4 2 unit

EXERCISE – 4.6

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

2. a. 3x + y = 10 2. b. x + 3y + 6 = 0 3. a. x –  3y + 6 = 0 3. b. x + 3y – 8 = 0

3. c. x + 3y = 8 3. d. i. x + 3y = 8 3. d. ii. No 4. a. Yes

4. b. 3x – y + 4 3 = 0 5. a. x + y = 2 2, x – y = 2 2 5. b. x + y ± 5 2 = 0 Answers

6. a. 4x + 3y = 25, 4x + 3y = – 25 6. b. 5x + 12y = 39, 5x + 12y = – 39

EXERCISE – 4.7

1. a. 13, 34 1. b. – 43, 3 1. c. 0, 0 1. d. 0, 3

Answers ~ 305

1. e. – 1 , 4 1. f. 1 , – 32 2. c. – 2, 2 2. d. 12, – 4 3
33 3

3. a. x cos 60° + y sin 60° = 8, a = 60°, p = 8 3. b. x cos 225° + y sin 225° = 5, a = 225°, p = 5

3. c. x cos 150° + y sin 150° = 1, a = 150°, p = 1 3. d. x cos 240° + y sin 240° = 2, a = 240°, p = 2

4. a. y = – ba x – bc, – ba, – bc 4. b. y = – ba x + b, – ba, b
4. c. y = – cot a x + p cosec a; – cot a, p cosec a

5. a. –x mc + y = 1, – mc , c 5. b. x + y = 1, p a, p a
c p p cos sin

cos a sin a

6. a. m 1 x – 1 1 y = c 1 6. b. b b2 x + a2 a b2 y = ab
m2 + m2 + m2 + a2 + + a2 + b2

7. a. a = 2 sin q, b = 2 cos q 8. a. i. y = 43 x – 54, x + y = 1, 43 x – 54 y = 1
– 45
5
3

8. a. ii. y = –  3 x + 121, x + y = 1, 3  x + 12 y = 141 8. b. p = 3, a = 210° 9. a. 9 sq. units
11 11 2

23 2

9. b. 356 sq. units 10. a. 52  46 sq. units 10. b. 7 sq. units

EXERCISE – 4.8

1. a. 3y + x + 7 = 0 1. b. 3x – 3y = 8 1. c. x –  3y + 4 + 3 = 0 1. d. 9x – 15y + 7 = 0

2. a. y + 5x – 11 = 0 2. b. 36x – 13y + 40 = 0 2. c. 5x + 8y – 41 = 0 2. d. x + y + 3 = 0

3. a. 3x + y – 2 – 3 3 = 0 3. b. 4x + 3y = 24 4. a. x – 2y + 13 = 0 4. b. 3 x – y + 2 3 + 1 = 0

5. a. 3x – 5y + 17 = 0 5. b. 3x – 4y = 0 6. a. y = x, x + y = – 6 6. b. 2x – y – 7 = 0, 47, – 72

7. a. bx + ay – 3ab = 0 7. b. 3x + 4y – 12 = 0, 5 8. a. 3 241, 5x + 4y – 22 = 0
8. b. 4 2, x + y – 6 = 0
9. a. 10 : 1 9. b. 10 : 1 10. a. 3 : 2

10. b. 2 : 5 externally 11. a. x + y – 5 = 0 11. b. 5x – 8y = 0 12. a. x – 5y + 18 = 0

12. b. x + y = 3

EXERCISE – 4.9

1. a. 9  units 1. b. 1730 units 1. c. 29  units 1. d. 3 units
2 34

2. a. 2 units 2. b. 3 units 2. c. 3 10 units 2. d. 14 units
5

3. a. 4 units 3. b. 3 units 3. c. 1 unit 3. d. 3 units
2 10

4. a. 7 or – 539 4. b. 12 or – 48 4. c. 5 or – 45 5. a. c 1 units
5. b. ab units m2 +

a2 + b2 5. c. p units

306 ~ Perfect Optional Mathematics Class 9

EXERCISE – 4.10A

1. a. 15 sq. units 1. b. 8 sq. units 1. c. 25.5 sq. units 2. a. 5 sq. units
2. b. 14 sq. units 2. c. 23 pq sq. units
4. a. 10 4. b. – 10 2. d. 12 (ab + bc + ca – a2 – b2 – c2) sq. units
5. b. 12 or 1
7. a. 40 sq. units 4. c. – 87 5. a. 0

7. b. 1821 sq. units

EXERCISE – 4.10B

3. a. 181 3. b. (7, 2) 4. a. (2, 3), (4, 6)
4. c. 2 : 1 : 1
4. b. 14 sq. units, 7 sq. units, 7 sq. units

5. a. 4 sq. units 5. b. 12 sq. units

EXERCISE – 5.1A

1. a. 162000" 1. b. 54020" 1. c. 253210" 2. a. 25.258º
2. b. 15.419º 3. b. 29.0308g
3. c. 4.8029g 2. c. 6.069º 3. a. 42.4003g 4. c. 282.04'
5. a. 25º 19' 37.8" 6. a. 60º, 66.67g
4. a. 4028' 4. b. 2103.48' 7. b. 34pc
8. c. 0.0351pc
5. b. 9º 4' 32.8" 5. c. 73º 27' 37.8" 10. a. 70g, 30g, 100g
12. a. 108º
6. b. 135º, 150g 6. c. 150°, 166.67g 7. a. p4c 12. e. 140º
13. c. 10
7. c. pc 8. a. 0.2241pc 8. b. 0.1266pc
3

9. a. 0.243 9. b. 0.449 9. c. 0.571

10. b. 63º, 72º, 45º 10. c. 36º, 63º, 81º 10. d. 90º, 63º, 27º

12. b. 120º 12. c. 128.571º 12. d. 135º

12. f. 144º 13. a. 5 13. b. 8

EXERCISE – 5.1B

1. a. 45º, 45º, 90º 1. b. 5090g, 6090g, 7090g 1. c. p6c, p3c, p2c 1. d. i. 80º, 55º, 45º

1. d. ii. 8889g, 6119g, 50g 1. d. iii. 49pc, 1316pc, pc
4

2. a. 27°, 63°, 90°; 30g, 70g, 100g; 32p0c, 72p0c, p2c 2. b. 36º, 72º, 108º, 144º, 40g, 80g, 120g, 160g Answers

3. a. 96º, 60º, 24º 3. b. 67º, 65º, 48º 3. c. 70g, 50g, 80g 4. a. 70g, 50g, 80g

4. b. 70º, 71p8c 4. c. 29pc, p3c, 49pc 5. a. 92p0c, 31p0c 5. b. p4c, 71p2c, p6c
6. a. 10 6. b. 8 7. a. 6, 8 7. b. 5, 10

Answers ~ 307

7. c. 24, 8 8. a. 51p2c 8. b. 97.5° 9. a. 72º, 63º, 45º

EXERCISE – 5.2A 1. d. 22cm
2. b. 2.25º
1. a. 14.67m 1. b. 11cm 1. c. 44cm 3. b. 38.18cm
1. e. 13.44m 1. f. 25cm 2. a. 30º
2. c. 60° 2. d. 8.59º 3. a. 8.4m 1. d. 76°21'49"
3. c. 35m 2. d. 5.5 km/hr
4. a. 1:2
EXERCISE – 5.2A 5. c. 22cm

1. a. 24º45', 27g 50' 1. b. 7º 1. c. 25°12'
2. a. 140m 2. b. 2m 2. c. 3.84m
2. e. 2793650.79km 3. a. 252m 3. b. 388800km
4. b. 3:1 5. a. 10.48cm 5. b. 28cm
5. d. 29.33cm

EXERCISE – 5.3A

1. a. RPRQ, RPRQ 1. b. 153, 152 1. c. 35, 34, 53, 54 2. a. ii. 45, 3
4. a. cos4f – sin4f 4. b. sin3q – cos3q 5

2. b. 2 , 1 , 1 4. c. 9 – 5 sec2q
3 2 2

4. d. 1 – tan8q 5. a. (tan A + cot A) (tan2A – tan A cot A + cot2A)

5. b. (1 + cos q) (1 – cos q) (1 + cos2q) 5. c. (cos a – 3) (cos a – 2)

5. d. (tan q + 2) (3 tan q – 4)

EXERCISE – 5.4

1. a. 153, 152 1. b. n 1. c. 34, 54 1. d. 2
7. a. ± 1 m2 – n2

7. e. 45 7. b. ± 1 7. c. 1 ± 2 7. d. ±2 5
7. i. ± 12 2

7. f. 494 7. g. ±  12 7. h. ±  3
7. j. ±  12 7. k. 1
7. l. ∞
EXERCISE – 5.5A

1. a. – sin q 1. b. cos q 1. c. – tan q 1. d. cos a
1. e. cot b 1. f. sin q 2. a. sin q 2. b. – cos q
2. c. – tan q 2. d. – cos b 2. e. cos q 2. f. – tan a

308 ~ Perfect Optional Mathematics Class 9

3. a. 23 3. b. – 23 3. c. 1 3. d. 1
3. e. – 12 3. f. – 1
4. c. 21 4. d. – 23 4. a. 23 4. b. 3
5. a. 0 5. b. 0 2
5. e. 0 5. f. 0
6. c. 35 4. e. 2 4. f. 3
8. b. 0 7. a. 1
8. c. 1 5. c. 0 5. d. 0
9. b. 1
10. b. – 31 6. a. 1 6. b. 1
10. f. 34
7. b. 2 2+ 3 8. a. 0
3. a. tan q 4 9. a. – 1
4. b. 1 10. a. 14
8. d. ∞ 10. e. – 4
4. f. – sec2­ q
9. c. 1 9. d. 1
10. c. 13 10. d. 27
10. g. 1 10. h. 4

EXERCISE – 5.5B

3. b. sec q. tan q 3. c. cot2q 4. a. 0
4. c. tan2a 4. d. cot2a 4. e. – 1

EXERCISE – 5.6A

1. a. 11 1. b. 3 1. c. 32 1. d. 1
2 25 2. c. – 1
4. a. 1
2. a. 7 2. b. 12 2. d. 3
3. b. 1 2

3. a. 14 4. b. 0

8. a. 15m 8. b. 45° 8. c. 40m

EXERCISE – 5.6B

4. a. 2 3 4. b. 2 3 4. c. 2 4. d. 3
4. e. 3 4. f. 4 4. g. 1 4. h. – 2

EXERCISE – 5.7A

1. a. 3 +1 1. b. 3 –1 1. c. 2 – 3 1. d. 2 + 3 Answers
2 2 2 2

1. e. 3 +1 1. f. – (2 + 3) 2. g. 3 +1 1. h. 3–1
2 2 2 2 22

EXERCISE – 5.7B

Answers ~ 309

4. a. 6565, 3563 4. b. 1, 0

8. a. sin A cos B cos C + cos A sin B cos C + cos A cos B sin C – sin A sin B sin C

8. b. 1ta–ntAan+BtatannBC+–tatannCC–tatannAA–tatannBAtatannCB

EXERCISE – 6.1A

2. a. x-component, y-component 2. b. x 2 – x1
y2 – y1

3. →a = 2 , →b = 5 , →c = 2 , →d = 4 , →e = – 3 , →r = – 6 , →g = 6 4. a. 4
5 – 1 – 6 2 7 – 2 2 1

4. b. 7 4. c. – 4 4. d. – 6 4. e. – 3
– 1 0 2 6

4. f. 6 5. a. – 1 5. b. 7 5. c. – 4
2 1 –  12 9

5. d. – 8 7. a. 12, 8 7. b. 4, 8 7. c. 18, 12
– 7

7. d. 6, 12 8. a. (4, 1) 8. b. (8, 2) 8. c. (6, 3)

8. d. (16, 4)

EXERCISE – 6.1B

1. 4 units 1. b. a2 + b2 units 1. c. 2 1. d. 30°

1. e. 45° 2. a. – 4 2. b. 41 2. c. 53.34°
– 5

3. a. 5 units, 37° 3. b. 13 units, 337.38° 3. c. 1 unit, 53° 3. d. 3 5 units, 243°

4. a. – 4 4. b. 2 4. c. – 8 4. d. 4
– 3 – 5 3 7

5. a. – 3 5. b. 73 units 5. c. 111° 6. c. 14, – 1
8

7. a. →a + →b 7. b. →c + →d 7. c. →b + →c + →d 7. d. →a + →b + →c + →d
8. b. →TO 8. c. →QP , →RQ
8. a. →TS , →QR 9. a. 8, 8

9. b. 2, 12 9. c. 12, 8 10. a. 2 10. b. ± 4, ± 3
2

EXERCISE – 6.1C

1. b. →a = k →b 1. c. 2m 2. b. | →a | 2. c. 3/5
3m →a 4/5

3. a. 4 , 41 units, 309°, 4/ 41 3. b. 41 units, 51°
– 5 – 5/ 41

4. b. 5, – 1, 13 units 4. c. (– 2, – 10), 3 , – 5
– 4 –  16

7. a. True 7. b. False 8. a. 3 , 2 , – 1
8. b. 10, 20, 10 1 4 3

EXERCISE – 6.2A

310 ~ Perfect Optional Mathematics Class 9

1. a. →AB + →BC = →AC 1. b. kx 2. a. 6 2. b. – 2
ky 8 – 2

2. c. 2 2. d. 4 3. b. – 3
8 0 5

4. a. 13 4. b. 7 , 7  2 4. c. 11 , 122 4. d. – 8 , 10 units
7 7 6

5. a. 10 5. b. 5→i – 4→j , 41 units 6. a. i. 9 6. a. ii. –   14
– 1 0

6. b. i. 4 6. b. ii. 3 7. a. i. 3 →i – 6 →j 7. b. 8 →i
– 2 8. b. 3 5

8. a. 2

EXERCISE – 6.2B

1. a. 7 , 5 2 units, 352° 1. b. 5 , 34 units, 329°
– 1 – 3

2. a. 16 2. b. 20 2. c. 37° 3. a. 4
12 8

3. b. 4 4. b. 153→i + 1123→j 5. a. 2 units, 135°, –  1  →i + 1  →j
8 22

5. b. 26 units, 1  →i + 5  →j 5. c. 3 units, 270°, 0
26 26 – 1

5. d. 218 units, 118°, 1 – 7 6. a. →AC = 6 , →DC = 4 , →EC = – 3 , →BE = 5
218 13 6 2 8 – 3

6. b. M→R = – 3 , →RU = 5 , →US = 1 7. a. →UT = →b , →QP = →a – →b, →US = 2→b – →a
– 4 5 – 4

7. b. →AC = →a + →b, A→D = 2→b, →CD = →b – →a, →DE = – →a , →EF = – →b, →FA = →a – →b

8. a. →a – →b, 2(→a – →b), 2→a – →b, →a – 2→b

8. b. →PR = →p + →q, →PO = 45 (→p + →q), →RO = – 45 (→p + →q), Q→O = 21 (→q – →p)

9. a. →u – →v, →u + →v 9. b. →QR = 9→v, →PR = 9→v – →u, →QS = 5→v+ →u, →RS = – 4→v + →u

11. a. 21 (→a + →b) 11. b. 13 (2→a + →b) 13. b. →a – →b + →c

EXERCISE – 7.1A

4. a. (– 3, 2) 4. b. A'(1, – 2), B'(– 3, – 5) 4. c. M'(– 1, 2)

5. a. P'(– 2, 3), Q'(4, – 3) 5. b. P'(2, – 3), Q'(– 4, 3) 5. c. P'(3, 2), Q'(– 3, – 4) 5. d. P'(– 3, – 2), Q'(3, 4)

5. e. P'(– 8, 3), Q'(– 2, – 3) 5. f. P'(2, 1), Q'(– 4, 7) 6. a. x = 3 6. b. x = 0

6. c. y = 0 6. d. x = y 6. e. y = 2 6. f. y = – x

EXERCISE – 7.1B Answers

1. a. A'(– 1, 1), B'(0, 5), C'(– 4, 3) 1. b. P'(1, – 3), Q'(4, – 3), R'(5, – 1), S'(3, – 1)

2. a. A'(6, 8), B'(– 4, 6), C'(0, 2) 2. b. A'(7, 6), B'(4, 5), C'(8, 2)

2. c. A'(– 1, 0), B'(1, 0), C'(3, 2), D'(1, 4), E'(– 1, 4), F'(– 2, 2)

Answers ~ 311

3. a. A'(– 1, 2), B'(– 1, – 1), C'(– 3, – 3), D'(– 5, – 1), E'(– 4, 2)

3. b. P'(1, 3), Q'(3, 4), R'(4, 1) 3. c. D'(4, – 7), E'(7, – 3), F'(8, – 5)

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

4. b. P'(2, 1), Q'(0, 1), R'(0, 3); P"(– 1, – 2), Q"(– 1, 0), R"(– 3, 0)

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

5. c. A'(– 6, 4), B'(– 3, 2), C'(– 7, 2) 5. d. A'(2, 2), B'(– 1, 4), C'(3, 4)

6. a. B'(4, – 4), C'(3, – 7) 6. b. B'(– 6, – 3), C'(– 3, – 9) 6. c. B'(– 4, 5), C'(– 6, 2)

7. a. A(2, 5), B(3, 2), C'(1, 3) 7. b. A(– 2, 3), B(– 4, 1), C(– 6, 4)

8. a. P(2, 3), Q(6, 2), R(4, 5) 8. b. A(2, 4), B(8, 2), C(4, – 2)

EXERCISE – 7.2A

2. a. – 100° 2. b. 130º 5. a. A'(– 3, 4) 5. b. P'(7, 2), Q'(3, 4)
6. a. M'(8, 0) 6. b. A'(– 1, – 1) 7. a. (– 2, 8) 7. b. (14, 4)
7. c. (– 11, – 1) 8. a. A(4, – 2) 8. b. B'(8, – 2) 8. c. M'(4, 1)
8. d. N'(– 4, 3) 8. e. P'(6, 4) 8. f. Q'(b, – a) 9. a. P'(5, 8)
9. b. Q'(9, 9) 9. c. M'(5, 3) 9. d. N'(– 2, 9) 9. e. S'(– 4, 0)
9. f. A'(– q + 1, p + 5)

EXERCISE – 7.2b

1. a. A'(– 5, 2), B'(– 7, 4), C'(– 4, 7) 1. b. A'(– 4, – 6), B'(– 2, – 3), C'(– 4, 0), D'(– 6, – 3)

2. a. P'(– 2, 1), Q'(– 4, 3), R'(– 1, 6) 2. b. P'(2, – 1), Q'(4, – 3), R'(1, – 6)

2. c. P'(– 1, – 2), Q'(– 3, – 4), R'(– 6, – 1) 3. a. A'(– 2, 2), B'(0, 3), C'(– 1, 7), D'(– 4, 5)

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

4. a. M'(– 1, 8), N'(1, 10), S'(4, 9) 4. b. B'(– 3, 4), C'(– 6, 4)

5. a. A'(1, 0), B'(4, – 2), C'(2, – 5), D'(5, – 6) 5. b. A'(– 1, 0), B'(– 4, 2), C'(– 2, 5), D'(– 5, 6)

5. c. A(0, – 1), B(– 2, – 4), C'(– 5, – 2), D'(– 6, – 5) 6. a. + 90º 6. b. B' (– 1, – 2), C'(2, 4)

7. a. (– 2, 1), – 90º 7. b. (2, 0), 90º

8. a. A'(2, 1), B'(5, 1), C'(1, 3) 8. b. A"(– 2, – 1), B"(– 5, – 1), C"(– 1, – 3)

9. a. P'(– 1, 2), Q'(– 3, 3), R'(– 2, 5) 9. b. P"(1, 6), Q"(2, 8), R"(4, 7)

10. a. – 90°, 0, 0 10. b. – 90°, 2, 2

EXERCISE – 7.3A

2. a. A'(4, 1) 2. b. P'(– 1, – 6) 2. c. M'(0, 2) 2. d. B'(– 2, – 1)
2. e. C'(6, – 11) 2. f. F'(2, – 4)
3. a. A→A' = – 4 3. b. 6
7 – 5

312 ~ Perfect Optional Mathematics Class 9

3. c. a 4. a. –  13 4. b. – 5 4. c. – 9
b 0 2 1

4. d. 9 5. a. – 1 5. b. A'(5, 0), A"(0, 5)
– 1 – 8

EXERCISE – 7.3B

1. a. P'(3, – 5), Q'(6, – 1) 1. b. X'(0, – 1), Y'(2, 2), Z'(6, 0)

1. c. W'(– 2, – 3), X'(0, 1), Y'(3, 0), Z'(1, – 4) 2. a. 3 , B'(7, 7), C'(1, – 2)
3

2. b. – 1 , A'(0, 4), B'(1, 2) 2. c. A'(– 4, – 7), B'(0, – 8), C'(– 2, – 4)
5

3. a. A'(4, 3), B'(10, 9), C'(6, 1) 3. b. A'(5, – 4), B'(11, 2), C'(7, – 6)
4. a. P'(2, – 3), Q'(4, – 3), R'(6, – 1), S'(3, – 1)
3. c. A'(0, – 2), B'(6, 4), C'(2, – 4) 5. a. A'(3, – 4), B'(5, – 5), C'(7, – 2)

4. b. P"(– 3, 0), Q"(– 1, 0), R"(1, 2), S"(– 2, 2)

5. b. A"(4, – 3), B"(5, – 5), C"(2, – 7) 5. c. A'''(– 3, – 4), B'''(– 5, – 5), C'''(– 7, – 2)

EXERCISE – 7.4A

2. a. k = 2 2. b. k = 12 3. a. A'(8, 4) 3. b. A'(2, 1)
3. c. A'(– 2, – 1) 3. d. A'(6, 3) 4. a. P'(3, 0) 4. b. P'(5, – 9)

5. a. A'(4, 6), B'(– 2, 8) 5. b. (1, – 2) 5. c. a = 4, b = – 2

EXERCISE – 7.4B

1. a. P'(– 4, – 10), Q'(6, – 8) 1. b. A'(2, 6), B'(8, 2), C'(6, 10) Answers
2. a. P'(– 8, 12), Q'(– 4, 4), R'(0, 8) 2. b. P'(4, – 6), Q'(2, – 2), R'(0, – 4)
2. c. P'(– 2, 3), Q'(– 1, 1), R'(0, 2) 2. d. P'(8, – 12), Q'(4, – 4), R'(0, – 8)
3. a. A'(12, 6), B'(12, 12), C'(6, 6), D'(6, 0) 3. b. A'(– 12, – 6), B'(– 12, – 12,), C'(– 6, – 6), D'(– 6, 0)
3. c. A'(2, 1), B'(2, 2), C'(1, 1), D'(1, 0) 3. d. A'(– 2, – 1), B'(– 2, – 2), C'(– 1, – 1), D'(– 1, 0)
4. a. A'(1, 3), B'(7, 3), C'(5, 7) 4. b. A"(– 3, 3), B"(– 6, 3), C"(– 5, 1)
5. a. A'(3, 8), B'(– 3, 10), C'(9, 6), D'(11, 4) 5. b. P'(– 1, 4), Q'(– 3, 0), R'(– 9, – 2), S'(– 7, 2)
6. a. A(3, 1), B(– 2, 2), C(2, 0) 6. b. X(3, 2), Y(9, 2), Z(5, 6)
7. a. 21, (3, – 1)
8. a. A'(1, – 3), B'(1, 3), C'(7, – 1) 7. b. (4, 1), 3
9. a. P'(– 3, – 1), Q'(– 1, – 1), R'(– 2, – 2) 8. b. A"(– 1, – 1), B"(– 1, 2), C"(2, 0)
9. c. P1(– 2, 5), Q1(– 2, 7), R1(– 1, 6) 9. b. P"(2, 3), Q"(4, 3), R"(3, 2)
9. d. P2(5, – 2), Q2(7, – 2), R2(6, – 1)

EXERCISE – 8.1A

1. a. 3nd, 6th, 9th 1. b. 2.5th, 7.5th 2. a. 7.2th 2. b. 28.8th
3. a. 30 3. b. 66 4. a. 7 4. b. 65

Answers ~ 313

5. a. 11.5 5. b. 64.5 6. a. 5.11 6. b. 14.5, 20

7. a. 39.1, 49, 66.5 7. b. 16.1, 18.5, 27.9 8. a. 10.16, 13.04, 16.16, 25.36

8. b. 22.25, 24.35, 33.45, 42.7 9. a. 12, 10.8 9. b. 29.5, 25.6

10. a. 26.2, 23.56 10. b. 14.5, 11.4 11. a. 12 11. b. 11

12. a. 7 12. b. 4 12. c. 10

EXERCISE – 8.1B

1. a. 260 1. b. 20 2. a. 14, 18 2. b. 25, 28
3. a. 30, 35 3. b. 62, 70 4. a. 140, 142, 152 4. b. 20, 40, 60
5. a. 25, 28, 20, 25 5. b. 15, 16, 10, 15

EXERCISE – 8.2

1. a. 7.5, 0.333 1. b. 36.5, 0.448 2. a. 1.75, 0.17 2. b. 57.5

2. c. 13, 0.071 2. d. 0.11 3. a. 75, 0.12 3. b. 1.5, 0.057

4. a. 4.5, 0.15 4. b. 4.29, 0.381 5. a. 4.5, 0.153 5. b. 3.45, 0.049

6. a. 4.375, 0.357; 4, 0.381 6. b. 24.17, 0.366; 23.5, 0.324

7. a. 10.4, 0.096 7. b. 13, 0.104 8. a. 97.54, ww0.125 8. b. 3.107, 0.176

9. a. 57.29, 0.073 9. b. 2.48, 0.165 10. a. 6.47, 0.199; 6.39, 0.183

10. b. 2.58, 0.044; 2.58, 0.044 11. a. 2.24, 0.187 11. b. 4.48, 0.1867

12. a. 11.92, 0.351 12. b. 9.85, 0.391 13. a. 6.22, 0.33 13. b. 1, 0.083

14. a. 3.07, 0.384 14. b. 4.23, 0.253 15. a. 8.12, 0.568 15. b. 3.44, 0.388

Contents Knowledge Specification Grid Higher ability Total Ques- Total Marks
tions
Algebra Each of 1 mark Understanding Application Each of 5 marks 21
Limits 8 5
Matrix 2 Each of 2 marks Each of 4 marks 1 2 9
Co-ordinate 1 – 4 15
Trigonometry 1 32 – 6 20
Vector 2 –1 1 8 10
Transformation 2 21 – 3 10
Statistics 1 21 1 3 10
Total 1 33 1 3
Marks – 2– – 38 100
10 –1 4 100
10 12 20
13 11
26 44

314 ~ Perfect Optional Mathematics Class 9

Model Evaluation

Optional Mathematics

Class : 9 Time : 3 hrs Full Marks : 100
Attempt all the questions:
[10×1 = 10]
Group ‘A'
P(4, – 2)
1. a. If f(x) = 3x – 2, then find f(– 2).

b. Denote the series 2 + 4 + 6 + 8 + 10 by S notation.

2. a. Evaluate : nl→im2   3x – 1
2

b. The elements of a matrix is defined as aij = i + j, find a22.

3. a. Find the length of PQ from the given figure. Q 3x – 4y + 5 = 0

b. If a straight line makes an angle of 30° in the positive side of X-axis, find the slope of the line.

4 a. Convert 80g into degrees.
b. If sin q = 53, find the value of cos q.
5. a. If →a = 2 and →b = 4 →a →b
5 10 , show that and are parallel.

b. If P'(4, – 5) is the image of P(x, y) after reflection on X-axis, then find P(x, y).

Group ‘B' [13×2 = 26]

6. a. If (2x, 2x + 3y) and (x + 4, 20) are equal ordered pairs, find the values of x and y.

b. If P(x) = 2x3 + 6x2 – 7x + 10 and Q(x) = 2x2 + 4x – 6, then find P(x) – 2Q(x).

c. If Sn = n2 + 1, then find tn.

7. a. If A – B = 5 6 and B = 4 5 , then find A.
7 8 –  1 3
2 – 3 1 2
b. If A = 4 5 and B = 3 4 , then find AB.

8. a. Prove that the points A(– 1, – 1), B(– 3, 0) and C(3, – 3) are collinear points.

b. Find the equation of the locus of a moving point which moves so that it is equidistant from two

fixed points (5, 1) and (3, – 2). O

9. a. The figure is a part of the circle of centre O and arc AB. If OB = 14cm pc
and AOB = p4c, calculate the length of the arc AB. 4

14cm

b. Prove that : 1 – cos4q = 1 + 2 cot2q. A B Answers
sin4q
?
10. a. If the magnitude of →a = m is 5 units, find the value of m.
4

Answers ~ 315

b. In triangle ABC, M is the mid point of BC, then prove that:
→AB + →AC = 2→AB

c. 7, 12, x + 5, 2x – 3, 2x, 3x – 3 and 47 are in ascending order. If its median is 21 find x.

Group ‘C' [11×4=44]

11. If f(m + 2) = f(m) + f(2), prove that : f(0) = 0 and f(– 2) = – f(2).

12. Find the product of f(x) = x2 + 3 and g(x) = x2 + x – 1.

13. Evaluate : lim x2 + 3x – 4
x→1 x –1

14. If A + B = –  1 9 and 2A – B = 7 0 , then find A and B.
7 4 2 8

15. The four vertices of a parallelogram ABCD are A(a, b), B(6, 3), C(9, 6) and D(6, 5). Find the

co-ordinates of A which is opposite to the vertex D.

16. If 3 sin q + 4 cos q = 5, then prove that tan q = 43.

17. Find the value of x: cosec (90° + a) + x cos (– a) . cot (90° + a) = sin (90° + a).

18. If A + B = pc , prove that: (1 + tan A) (1 + tan B) = 2.
4

19. P(2, 2), Q(5, 2), R(8, 4) and S(1, 4) are the vertices of a trapezium. Find its image under enlargement

with centre at (4, – 1) and scale factor 2.

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

Weight (in kg) 10 20 30 40 50

No. of workers 4 5 6 3 2

21. Calculate the values of D3 and P13, from given data :
16, 18, 20, 13, 25, 27, 33, 17, 25, 38, 40, 15, 18

Group ‘D' [4×5 = 40]

22. In the following pattern of numbers:

a. add one more figure in the same pattern.

b. find the formula for the nth term.

23. If P(a, b) lies on the line 6x – y = 1 and Q(b, a) lies on the line 2x – 5y = 5, find the equation of PQ.

24. In the given figure, ABCDEF is a regular hexagun, then shown that:
→AC + A→D + →EA + →FA = 3→AB

25. A(5, 6), B(1, 2) and C(4, 2) are the vertices of triangle ABC. Find the vertices of the image of
triangle ABC under the rotation through + 90° about the origin. Also draw both triangle on the same
graph.
THE END

316 ~ Perfect Optional Mathematics Class 9