Answers part i

268 MATHEMATICS

ANSWERS
EXERCISE 1.1
1. (i) Neither reflexive nor symmetric nor transitive.
(ii) Neither reflexive nor symmetric nor transitive.
(iii) Reflexive and transitive but not symmetric.
(iv) Reflexive, symmetric and transitive.
(v) (a) Reflexive, symmetric and transitive.
(b) Reflexive, symmetric and transitive.
(c) Neither reflexive nor symmetric nor transitive.
(d) Neither reflexive nor symmetric nor transitive.
(e) Neither reflexive nor symmetric nor transitive.
3. Neither reflexive nor symmetric nor transitive.
5. Neither reflexive nor symmetric nor transitive.
9. (i) {1, 5, 9}, (ii) {1} 12. T1 is related to T3.
13. The set of all triangles 14. The set of all lines y = 2x + c, c ∈ R
15. B 16. C

EXERCISE 1.2
1. No
2. (i) Injective but not surjective (ii) Neither injective nor surjective
(iii) Neither injective nor surjective (iv) Injective but not surjective
(v) Injective but not surjective
7. (i) One-one and onto (ii) Neither one-one nor onto.
9. No 10. Yes 11. D 12. A

EXERCISE 1.3
1. gof = {(1, 3), (3,1), (4,3)}
3. (i) (gof ) (x) = | 5 | x |– 2|, (fog) (x) = |5x – 2|
(ii) (g o f ) (x) = 2x, (f o g) (x) = 8x
4. Inverse of f is f itself

ANSWERS 269

5. (i) No, since f is many-one (ii) No, since g is many-one.
(iii) Yes, since h is one-one-onto.
2y y −3
6. f –1 is given by f –1 (y) = , y ≠ 1 7. f –1 is given by f –1 (y) =
1− y 4
–1
11. f is given by f –1 (a) = 1, f –1 (b) = 2 and f –1 (c) = 3.
13. (C) 14. (B)

EXERCISE 1.4
1. (i) No (ii) Yes (iii) Yes (iv) Yes (v) Yes
2. (i) ∗ is neither commutative nor associative
(ii) ∗ is commutative but not associative
(iii) ∗ is both commutative and associative
(iv) ∗ is commutative but not associative
(v) ∗ is neither commutative nor associative
(vi) ∗ is neither commutative nor associative

3. Λ 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5

4. (i) (2 * 3) * 4 = 1 and 2 * (3 * 4) = 1 (ii) Yes (iii) 1
5. Yes
6. (i) 5 * 7 = 35, 20 * 16 = 80 (ii) Yes (iii) Yes (iv) 1 (v) 1
7. No 8. ∗ is both commutative and associative; ∗ does not have any identity in N
9. (ii) , (iv), (v) are commutative; (v) is associative.
11. Identity element does not exist.
12. (ii) False (ii) True 13. B

270 MATHEMATICS

Miscellaneous Exercise on Chapter 1

y−7
1. g ( y) = 2. The inverse of f is f itself
10
3. x4 – 6x3 + 10x2 – 3x 8. No 10. n!
11. (i) F–1 = {(3, a), (2, b), (1, c)}, (ii) F–1 does not exist 12. No
15. Yes 16. A 17. B 18. No
19. B

EXERCISE 2.1
−π π π −π
1. 2. 3. 4.
6 6 6 3
2π π π π
5. 6. − 7. 8.
3 4 6 6
3π −π 3π 2π
9. 10. 11. 12.
4 4 4 3
13. B 14. B

EXERCISE 2.2
1 −1 π x π
5. tan x 6. – sec–1 x 7. 8. −x
2 2 2 4
−1 x −1 x π
9. sin 10. 3tan 11. 12. 0
a a 4
x+ y 1 1 π
13. 1 − xy 14. 15. ± 16.
5 2 3
−π 17
17. 18. 19. B 20. D
4 6
21. B

π π π 1
1. 2. 13. x= 14. x=
6 6 4 3
15. D 16. C 17. C

ANSWERS 271

EXERCISE 3.1
5
1. (i) 3 × 4 (ii) 12 (iii) 19, 35, – 5, 12,
2
2. 1 × 24, 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3, 12 × 2, 24 × 1; 1 × 13, 13 × 1
3. 1 × 18, 2 × 9, 3 × 6, 6 × 3, 9 × 2, 18 × 1; 1 × 5, 5 × 1
⎡ 9⎤
⎢2 ⎡ 1⎤ ⎡9 25 ⎤
2⎥ 1
4. (i) ⎢ ⎥ (ii) ⎢ 2⎥ (iii) ⎢ 2 2⎥
⎢9 ⎢ ⎥ ⎢ ⎥
8⎥ ⎣2 1⎦ ⎣8 18 ⎦
⎢2
⎣ ⎥
⎦
⎡ 1 1⎤
⎢1 2 0 2 ⎥
⎢ ⎥ ⎡1 0 −1 −2⎤
⎢ 5 2 3 1 ⎥ (ii) ⎢ 3 2 1 0 ⎥
5. (i) ⎢
2 2 ⎥ ⎢ ⎥
⎢ ⎥ ⎢5 4 3 2 ⎥
⎣ ⎦
⎢4 7 3 5 ⎥
⎢
⎣ 2 2⎥⎦
6. (i) x = 1, y = 4, z = 3
(ii) x = 4, y = 2, z = 0 or x = 2, y = 4, z = 0
(iii) x = 2, y = 4, z=3
7. a = 1, b = 2, c = 3, d = 4
8. C 9. B 10. D

EXERCISE 3.2

⎡3 7⎤ ⎡1 1 ⎤
1. (i) A + B = ⎢ (ii) A − B = ⎢ ⎥
⎣1 7⎥
⎦ ⎣5 −3⎦
⎡8 7⎤ ⎡ −6 26⎤ ⎡11 10⎤
(iii) 3A − C = ⎢
2⎥
(iv) AB =⎢ ⎥ (v) BA = ⎢ ⎥
⎣6 ⎦ ⎣ 1 19 ⎦ ⎣11 2 ⎦
⎡ 2a 2b ⎤ ⎡ ( a + b )2 (b + c) 2 ⎤
2. (i) ⎢ ⎥ (ii) ⎢ ⎥
⎢ (a − c ) (a − b ) 2 ⎥
2
⎣ 0 2a ⎦ ⎣ ⎦
⎡11 11 0 ⎤
⎢ ⎥ ⎡1 1⎤
(iii) ⎢16 5 21⎥ (iv) ⎢1 1⎥
⎢ 5 10 9 ⎥ ⎣ ⎦
⎣ ⎦

272 MATHEMATICS

⎡2 3 4 ⎤
⎡ a 2 + b2 0⎤ ⎢ ⎥ ⎡ − 3 − 4 1⎤
3. (i) ⎢ ⎥ (ii) ⎢ 4 6 8 ⎥ (iii) ⎢8 13 9⎥
⎢ 0
⎣ a +b ⎥
2 2
⎦ ⎣ ⎦
⎢ 6 9 12⎥
⎣ ⎦
⎡14 0 42⎤ ⎡ 1 2 3⎤
⎢18 −1 56 ⎥ ⎢ 1 4 5⎥ ⎡14 −6⎤
(iv) ⎢ ⎥ (v) ⎢ ⎥ (vi) ⎢4 5⎥
⎢ 22 −2 70 ⎥ ⎢ −2 2 0⎥ ⎣ ⎦
⎣ ⎦ ⎣ ⎦
⎡ 4 1 −1⎤ ⎡ −1 −2 0⎤
4. A+B = ⎢ 9 2 7 ⎥ , B − C = ⎢ 4 −1 3⎥
⎢ ⎥ ⎢ ⎥
⎢ 3 −1 4 ⎥
⎣ ⎦ ⎢ 1 2 0⎥
⎣ ⎦
⎡0 0 0 ⎤
⎢0 0 0 ⎥ ⎡1 0 ⎤
5. ⎢ ⎥ 6. ⎢0 1 ⎥
⎢0 0 0 ⎥ ⎣ ⎦
⎣ ⎦
⎡ 2 −12 ⎤ ⎡2 13 ⎤
⎡5 0 ⎤ ⎡ 2 0⎤ ⎢ 5 ⎥, ⎢5 5⎥
7. (i) X=⎢ ⎥ , Y = ⎢1 1 ⎥ (ii) X = ⎢ 5 ⎥ Y= ⎢ ⎥
⎣1 4 ⎦ ⎣ ⎦ ⎢ −11 3 ⎥ ⎢14 −2⎥
⎢ 5
⎣ ⎥
⎦ ⎢5
⎣ ⎥
⎦
⎡ −1 −1⎤
8. X=⎢ ⎥ 9. x = 3, y = 3 10. x = 3, y = 6, z = 9, t = 6
⎣ −2 −1⎦
11. x = 3, y = – 4 12. x = 2, y = 4, w = 3, z = 1
⎡ 1 −1 −3 ⎤
⎢ −1 −1 −10⎥
15. ⎢ ⎥ 17. k = 1
⎢ −5 4
⎣ 4 ⎥
⎦
19. (a) Rs 15000, Rs 15000 (b) Rs 5000, Rs 25000
20. Rs 20160 21. A 22. B

EXERCISE 3.3

⎡ −1 3 2⎤
⎡ 1 ⎤ ⎡ 1 2⎤ ⎢ ⎥
1. (i) ⎢5 2 −1⎥ (ii) ⎢ ⎥ (iii) ⎢5 5 3⎥
⎣ ⎦ ⎣ −1 3⎦ ⎢6
⎣ 6 −1⎥
⎦

ANSWERS 273

⎡ 0 0 0⎤ ⎡ 0 a b⎤
⎡ − 4 5⎤ ⎢ 0 0 0⎥ , ⎢ − a 0 c ⎥
4. ⎢ 1 6⎥ 9.
⎢ ⎥ ⎢ ⎥
⎣ ⎦
⎢ 0 0 0⎥ ⎢ −b − c 0⎥
⎣ ⎦ ⎣ ⎦

⎡3 3 ⎤ ⎡ 0 2⎤
10. (i) A = ⎢ ⎥+ ⎢ −2 0 ⎥
⎣3 −1⎦ ⎣ ⎦

⎡ 6 −2 2 ⎤ ⎡0 0 0⎤
(ii) A = ⎢−2
⎢ 3 ⎥ + ⎢0
− 1⎥ ⎢ 0 0⎥
⎥
⎢ 2
⎣ −1 3 ⎥
⎦ ⎢0
⎣ 0 0⎥
⎦

⎡ 1 −5 ⎤ ⎡ 5 3⎤
⎢ 3 2 2⎥ ⎢ 0 2 2⎥
⎢ ⎥ ⎢ ⎥
(iii) A = ⎢
1
−2 −2 ⎥ + ⎢ −5 3⎥
⎡ 1 2⎤ ⎡ 0 3 ⎤
A=⎢
⎢ 2 ⎥ ⎢ 2
0
⎥
(iv) ⎥+⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎣ 2 2⎦ ⎣ −3 0 ⎦
⎢ −5 −2 2⎥ ⎢ −3 −3 0⎥
⎢ 2
⎣ ⎥
⎦ ⎢
⎣ 2 ⎥
⎦
11. A 12. B

EXERCISE 3.4

⎡ 3 1⎤
⎢ 5 5⎥ ⎡ 1 −1⎤ ⎡ 7 −3⎤
1. ⎢ ⎥ 2. ⎢ −1 2 ⎥ 3. ⎢ −2 1 ⎥
⎢ −2 1⎥ ⎣ ⎦ ⎣ ⎦
⎢ 5
⎣ 5⎥
⎦
⎡ −7 3 ⎤ ⎡ 4 −1⎤ ⎡ 3 −5 ⎤
4. ⎢ 5 −2 ⎥ 5. ⎢ −7 2 ⎥ 6. ⎢ −1 2 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ 2 −1⎤ ⎡ 4 −5 ⎤ ⎡ 7 −10⎤
7. ⎢ −5 3 ⎥ 8. ⎢ −3 4 ⎥ 9. ⎢ −2 3 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ 1⎤
⎢1 ⎡ −1 3⎤
2⎥
10. ⎢ ⎥ 11. ⎢ −1 ⎥ 12. Inverse does not exist.
⎢2 3⎥ ⎢ 1⎥
⎢
⎣ 2⎥
⎦ ⎣2 ⎦

274 MATHEMATICS

⎡2 3⎤
13. ⎢1 2⎥
14. Inverse does not exist.
⎣ ⎦

⎡ −2 3⎤ ⎡ −2 −3 ⎤
⎢5 0 ⎢1
5⎥ 5 5⎥ −1 1 ⎤
⎢ ⎥ ⎢ ⎥ ⎡ 3
⎢ −1
15. ⎢
1
0⎥ ⎢ −2
16. ⎢
4 11 ⎥
17.
⎢ −15 6 −5⎥
5 5 ⎥ 5 25 25 ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ 5 −2 2 ⎥
⎣ ⎦
⎢ 2 1 −2 ⎥ ⎢ −3 1 9⎥
⎢5
⎣ 5 5⎥ ⎦ ⎢5
⎣ 25 25 ⎥
⎦

18. D


1 1 1
6. x=± ,y=± ,z=±
2 6 3

7. x = – 1 9. x = ± 4 3
10. (a) Total revenue in the market - I = Rs 46000
Total revenue in the market - II = Rs 53000
(b) Rs 15000, Rs 17000

⎡ 1 −2 ⎤
11. X=⎢ ⎥ 13. C 14. B 15. C
⎣2 0 ⎦

EXERCISE 4.1
1. (i) 18 2. (i) 1, (ii) x3 – x2 + 2
5. (i) – 12, (ii) 46, (iii) 0, (iv) 5 6. 0

7. (i) x = ± 3 , (ii) x = 2 8. (B)

EXERCISE 4.2
15. C 16. C

ANSWERS 275

EXERCISE 4.3

15 47
1. (i) , (ii) , (iii) 15
2 2
3. (i) 0, 8, (ii) 0, 8 4. (i) y = 2x, (ii) x – 3y = 0 5. (D)

EXERCISE 4.4
1. (i) M11 = 3, M12 = 0, M21 = – 4, M22 = 2, A11 = 3, A12 = 0, A21 = 4, A22 = 2
(ii) M11 = d, M12 = b, M21 = c, M22 = a
A11 = d, A12= – b, A21 = – c, A22 = a
2. (i) M11= 1, M12= 0, M13 = 0, M21 = 0, M22 = 1, M23 = 0, M31 = 0, M32 = 0, M33 = 1,
A11= 1, A12= 0, A13= 0, A21= 0, A22= 1, A23= 0, A31= 0, A32= 0, A33= 1
(ii) M11= 11, M12= 6, M13= 3, M21= –4, M22= 2, M23= 1, M31= –20, M32= –13, M33= 5
A11=11, A12= – 6, A13= 3, A21= 4, A22= 2, A23= –1, A31= –20, A32= 13, A33= 5
3. 7 4. (x – y) (y – z) (z – x) 5. (D)

EXERCISE 4.5

⎡ 3 1 −11⎤
⎡ 4 −2⎤ ⎢ −12 5 −1 ⎥ 1 ⎡ 3 2⎤
1. ⎢ −3 1 ⎥ 2. ⎢ ⎥ 5.
14 ⎢ − 4 2 ⎥
⎣ ⎦ ⎣ ⎦
⎢ 6 2 5 ⎥
⎣ ⎦

⎡10 −10 2 ⎤ ⎡ −3 0 0⎤
1 ⎡ 2 −5⎤ 1 ⎢ −1 ⎢
0 5 − 4⎥ 3 −1 0⎥
13 ⎢ 3 −1⎥
6. 7. 8.
⎣ ⎦ 10 ⎢ ⎥ 3 ⎢ ⎥
⎢0
⎣ 0 2⎥⎦ ⎢ −9 −2 3⎥
⎣ ⎦

⎡ −1 5 3⎤ ⎡ −2 0 1 ⎤ ⎡1 0 0 ⎤
−1 ⎢ ⎢ 9 2 −3⎥ ⎢ 0 cos α
9. − 4 23 12 ⎥ 10. 11. ⎢ sin α ⎥
3 ⎢ ⎥ ⎢ ⎥ ⎥
⎢ 1 −11 − 6⎥
⎣ ⎦ ⎢ 6 1 −2⎥
⎣ ⎦ ⎢ 0 sin α
⎣ – cos α ⎥
⎦

⎡ −3 4 5⎤
⎡ 2 −1⎤
A = ⎢ 9 −1 − 4 ⎥
1 −11 ⎢
13. ⎢1 3 ⎥ 14. a = – 4, b = 1 15. ⎥
7 ⎣ ⎦ 11
⎢ 5 −3 −1 ⎥
⎣ ⎦

276 MATHEMATICS

⎡ 3 1 −1⎤
1⎢
1 3 1⎥
16. 4⎢ ⎥ 17. B 18. B
⎢−1 1 3⎥
⎣ ⎦

EXERCISE 4.6
1. Consistent 2. Consistent 3. Inconsistent
4. Consistent 5. Inconsistent 6. Consistent

−5 12 −6 −19
7. x = 2, y = – 3 8. x= , y= 9. x= , y=
11 11 11 11

1 −3
10. x = –1, y = 4 11. x = 1, y = , z =
2 2
12. x = 2, y = –1, z = 1 13. x = 1, y = 2, z = –1
14. x = 2, y = 1, z = 3

⎡ 0 1 −2 ⎤
⎢ −2 9 −23⎥
15. ⎢ ⎥ , x = 1, y = 2, z = 3
⎢ −1 5 −13⎥
⎣ ⎦

16. cost of onions per kg = Rs 5
cost of wheat per kg = Rs 8
cost of rice per kg = Rs 8


⎡9 −3 5 ⎤
−a ⎢−2 1 0⎥
3. 1 5. x= 7. ⎢ ⎥
3 ⎢1
⎣ 0 2⎥⎦
9. – 2(x3 + y3) 10. xy 16. x = 2, y = 3, z = 5
17. A 18. A 19. D

ANSWERS 277

EXERCISE 5.1
2. f is continuous at x = 3
3. (a), (b), (c) and (d) are all continuous functions
5. f is continuous at x = 0 and x = 2; Not continuous at x = 1
6. Discontinuous at x = 2 7. Discontinuous at x = 3
8. Discontinuous at x = 0 9. No point of discontinuity
10. No point of discontinuity 11. No point of discontinuity
12. f is continuous at x = 1 13. f is not continuous at x = 1
14. f is not continuous at x = 1 and x = 3
15. x = 1 is the only point of discontinuity
2
16. Continuous 17. a =b+
3
18. For no value of λ, f is continuous at x = 0 but f is continuous at x = 1 for any
value of λ.
20. f is continuous at x = π 21. (a), (b) and (c) are all continuous
22. Cosine function is continuous for all x ∈ R; cosecant is continuous except for
π
x = nπ, n ∈ Z; secant is continuous except for x = (2n + 1) , n ∈ Z and
2
cotangent function is continuous except for x = nπ, n ∈ Z
23. There is no point of discontinuity.
24. Yes, f is continuous for all x ∈ R 25. f is continuous for all x ∈ R
3 −2
26. k = 6 27. k= 28. k=
4 π
9
29. k= 30. a = 2, b = 1
5
34. There is no point of discontinuity.

EXERCISE 5.2
2
1. 2x cos (x + 5) 2. – cos x sin (sin x) 3. a cos (ax + b)
sec (tan x).tan (tan x ).sec 2 x
4.
2 x

5. a cos (ax + b) sec (cx + d) + c sin (ax + b) tan (cx + d) sec (cx + d)
6. 10x4 sinx5 cosx5 cosx3 – 3x2 sinx3 sin2 x5

278 MATHEMATICS

−2 2 x sin x
7. 2 2 8. −
sin x sin 2 x 2 x

EXERCISE 5.3

cosx − 2 2 a
1. 2. 3. −
3 cos y − 3 2by + sin y

sec 2 x − y (2 x + y ) (3x 2 + 2 xy + y 2 )
4. 5. − 6. −
x + 2y −1 ( x + 2 y) ( x 2 + 2 xy + 3 y 2 )

y sin xy sin 2 x 2 3
7. sin 2 y − x sin xy 8. 9. 10.
sin 2 y 1 + x2 1 + x2

2 −2 −2 2
11. 12. 13. 14.
1 + x2 1 + x2 1 + x2 1 − x2

2
15. −
1 − x2

EXERCISE 5.4

e x (sin x − cosx) esin −1 x
1. , x ≠ nπ, n ∈ Z 2. , x ∈( − 1,1)
sin 2 x 1 − x2

e − x cos (tan −1 e – x )
−
3
3. 3 x2 e x 4.
1+ e −2 x

π x2
, n ∈N 6. e x + 2 x e + 3 x 2 e x + 4 x 3e x + 5 x 4 e x
3 4 5
5. – ex tan ex, e x ≠ (2n + 1)
2
x
e 1
7. ,x>0 8. ,x>1
x x log x
4 xe

( x sin x ⋅ log x + cos x) , ⎛1 ⎞
9. − x > 0 10. − ⎜ + e x ⎟ sin (log x + e x ), x > 0
x (log x) 2 ⎝x ⎠

ANSWERS 279

EXERCISE 5.5
1. – cos x cos 2x cos 3x [tan x + 2 tan 2x + 3 tan 3x]
1 ( x − 1) ( x − 2) ⎡ 1 1 1 1 1 ⎤
2. ⎢ x −1+ x − 2 − x − 3 − x − 4 − x −5⎥
2 ( x − 3)( x − 4)( x − 5) ⎣ ⎦
⎡ cos x ⎤
3. (log x )cos x ⎢ − sin x log (log x) ⎥
⎣ x log x ⎦
4. xx (1 + log x) – 2sin x cos x log 2
5. (x + 3) (x + 4)2 (x + 5)3 (9x2 + 70x + 133)

1 ⎞ ⎡ x 2 −1 1 ⎤
x 1
⎛ 1+ ⎛ x + 1 − log x ⎞
6. ⎜ x + ⎟ ⎢ 2 + log ( x + ) ⎥ + x x ⎜ ⎟
⎝ x ⎠ ⎣ x +1 x ⎦ ⎝ x2 ⎠
x-1 logx–1
7. (log x) [1 + log x . log (log x)] + 2x . logx
1 1
8. (sin x)x (x cot x + log sin x) +
2 x − x2
⎡ sin x ⎤
9. x sinx ⎢ + cos x log x ⎥ + (sin x)cos x [cos x cot x – sin x log sin x]
⎣ x ⎦
4x
10. x x cosx [cos x . (1 + log x) – x sin x log x] –
( x − 1) 2
2

1
11. (x cos x)x [1 – x tan x + log (x cos x)] + (x sin x) x ⎡ x cot x + 1 − log ( x sin x ) ⎤
⎢ ⎥
⎣ x2 ⎦
yx y −1 + y x log y y ⎛ y − x log y ⎞
12. − 13. ⎜ ⎟
x y log x + xy x −1 x ⎝ x − y log x ⎠
y tan x + log cos y y ( x −1)
14. 15.
x tan y + log cos x x ( y + 1)
⎡ 1 2x 4 x3 8x7 ⎤
16. (1 + x) (1 + x2) (1 +x4) (1 + x8) ⎢ + + + ⎥ ; f ′(1) = 120
⎣ 1 + x 1 + x 2 1 + x 4 1 + x8 ⎦
17. 5x4 – 20x3 + 45x2 – 52x + 11

EXERCISE 5.6

b 1
1. 2t2 2. 3. – 4 sin t 4. −
a t2

280 MATHEMATICS

cos θ − 2cos 2θ θ
5. 6. − cot 7. – cot 3t 8. tan t
2sin 2θ − sin θ 2
b
9. cosec θ 10. tan θ
a

EXERCISE 5.7
1. 2 2. 380 x18 3. – x cos x – 2 sin x
1
4. − 5. x(5 + 6 log x) 6. 2ex (5 cos 5x – 12 sin 5x)
x2
2x
7. 9 e6x (3 cos 3x – 4 sin 3x) 8. −
(1 + x 2 ) 2
(1 + log x) sin (log x) + cos (log x)
9. − 10. −
( x log x) 2 x2
12. – cot y cosec2 y

1. 27 (3x2 – 9x + 5)8 (2x – 3) 2. 3sinx cosx (sinx – 2 cos4 x)

3. (5 x ) 3cos 2 x ⎡ ⎤
3cos 2 x
⎢ − 6sin 2 x log 5 x ⎥
⎣ x ⎦
⎡ x ⎤
3 x ⎢ cos −1 ⎥
1 2
4. 5. − ⎢ + ⎥
2 1 − x3 3

⎢ 4 − x 2 x + 7 (2 x + 7) 2
2
⎥
⎣ ⎦
1 log x ⎡ 1 log (log x) ⎤
6.
2
7. (log x ) ⎢x +
⎣ x ⎥, x > 1
⎦
8. (a sin x – b cos x) sin (a cos x + b sin x)
9. (sinx – cosx)sin x – cos x (cosx + sinx) (1 + log (sinx – cos x)), sinx > cosx
10. xx (1 + log x) + ax a–1 + ax log a
2
−3 ⎡ x2 − 3 ⎤ 2 ⎡ x2 ⎤
11. xx ⎢ + 2 x log x ⎥ + ( x − 3) x ⎢ + 2 x log( x − 3)⎥
⎣ x ⎦ ⎣x −3 ⎦

ANSWERS 281

6 t sec3 t π
12. cot 13. 0 17. ,0 < t <
5 2 at 2

EXERCISE 6.1
2
1. (a) 6π cm /s (b) 8π cm2/s
8
2. cm2/s 3. 60π cm2/s 4. 900 cm3/s
3

5. 80π cm2/s 6. 1.4π cm/s
7. (a) –2 cm/min (b) 2 cm2/min
1 8
8. cm/s 9. 400π cm3/s 10. cm/s
π 3

⎛ −31 ⎞
11. (4, 11) and ⎜ − 4, ⎟ 12. 2π cm3/s
⎝ 3 ⎠

27 1
13. π (2 x + 1) 2 14. cm/s 15. Rs 20.967
8 48π

16. Rs 208 17. B 18. D

EXERCISE 6.2

⎛3 ⎞ ⎛ 3⎞
4. (a) ⎜ , ∞ ⎟ (b) ⎜ − ∞, ⎟
⎝4 ⎠ ⎝ 4⎠

5. (a) (– ∞, – 2) and (3, ∞) (b) (– 2, 3)
6. (a) Strictly decreasing for x < – 1 and strictly increasing for x > – 1

3 3
(b) Strictly decreasing for x > − and strictly increasing for x < −
2 2
(c) Strictly increasing for – 2 < x < – 1 and strictly decreasing for x < – 2 and
x>–1
9 9
(d) Strictly increasing for x < − and strictly decreasing for x > −
2 2

282 MATHEMATICS

(e) Strictly increasing in (1, 3) and (3, ∞), strictly decreasing in (– ∞, –1)
and (– 1, 1).
8. 0 < x < 1 and x > 2 12. A, B
13. D 14. a = – 2 19. D

EXERCISE 6.3

−1
1. 764 2. 3. 11 4. 24
64
−a
5. 1 6. 7. (3, – 20) and (–1, 12)
2b
8. (3, 1) 9. (2, – 9)
10. (i) y + x +1 = 0 and y + x – 3 = 0
11. No tangent to the curve which has slope 2.
1
12. y= 13. (i) (0, ± 4) (ii) (± 3, 0)
2
14. (i) Tangent: 10x + y = 5; Normal: x – 10y + 50 = 0
(ii) Tangent: y = 2x + 1; Normal: x + 2y – 7 = 0
(iii) Tangent: y = 3x – 2; Normal: x + 3y – 4 = 0
(iv) Tangent: y = 0; Normal: x = 0
(v) Tangent: x + y − 2 = 0; Normal x = y
15. (a) y – 2x – 3 = 0 (b) 36 y + 12x – 227 = 0
17. (0, 0), (3, 27) 18. (0, 0), (1, 2), (–1, –2)
19. (1, ± 2) 20. 2x + 3my – am2 (2 + 3m2) = 0
21. x + 14y – 254 = 0, x + 14y + 86 = 0
22. ty = x + at2, y = – tx + 2at + at3
x x0 y y0 y− y x−x
24. 2
− 2 = 1, 2 0 + 2 0 = 0
a b a y0 b x0
25. 48x – 24y = 23 26. D 27. A

EXERCISE 6.4
1. (i) 5.03 (ii) 7.035 (iii) 0.8
(iv) 0.208 (v) 0.9999 (vi) 1.96875

ANSWERS 283

(vii) 2.9629 (viii) 3.9961 (ix) 3.009
(x) 20.025 (xi) 0.06083 (xii) 2.948
(xiii) 3.0046 (xiv) 7.904 (xv) 2.00187
2. 28.21 3. – 34.995 4. 0.03 x3 m3
5. 0.12 x2 m2 6. 3.92 π m3 7. 2.16 π m3
8. D 9. C

EXERCISE 6.5
1. (i) Minimum Value = 3 (ii) Minimum Value = – 2
(iii) Maximum Value = 10 (iv) Neither minimum nor maximum value
2. (i) Minimum Value = – 1; No maximum value
(ii) Maximum Value = 3; No minimum value
(iii) Minimum Value = 4; Maximum Value = 6
(iv) Minimum Value = 2; Maximum Value = 4
(v) Neither minimum nor Maximum Value
3. (i) local minimum at x = 0, local minimum value = 0
(ii) local minimum at x = 1, local minimum value = – 2
local maximum at x = – 1, local maximum value = 2

π
(iii) local maximum at x = , local maximum value = 2
4

3π
(iv) local maximum at x = , local maximum value = 2
4

7π
local minimum at x = , local minimum value = – 2
4
(v) local maximum at x = 1, local maximum value = 19
local minimum at x = 3, local minimum value = 15
(vi) local minimum at x = 2, local minimum value = 2

284 MATHEMATICS

1
(vii) local maximum at x = 0, local maximum value =
2

2 2 3
(viii) local maximum at x = , local maximum value =
3 9

5. (i) Absolute minimum value = – 8, absolute maximum value = 8
(ii) Absolute minimum value = – 1, absolute maximum value = 2
(iii) Absolute minimum value = – 10, absolute maximum value = 8
(iv) Absolute minimum value = 19, absolute maximum value = 3
6. Maximum profit = 49 unit.
7. Minima at x = 2, minimum value = – 39, Maxima at x = 0, maximum value = 25.

π 5π
8. At x = and 9. Maximum value = 2
4 4
10. Maximum at x = 3, maximum value 89; maximum at x = – 2, maximum value = 139
11. a = 120
12. Maximum at x = 2π, maximum value = 2π; Minimum at x = 0, minimum value = 0
13. 12, 12 14. 45, 15 15. 25, 10 16. 8, 8
17. 3 cm 18. x = 5 cm

1 1
⎛ 50 ⎞ 3 ⎛ 50 ⎞ 3
21. radius = ⎜ ⎟ cm and height = 2 ⎜ ⎟ cm
⎝ π⎠ ⎝ π⎠

112 28π
22. cm, cm 27. A 28. D 29. C
π+4 π+4

1. (a) 0.677 (b) 0.497
3. b 3 cm2/s 4. x + y – 3 = 0

ANSWERS 285

π 3π π 3π
6. (i) 0 < x < and < x < 2π (ii) <x<
2 2 2 2
7. (i) x < –1 and x > 1 (ii) – 1 < x < 1

3 3
8. ab 9. Rs 1000
4

20 10
11. length = m, breadth = m
π+4 π+4
2
13. (i) local maxima at x = 2 (ii) local minima at x =
7
(iii) point of inflection at x = –1
5
14. Absolute maximum = , Absolute minimum = 1
4
4π R 3
17. 19. A 20. B 21. A
3 3
22. B 23. A 24. A

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