Page # 126 TANGENT & NORMAL

TANGENT & NORMAL

Answer Ex–I SINGLE CORRECT (OBJECTIVE QUESTIONS)

1. B 2. A 3. B 4. D 5. C 6. B 7. A 8. A
9. A 10. B 11. B 12. D 13. B 14. D 15. B 16. B
17. B 18. B 19. A 20. B 21. A 22. B 23. B 24. B
25. B 26. B 27. B 28. A 29. B 30. A

Answer Ex–II MULTIPLE CORRECT (OBJECTIVE QUESTIONS)

1. AC 2. AB 3. AD 4. ABD 5. AB 6. AC 7. ABC 8. AB
9. AC 10. AC 11. AD 12. AB 13. BC 14. ACD 15. AC

Answer Ex–III SUBJECTIVE QUESTIONS

1. (0, 0) ; (3, 27) 2. 2x + y = 2 3. Tangent : x + y = 6, Normal x – y = 0
4. y = x 5. (a) y – 2x – 3 = 0 (b) 2x + y – 7 = 0


8. (4, 11) & (–4, –31/3) 9. (0, 0), (1, 2), (–1, –2) 10. (9/4, 3/8) 11.
3
2
12. 3 2 – 1 13. (–6, 3) 15. (i) –2 cm/min (ii) 2 cm /min

c
16. zero 17. –1 18. –1500 ft/sec 19. ±
2

 13  8b
20. p  (0, 1/e) 21. 8 22. a    ,3  23.
 4  27

Answer Ex–IV ADVANCED SUBJECTIVE QUESTIONS

1. (0, 1) 2. x = 1 when t = 1, m ; 5x – 4y = 1 if t  1, m = 1/3

m m
3. T : x – 2y = 0 ; N : 2x + y = 0 4. x + 2y = /2 & x + 2y = –3 /2 10.
2
1
11. (a) n = –2 12. ± 13. (i) 6 km/h (ii) 2 km/hr 14. 1 + 36 cu. cm/sec
2 2
66 1
15. 1/48  cm/s 16. 0.05 cm/sec 17. 18. cm/sec.
7 4
1 5 1/4
19. (a) – m/min., (b) – m/min. 20. (a) r = (1 + t) , (b) t = 80
24 288

Answer Ex–V JEE PROBLEMS

1. 2 x + y – 2 2 = 0 or 2x–y–2 2 =0 2. D 3. D 4. D 5. A

394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564
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 MONOTONOCITY Page # 127

MONOTONOCITY
Answer Ex–I SINGLE CORRECT (OBJECTIVE QUESTIONS)

1. C 2. C 3. A 4. B 5. D 6. B 7. D 8. C
9. D 10. A 11. C 12. D 13. B 14. C 15. D 16. C
17. B 18. C 19. A 20. B 21. B 22. B 23. D 24. B
25. A 26. B 27. D 28. D 29. C 30. B 31. A 32. B
33. B 34. B 35. C 36. D 37. D 38. A

Answer Ex–II MULTIPLE CORRECT (OBJECTIVE QUESTIONS)

1. AB 2. BC 3. AD 4. BC 5. BC 6. ABC 7. ACD
8. AD 9. AB 10. AD 11. AB 12. AD 13. AC 14. ABCD
15. AB 16. CD 17. AB 18. AC

Answer Ex–III SUBJECTIVE QUESTIONS

+
1. a  R 3. [1  6 , 1  6 ]

4. (i) M.D. in (–, –3]  [0, 2] M.. in (–3, 0]  [2, )

 2 2   2 2 
(ii) M.. in  ,  , n  Z M.D. in  4n  1, 4n  1 , n  Z
 4n  3 4n  1  

 1   1 
(iii) M.D. in  0,  M.. in  ,  
 3  3 

5. (– , –3] 6. (/6) + (1/2). n 3, (/3) – (1/2) n 3 10. 2 sin x + tan x > 3x, limit = 0

11. [ 5 , 10 ] 12. f(x) is injective  x  [0, )

   
14. increasig when x   ,  decreasin when x   0, .
4 2  4

16. a < – (2 + 5 ) or a > 5 17. (–, – 2)  (0, ) 18. ln (1 + x)

20. Neither increasing nor decreasing at x = –1, increasing at x = 0, 1.

21. [1, ) 24. [–7, –1]  [2, 3]

394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564
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 Page # 128 MONOTONOCITY

26. (a)  in (2, ) & D in (–, 2) (b)  in (1, ) & D in (–, 0)  (0, 1)

1 1 1 1
(c)  in (0, 2) & D in (–, 2)  (2, ) (d)  for x > or – < x < 0 & D for x < – or 0 < x<
2 2 2 2
27. (–2, 0)  (2, )

28. (a)  in [0, 3/4)  (7/4, 2] & D in (3/4, 7/4)

(b)  in [0, /6)  (/2, 5/6)  (3/2, 2] & D in (/6, /2)  (5/6, 3/2)
(c)  in [0, /2)  (3/2, 2) & D in (/2, 3/2)

29. continuous but not diff. at x = 1

30. (a) Maximum at x = 1 and f(–1) =18; Minimum at x = 1/8 and f(1/8) = –9/4
(b) 2 & –10

3
31. a  (–, –3]  [1, ) 32. 0a 33.  in (3, ) and  in (1, 3)
2
34. a0

 5 5  5
35. (a) (–, 0] (b)  in  1,  and  in (–, 1)   ,   – {–3} (c) x =
 3 3  3

(d) removable discont. at x = –3 (missing point ) and non removable discont. at x = 1 (infinite type)
(e) –2
3
38. (–1, 0)  (0, ) 39. (b –a) /4

Answer Ex–IV ADVANCED SUBJECTIVE QUESTIONS

mb  na
11. c= which lies between n & b 16. a = 3, b = 4 and m = 1
mn

17. y = –5x – 9 and y = 5x + 11.

Answer Ex–V JEE PROBLEMS

1 
1. (a) B ; (b) D ; (c) C 2. (a) A, (b) cos  cos 1 p  3. A 5. (a) D; (b) C
3 

7. D 8. (a) B; (b) (i) B, (ii) A, (iii) A; (c) (A)–P, Q, R ; (B)–P, S ; (C)–R, S ; (D)–P, Q

9. (a) C, (b) A, B, C, D 10. B, C, D 11. B,C 12. B 13. C

394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564
IVRS No. 0744-2439051, 0744-2439052, 0744-2439053 www.motioniitjee.com, email-hr.motioniitjee@gmail.com
 MAXIMA & MINIMA Page # 129

MAXIMA & MINIMA

Answer Ex–I SINGLE CORRECT (OBJECTIVE QUESTIONS)
1. C 2. D 3. C 4. D 5. A 6. C 7. B 8. A
9. D 10. D 11. C 12. A 13. A 14. B 15. D 16. C
17. A 18. B 19. C 20. C 21. B 22. A 23. B 24. D
25. D 26. D 27. A 28. B 29. C 30. B 31. C 32. C
33. A 34. D 35. A 36. B 37. D 38. A 39. B 40. C
41. A 42. C 43. A 44. B 45. A 46. B 47. D 48. C
49. A 50. D 51. C 52. B 53. D

Answer Ex–II MULTIPLE CORRECT (OBJECTIVE QUESTIONS)

1. D 2. ACD 3. B 4. ACD 5. AC 6. BD 7. AC 8. BD
9. AC 10. AD 11. AC 12. AC 13. AC 14. ABD 15. BC 16. ACD
17. BC 18. AC 19. ACD

Answer Ex–III SUBJECTIVE QUESTIONS

1. (i) maxima (ii) minima (iii) neither maxima nor minima (iv) neither maxima nor minima
(v) neither maxima nor minima (vi) maxima 2. a=b=3
3. (i) local max at x = 1, local min at x = 6(ii) local max. at x = – 1/5, local min. at x = –1
1
(iii) local minima at x = , No local maxima 4. minima at x = 0
e
 3
5. (i) 3 points, x = 0, –3, –5 (ii)  points, x  [–1, 2] (iii) 2 points, x = ,
4 4
6. local max. at x = 1, local min. at x = 2
7. (i) max. = 8, min. = –8 (ii) max. = 2 , min. = –1 (iii) max. = 8, min. = –10
(iv) max. = 25, min. = –39 (v) max. at x = /6, max. value = 3/4; min. at x = 0 & /2, min. value = 1/2
9. Greatest and least values are not defined 10. b  (0, e]
11. (i) local max. at x = –1, Maxima of f(x) = –2, local min. at x = 1, Minima of f(x) = 2

(ii) Local Minima at x = + 2n, n , Minima of f(x) = 1,
2

Local Maxima at x = – + 2n, n , Maxima of f(x) = –1
2
80 20
12. It is a global maxima. 14. , 17. 12cm, 6 cm 18. 40 mph
4 4
1 5
19. 27 3 sq. cms. 20. width 2 3 m, length 3 3 m 22.  , 
2 4
12 5 2 6 4r 2
23. f(x) = 2 x –
4
x  x 24. 25. Rs. 400 26. 5 km. from B towards A
5 3 3 3

4 3
27. cos A = 0.8 28. 29. 2/3 30. side 10’, height 10’
9

 1  1 3 3 2
31. Global maximum  n 3 , Global minima  n 3 32. a
6 4 3 4 8

394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564
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 Page # 130 MAXIMA & MINIMA

Answer Ex–IV ADVANCED SUBJECTIVE QUESTIONS

3 2
1. f(x) = x + x – x + 2 2. (a) Max. at x = 2, Max. value = 2, Min. at x = 0, Min. value of = 0
(b) Max. at x= /6 & also at x = 5 /6 and Max. value = 3/2, Min. at x = /2, Min. value = –3

2 6 12 5  
Pmax. = a 1  cos ec
4 
3. f(x) =
3
x –
5
x + 2x 4.
 2
5. 75 3 sq. units.

2A 2A
6. r= ,s= 8. 3x + 4y – 9 = 0 ; 3x – 4y + 9 = 0 9. 4 2 m
4 4
10. 1/ cu m 11. 110, 70 12. side 10, height 10 13. 32 sq. units 14. = 60º
15. a = 1, b = 0 17. width 2 3 m, length 3 3 m 18. |a – b|
2
19. (a) (–1, 0), (0, 5/6) (b) F(x) = (x – x), F(x) = 2x – 1
(c) increasing (–, 0)  (1, ) decreasing (0, 1); (d) (0, 5/6); (1, 2/3); (e) x = 1/2

d d 2
20. (a) x = y =
2
(b) x =
3
,y=
3
d 21. 6’ × 18’ 22. r= A ,  = 2 radians

3
23. (a) 0, 3 (c) , t = n 4 24. cos A = 0.8 26. (0, 0)
4

32p 3 32p 3
28. p<a<
27
+ p if p > 0;
27
+ p < a < p if p < 0 30. 4 when a = 2

2
31. (a) f is continuous at x = 0 (b) – (c) does not exist, does not exist; (d) point of inflection x = 1
e
 3
32. (a) x = –2 (b) no inflection point (c) maxima at x = and – and no minima,
2 2
3   1 1 
(d) x = and x = – , (e) – n 2 33. 4 34. m  ,  35.
2 2  32 16  4

1/ 3
 4V  
36. 320 37. H = x =   38. L/4 39.
 3 3
40. (a) increasing in (0, 2) and decreasing in (– , 0)  (2, ), local min. value = 0 local max. value = 2

(b) concave up for (– , 2 – 2 )  (2 + 2 , ) and concave down in (2 – 2, 2+ 2)

1 2.x 2
(e) f(x) = e .x
2

Answer Ex–V JEE PROBLEMS

1. A 2. 2ab 3. (a) D; (b) A 4. A 5. (a) (2, 1) ; (b) 5 6. (a) D 7. 4 65

8. (a) B, C ; (b) A (c) 6 solutions 9. (a) C ; (b) (i) A, (ii) A, (iii) B 10. (a) 0 ; (b) 7
11. (a) D, (b) 1 12. 2 13. 9 14. 5 15. A,B,C,D

394 - Rajeev Gandhi Nagar Kota, Ph. No. 0744-2209671, 93141-87482, 93527-21564
IVRS No. 0744-2439051, 0744-2439052, 0744-2439053, www.motioniitjee.com, email-info@motioniitjee.com