answers...answers 305 (iii) ( f g) x = 2x3 + x3 + 2x + 1 (iv)f g ⎛⎞ ⎜⎟ ⎝⎠ x = 221 1 x x...

ANSWERS

1.3 EXERCISE

1. (i) {2} (ii) {0, 1} (iii) {1, p}

2. (i) {0, –1, 1} (ii) 11

3

−⎧ ⎫⎨ ⎬⎩ ⎭

(iii) { }3, 2, 2, 3− −

3. {1, 2, 22, 23, ...2P – 1,(2p –1}

4. (i) True (ii) False (iii) True (iv) True

7. (i) {2, 4, 6, 8, ... , 98} (ii) (1,4, 9, 16, 25, 36, 49, 64, 81,}

8. (i) {4, 8, 12} (ii) {7, 8, 9} (iii)1 3

,1,2 2

⎧ ⎫⎨ ⎬⎩ ⎭

(iv) {0, 1, 2}

9. (i) {4, 5, 6, ....10} (ii) {5} (iii) {1, 2, 3, 4, 5}

10. 11.

13. True 14. False 15. True

16. True 17. True 22. T = {10}

24. (i) 2 (ii) 3 (iii) 3 (iv) 9

25. 25 26. 20 27. (a) 3300 (b) 4000

28. (i) 6, (ii) 3, (iii) 9, (iv) 1, (v) 2, (vi) 6, (vii) 30, (viii) 20 29. C

30. B 31. B 32. D 33. C

34. D 35. B 36. B 37. C

304 EXEMPLAR PROBLEMS – MATHEMATICS

38. C 39. C 40. A 41. B

42. B 43. C 44. [1,2] 45. 1

46. n (B) 47. A ∩ B′ 48. {φ, {1}, {2}, {1, 2}49. {0, 1, 2, 3, 4, 5, 6, 8} 50.(i) {1,5, 9, 10 } (ii) { 1, 2,3, 5, 6, 7, 9, 10 }

51. A ∪ ∪ ∪ ∪ ∪ Β′ 52. (i) ↔(b) (ii) ↔(c) (iii) ↔ (a) (iv) ↔(f) (v) ↔(d) (vi) ↔ (e)

53. True 54. False 55. False 56. True

57. True 58. False

2.3 EXERCISE

1. (i) {(– 1, 1), (–1, 3), (2, 1), (2, 3), (3, 1), (3, 3)}

(ii) {(1, –1), (1, 2), (1, 3), (3, –1), (3, 2), (3, 3)}

(iii) {(1, 1), (1, 3), (3, 1), (3, 3)}

(iv) {(–1, –1), (–1, 2), (–1, 3), (2, –1), (2, 2), (2, 3), (3, –1), (3, 2), (3, 3)}

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

3. (i) {(0, 3), (1, 3)}

(ii) {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1,5)}

4. (i)11

3a = and b =

2

3 (ii) a = 0 and b = – 2

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

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

(iii) { (4, 5), (5, 4), (5, 5)}

6. Domain of R = {0, 3, 4, 5} = Range of R

7. Domain of R1 = [–5, 5 ] and Range of R

1 = [–3, 17 ]

8. R2 = {(0, 8), (8, 0) (0,– 8), (– 8, 0)}

9. Domain of R3= R and range of R

3 = R+ ∪ {0}

10. (i) h is not a function (ii) f is a function (iii) g is a function (iv) s is afunction(v) t is a constant function

11. (a) 6 (b)1364

4(c) 13 (d) t2 _ 4 (e) t + 5

12. (a) x = 4 (b) x > 4

13. (i) (f + g) x = x2 + 2x + 2 (ii) (f – g) x = 2x – x2

ANSWERS 305

(iii) ( f g) x = 2x3 + x3 + 2x + 1 (iv)f

g

⎛ ⎞⎜ ⎟⎝ ⎠

x = 2

2 1

1

x

x

++

14. (i) f = {(–1, 0), (0, 1), (3, 28), (7, 344), (9, 730)}

15. x = –1 , 4

3

16. Yes, α ===== 2 , β = = = = = −117. (i) R – {2nπ : n∈Z} (ii) R+ (iii) R

(iv) R – {–1, 1} (v) R – {4}

18. (i) [3

2,∞) (ii) (– ∞, 1] (iii) [ 0 , ∞) (iv) [ – 2, 4]

19.

2 , 3 2

( ) 4 , 2 2

2 , 2 3

x x

f x x

x

− − ≤ <−⎧⎪= − ≤ <⎨⎪ ≤ ≤⎩

21. (i) (f + g) x = x + x (ii) (f – g ) x = x – x

(iii) (fg) x = 3

2x (iv) 1fx

g x

⎛ ⎞=⎜ ⎟

⎝ ⎠

22. Domain of f = (5, ∞) and Range of f = R+

24. D 25. D 26. B 27. C

28. B 29. B 30. A 31. C

32. C 33. A 34. B 35. A

36. {2, 3, 4, 5} 37. (a) ↔ (iii) (b) ↔ (iv) (c) ↔ (ii) (d) ↔ ↔ ↔ ↔ ↔ (i)3 8 .3 8 .3 8 .3 8 .3 8 . False 39. False 40. True 41. False

42. True.

3.3 EXERCISE

4.56

335.

2cos

cos 2

x

x


8.1

2 1+ 15. θ = nπ + (–1)n 4 4

π π−

16. θ = 2nπ + 7

4

π17. θ = 2nπ ±

3

π

18. θ = 5

,3 3

π π19. , ,

6 4 2x

π π π=

22. 1 25.23 3 1 1

17 2 2

⎛ ⎞−+⎜ ⎟⎝ ⎠

26.3

227. nπ ±

4

π

28.2 8

nπ π± 29. 2

4 12n

π πθ = π ± +

30. C 31. D 32. D 33. C

34. B 35. C 36. B 37. C

38. A 39. B 40. D 41. D

42. A 43. D 44. C 45. B

46. C 47. C 48. C 49. B

50. C 51. B 52. C 53. C

54. A 55. B 56. A 57. B

58. B 59. D 60. 1 61.1

8

62. tan β 63.1

4[4 – 3(a2 – 1)2] , 22 a−

64.2 2

1sin 2A

x x− + 65. 13 66. [–3, 3] 67. 2

68. True 69. False 70. False 71. True

72. False 73. True 74. True 75. True

76. (a) ↔ (iv) (b) ↔ (i) (c) ↔ (ii) (d) ↔ (iii)

ANSWERS 307

4.3 EXERCISE

1. P(n) : 2n < ∠n 2. P(n) : 1 + 2 + 3 + ... + n = ( 1)

2

n n+

26. A 27. B 28. A

29. 4 30. False

5.3 EXERCISE

1. 2n 2. –1 + i 3. (0, – 2) 4.2

5

5. (1, 0) 6. icot 2

θ11.

3– 2

2i 12.

1– 2

2i

13. 1:3 14.10 2

,0 ,3 3

⎛ ⎞⎜ ⎟⎝ ⎠ 15. 1 18. 0

21. 2 2, 2 2i i± − ± 22. –2 – i

23. 5 5

2 cos sin12 12

iπ π⎛ ⎞+⎜ ⎟⎝ ⎠

25. (i) ( )( )2 22 21 2a b z z+ + (ii) –15

(iii) –2 (iv) 0 (v) 1

2 2

i− (vi) 1z (vii) 0

(viii) 6 and 0 (ix) a circle (x) – 2 3 + 2i

26. (i) F (ii) F (iii) T (iv) T

(v) T (vi) T (vii) F (viii) F

27. (a) ↔ (v), (b) ↔ (iii), (c) ↔ (i),(d) ↔ (iv), (e) ↔ (ii), (f) ↔ (vi),

(g) ↔ (viii) and (h) ↔ (vii)


28.2 11

25 25i

−− 29. No 30.

2 4

2

( 1)

4 1

a

a

++ 31. – 2 3 + 2 i

32. 1 33.2

3

π34. Real axis

35. D 36. C 37. B 38. A

39. B 40. A 41. A 42. B

43. D 44. D 45. B 46. B

47. C 48. C 49. C 50. A

6.3 EXERCISE

1.1

13

x≤ ≤ 2. [0,1] ∪ [3,4] 3. ( , 5−∞ − )∪(–3, 3)∪[5, ∞∞∞∞∞)

4. [ 4, 2] [2,6]− − ∪ 5.34 22

,3 3

−⎡ ⎤⎢ ⎥⎣ ⎦

6. No Solution

7. More than 2000.

8. Between 7.77 and 8.77.

9. More than 230 litres but less than 920 litres.

10. Between 104 °F and 113 °F

11. 41 cm.

12. Between 8 km and 10 km

13. No Solution

14. 20, 3 2 48, 0, 0x y x y x y+ ≤ + ≤ ≥ ≥

15. 8, 4, 5, 5, 0, 0x y x y x y x y+ ≤ + ≥ ≤ ≤ ≥ ≥

17. No Solution.

19. C 20. C 21. A 22. B

23. D 24. C 25. B 26. A

27. D 28. B 29. A 30. B

31. (i) F (ii) F (iii) T (iv) F

(v) T (vi) F (vii) T (viii) F

ANSWERS 309

(ix) T (x) F (xi) T (xii) F

(xiii)F (xiv) T (xv) T.

32. (i) ≤ (ii) ≥ (iii) > (iv) >

(v) > (vi) > (vii) < , > (viii) ≤ .

7.3 EXERCISE

1. 1440 2. 481 3. 780 4. 144

5. 22 6. 3960 7. 4,68000 8. 200

9.3

3C ( – 2)!3!nr r−− 10. 14400 11. 112 15. r = 3

16. 192 17. 190 18. 8400 19. 3 20. 11

21. 3

18!

(6!) 22. (a) 11C4

(b) 6C2 × 5C

2 (c) 6C

4 + 5C

4

23. (i) 14C9 (ii) 14C

1124. 2(20C

5 × 20C

6)

25. (i) 21, (ii) 441 (iii) 91 26. A 27. B

28. C 29. B 30. C 31. A

32. B 33. D 34. B 35. C

36. D 37. A 38. C 39. B

40. B 41. n = 7 42. 0 43. nr

44. 1,51,200 45. 80 46. 56 47. 18

48. 35 49. 7800 50. 64 51. False

52. False 53. False 54. True 55. True

56. True 57. True 58. False 59. False

60. (a) ↔ (ii) (b) ↔ (iii) and (c) ↔ (i)

61. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (iv), (d) ↔ (ii)

62. (a) ↔ (iv) (b) ↔ (iii) (c) ↔ (ii), (d) ↔ (i)

63. (a) ↔ (i) (b) ↔ (iii) (c) ↔ (iv), (d) ↔ (ii)

64. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (ii)

8.3 EXERCISE

1.5

1510

1C

6⎛ ⎞⎜ ⎟⎝ ⎠ 2. k = ± 3 3. –19 4. –3003 (310) (25)


5. (i) –252 (ii) 17189

8x ; 1921

16x

− 6. –252 7. –1365 8. 5532252y x

9. r = 6 11. 990 12. 2p =± 14. n = 9

17. 17

5418. (C) 19. (A) 20. (C)

21. (D) 22. (B) 23. (B) 24. (C)

25. 3015C 26.

( )1 ( 2)

2

n n+ +27. 16

8C 28. n = 12

29. 1120

27a–6a4 30. 28 C

14 a56 b14 31. 1 32. Third term

33. 12 34. F 35. T 36. F

37. F 38. T 39. F 40. F

9.3 EXERCISE

2. Rs 1400 3. Rs 8080 , Rs 83520 5. 12 days

6. 3420° 7.15

8cm 8. 2480 m 9. Rs 725

11. (i) 4n3 + 9n2 + 6n (ii) 4960 12. Tr= 6r – 1 17. D

18. C 19. A 20. B 21. C

22. B 23. B 24. A 25. D

26. A 27.a b

orb c

28. First term + last term

29. 45 30. F 31. T 32. T

33. F 34. F

35. (a) ↔ ↔ ↔ ↔ ↔ (iii) (b) ↔ ↔ ↔ ↔ ↔ (i) (c) ↔ ↔ ↔ ↔ ↔ (ii)36. (a) ↔ ↔ ↔ ↔ ↔ (iii) (b) ↔ ↔ ↔ ↔ ↔ (i) (c) ↔ ↔ ↔ ↔ ↔ (ii)(d) ↔ ↔ ↔ ↔ ↔ (iv)

10.3 EXERCISE

1. x + y + 1 = 0 2. x – 4y + 3 = 0 3. 60° or 120°

ANSWERS 311

4. x + y = 7 or 16 8

x y+ = 5. (3, 1), (– 7, 11)

7. y – 3 x – 2 + 3 = 0 8. 3x + 4y + 3 = 0

9.8

3a

−= , b = 4 10. 8x – 5y + 60 = 0

11. 3x y+ = 8 12. x – 7y – 12 = 0

13.2

314. (1, 1)

15. 15° or 75° 17. 9x – 20y + 96 = 0

18. 3x – 4y + 6 = 0 and 4x – 3y + 1 = 0 20. (0, 2 + 5 3

2)

22. A 23. A 24. B 25. B

26. C 27. D 28. A 29. A

30. A 31. B 32. B 33. A

34. C 35. A 36. B 37. B

38. C 39. D 40. B 41. B

42. (1, – 2) 43. x + y + 1 = 0 44. 3x – y – 7 = 0, x + 3y – 9 = 0

45. opposite sides 46. 13 (x2 + y2) – 83 x + 64 y + 182 = 0

47. 4 x2 y2 = p2 (x2 + y2) 48. True 49. False

50. False 51. True 52. True 53. True

54. True 55. False 56. False

57. (a) ↔ (iii) (b) ↔ (i) and (c) ↔ (ii)

58. (a) ↔ (iv) (b) ↔ (iii) (c) ↔ (i), (d) ↔ (ii)

59. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (iv), (d) ↔ (ii)

11.3 EXERCISE

1. x2 + y2 – 2ax – 2ay + a2 = 0 3. ,2 2

a b⎛ ⎞⎜ ⎟⎝ ⎠


4. x2 + y2 – 2x – 4y + 1 = 0 5.3

4

6. x2 + y2 + 4x + 4y + 4 = 0 7. (1, 2)

8. x2 + y2 – 2x + 4y – 20 = 0 9. k ± 8

10. x2 + y2 – 6x + 12y – 15 = 0 11.3

2

12. ecentricity = 4

5and foci (4, 0) and (–4, 0) 13.

39

4

14.2 24 4

181 45

x y+ = 15. 18 16. (2, 4) , (2, – 4)

17. 2

4 cos

sin

a θθ

18. x2 + 8y = 32 19. m = 1 20. x2 – y2 = 32

21.13

222.

2 2 4

4 5 9

x y− = . 23. x2 + y2 – 2x + 2y = 47

24. x2 + y2 – 4x – 10y + 25 = 0 25. (x – 3)2 + (y + 1)2 = 38

26. x2 + y2 – 18x – 16y + 120 = 0 27. x2 + y2 – 8x – 6y + 16 = 0

28. (a) y2 = 12x – 36, (b) x2 = 32 – 8y, (c)4x2 + 4xy + y2 + 4x + 32y + 16 = 0

29. 3x2 + 4y2 – 36x = 0 30. 9x2 + 5y2 = 180

32. (a) 15x2 – y2 = 15 (b) 9x2 – 7y2 + 343 = 0, (c) y2 – x2 = 5

33. False 34. False 35. True 36. False

37. True 38. False 39. True 40. True

41. (x – 3)2 + (y + 4)2 = 2

45

13⎛ ⎞⎜ ⎟⎝ ⎠ 42. x2 + y2 – 46x + 22y = 0

43. 6 + 2 5 , 2 5 44.2 24 4

11 5

x y+ =

45. 4x2 + 4xy + y2 + 4x + 32y + 16 = 0 46.2 2

– 136 64

y x= and (0, ± 10).

47. (C) 48. (C) 49. (C) 50. (C)

ANSWERS 313

51. A 52. B 53. A 54. A

55. D 56. B 57. C 58. A

59. A

12.3 EXERCISE

2. (i) 1st octant (ii) 4th octant (iii) viiith octant (iv) vth octant (v) 2nd octant

(vi) 3rd octant (vii) viiith octant (viii) vith octant

3. (i) (3,0,0), (0,4,0), (0,0,2) (ii) (–5, 0, 0), (0,3,0), (0,0,7) (iii) (4,0,0), (0, –3, 0),(0,0,5)

4. (i) (3,4,0), (0,4,5), (3,0,5) (ii) (–5, 3, 0),(0,3,7), (–5, 0, 7) (iii) (4,–3, 0),(0,–3, –5), (4, 0, –5)

5. 5 6. 11 9. (2,–4, 16) 11. (–2, –2, –1)

12. (1, 1, –2) 13. (–3, 4, –7), (7, 2, 5) and (–3, 12, 17) 14. (4, 7, 6)

15. (4, –5, 1), (3, –2, –1) 16. a = –2, b = –8, c = 2

17.7 13

, ,92 2

⎛ ⎞⎜ ⎟⎝ ⎠ 18. 2:1 externally

19. vertices are (3,4,5), (–1,6,–7), (1,2,3) and centroid is (1,4,1

3 )

20. 1:3 externally

21. (2,0,0), (2,2,0), (0,2,0), (0,2,2) (0,0,2) (2,0,2), (0,0,0), (2,2,2)

22. A 23. B 24. A 25. B

26. A 27. B 28. B 29. A

30. A 31. B 32. A 33. D

34. A 35. Three cordinates planes 36. Three pairs

37. given point 38. Eight 39. (0, y, z) 40. x = 0

41. (0, 0, z) 42. x = 0 , y = 0 43. z- cordinates

44. (y, z cordinates) 45. yz-plane 46. x-axis 47. 333

48. a = 5 or –3 49. (1, 1, –2)

50. (a) ↔ (iii) (b) ↔ (i) (c) ↔ (ii) (d) ↔ (vi) (e) ↔ (iv) (f) ↔ (v) (g) ↔ (viii)(h) ↔ (vii) (i) ↔ (x) (j) ↔ (ix)


13.3 EXERCISE

1. 6 2. 2 3. 2

1

x4.

–2

312

3

5. 3 6. ( )3

25

22

a + 7. 7 8. 8

9.8

510. 1 11. 0 12.

1

15

13.7

214. n = 5 15.

3

716.

1

4

17. 2 18. 1 19.2

2

m

n20. 3

21. 2 22. 2 23. 1 24. 2 cosa a

25. 4 26.1

4 227. 0 28.

3

8k =

29.2

2

13 2 1 –x x

x+ + 30.

22 4

3 33 – – 3x

x x+

31. 2 23 sec 5sec 3tan 3x x x x+ + + 32. 22 tan secx x

33. ( )2

22

55 – 40 –15

5 – 7 9

x x

x x + 34.5 4

2

– cos 5sec sin 1

sin

x x x

x

+ +

35. ( )cosec 2 – cot2

xx x x

36. ( )( ) ( )( )2 2cot – sin cos 2 – cosax x q x p q x ax ec x+ + +

37. ( )2

cos sin

cos

bc x ad x db

c d x

+ +

+ 38. 2cos 2x

ANSWERS 315

39. ( )( )( )22 – 7 30 – 43 3 5x x x + 40. 2 cos 2 sin – 2sin 2x x x x x+

41.23

sin 2 cos24

x x 42.( )

( )22

– 2ax b

ax bx c

+

+ +

43. ( )2–2 sin 1x x + 44. ( )2

–ad bc

cx d+

45.1

–32

3x 46. cos – sinx x x

47. ( )sec tan 1x x x + 48. 2 2

2

–

αα β

49. –4 50.1

252. k = 6 53. c = 1

54. C 55. A 56. A 57. B

58. A 59. C 60. C 61. D

62. B 63. D 64. C 65. D

66. B 67. B 68. D 69. A

70. A 71. A 72. A 73. B

74. C 75. A 76. D 77. 1

78.2 3

3m = 79. y 80. 1

14.3 EXERCISE

1. (i) to (v) and (viii) to (x) are statements.

2. (i) p : Number 7 is prime (ii) p : Chennai is in Indiaq : Number 7 is odd q : Chennai is capital of Tamil Nadu

(iii)p : 100 is divisble by 3 (iv) p : Chandigarh is capital of Haryanaq : 100 is divisible by 11 q : Chandigarh is the capital of U.P

r : 100 is divisible by 5


(v) p : 7 is a rational number (vi) p : 0 is less than every positive integer

q : 7 is an irrational number q : 0 is less than every negative integer

(vii) p : plants use sunlight for photosynthesis

q : plants use water for photosynthesis

r : plants use carbondioxide for photosynthesis

(viii) p : two lines in a plane intersect at one point

q : two lines in a plane are parallel

(ix) p : a rectangle is a quadrilateral

q : a rectangle is a 5- sided polygons.

3. (i) Compound statement is true and its component statements are :

p : 57 is divisible by 2 and q : 57 is divisble by 3

(ii) component statement is true and its component statements are :

p : 24 is multiple of 4 and q : 24 is multiple of 6

(iii) component statement is false and is component statements are

p : All living things have two eyes

q : All living things have two legs

(iv) component statement is true and its component statements are :

p : 2 is an number ; q : 2 is a prime number

4. (i) The number 17 is not prime (ii) 2 + 7≠ 6 (iii) Violet are not blue

(iv) 5 is not a rational number (v) 2 is a prime number

(vi) There exists a real number which is not an irrational number

(vii) Cow has not four legs (viii) A leap year has not 366 days

(ix) There exist similar triangles which are not congruent

(x) Area of a circle is not same as the perimeter of the circle

5. (i) p ∧ q where p : Rahul passed in Hndi; q : Rahul passed in English

(ii) p ∧ q where p : x is even integer ; q : y is even integer(iii) p ∧ q ∧ r where p : 2 is factor of 12; q : 3 is factor of 12; r : 6 is factor

of 12(iv) p ∨ q where p : x is an odd integer ; q : x +1 is an odd integer

(v) p ∨ q where p : a number is divisible by 2, q : it is divisibe by 3(vi) p ∨ q where p : x = 2 is a root of 3x2 – x –10 = 0, q : x = 3 is a root of

3x2 –x –10 = 0

ANSWERS 317

(vii) p ∨ q where p : student can take Hindi as an optional paper and q :student can take English as an optional paper.

6. (i) It is false that all rational numbers are real and complex

(ii) It is false that all real numbers are rational or irrational

(iii) x = 2 is not a root of the quadratic equation x2– 5x + 6 = 0 or x = 3 is nota root of the quadratic equation x2–5x + 6 = 0

(iv) A triangle has neither 3-sides nor 4-sides

(v) 35 is not a prime number and it is not a complex number

(vi) It is false that all prime integers are either even or odd

(vii) x is not equal to x and it not eqaul to –x

(viii) 6 is not divisible by 2 or it is not divisible by 3.

7. (i) If the number is odd number then its square is odd number

(ii) If you take the dinner then you will get sweet dish

(iii) If you will not study then you will fail

(iv) If an integer is divisible by 5 then its unit digits are 0 or 5

(v) If the number is prime then its square is not prime

(vi) If a,b and c are in A.P then 2b = a + c.

8. (i) The unit digit of an integer is zero if and only if it is divisible by 5.

(ii) A natural number n is odd if and only if it is not divisible by 2.

(iii) A triangle is an equilateral triangle if and only if all three sides of triangleare equal.

9. (i) If 3x ≠ then x y≠ or 3y ≠

(ii) If n is not an integer then it is not a natural number.

(iii) If the triangle is not equilateral then all three sides of the triangle are notequal

(iv) If xy is not positive integer then either x or y is not negative integer.

(v) If natural number n is not divisible by 2 and 3 then n is not divisible by 6.

(vi) The weather will not be cold if it does not snow.

10. (i) If the rectangle R is rhombus then it is square.

(ii) If tomorrow is Tuesday then today is Monday.

(iii) If you must visit Taj Mahal you go to Agra.


(iv) If the triangle is right angle then sum of squares of two sides of a triangleis equal to the square of third side.

(v) If the triangle is equilateral then all three anlges of triangle are equal.

(vi) If 2x = 3y then x:y = 3:2

(vii) If the opposite angles of a quadrilaterals are supplementary then S iscyclic.

(viii) If x is neither positive nor negative than x is 0.

(ix) If the ratio of corresponding sides of two triangles are equal thentrianges are similar.

11. (i) There exists (ii) For all (iii) There exists (iv) For every (v) For all (vi)There exists (vii) For all (viii)There exists (ix) There exists (x) Thereexists

17.. C 18. D 19. B 20. D

21. C 22. B 23. A 24. B

25. C 26. A 27. C 28. B

29. A 30. C 31. B 32. A

33. C 34. A 35. C 36. D

37. (i), (ii) and (iv) are statement; (iii) and (v) are not statements.

15.3 EXERCISE

1. 0.32 2. 1.25 3.2 –1

4

n

n4.

4

n

5.2 –1

12

n6. 3.87 7.

2 2 21 1 2 2 1 2 1 2

21 2 1 2

( ) ( ) ( )

( )

n s n s n n x x

n n n n

+ −+

+ +

8. 5.59 9. 7 10. 1.38

11. Mean = 2.8, SD = 1.12 12. 8.9

13. 5000, 251600 14. Mean = 5.17, SD = 1.53

15. Mean = 5.5, Var. = 4.26 16. 0.99

17. 7.08 18. Mean = 239

40, SD = 2.85

ANSWERS 319

19. Var. = 1.16gm, S.D = 1.08 gm 20. Mean = ( 1)

2

d na

−+ ,

S.D = 2 1

12

nd

−

21. Hashina is more intelligent and consistent

22. 10.24 23. Mean = 42.3, Var. 43.81

24. B 25. B 26. B 27. C

28. A 29. C 30. C 31. A

32. C 33. A 34. D 35. D

36. A 37. D 38. A 39. A

40. SD 41. 0, less 42. 11 43. Independent

44. Minimum 45. Least 46. greater than or equal

16.3 EXERCISE

1.1

722.

2

33. 0.556

4. (a) 5k–1 elements (b) 5 1

4

k −5.

4

96. 0.93

7. (a) 0.65 (b) 0.55 (c) 0.8 (d) 0 (e) 0.35 (f) 0.2

8. (a) 0.35 (b) 0.77 (c) 0.51 (d) 0.57 9. (a) 2

9 (b)

5

9

10. (a)p(John promoted) = 1

8, p(Rita promoted) =

1

4, p(Aslam promoted) =

1

2,

p(Gurpreet promoted) = 1

8 (b) P(A) =

1

4

11. (a) 0.20 (b) 0.17 (c) 0.45 (d) 0.13 (e) 0.15 (f) 0.51

12. (a) { }1 2 1 2 1 2 1 2 1 2 1 1 2 2 2 1, , , , , , , , , ,S B B B W B B B W WB WB BW BW W B W W W B W W=

(b) 1

6 (c)

2

3


13. (a) 5

143 (b)

28

143 (c)

40

14314. (a)

2

143 (b)

2

143 (c)

25

26 (d)

15

26

15.7

13

16. (a) p(A) = .25, p(B) = .32 , p(A∩Β) = .17 (b) p(A∪B) = .40 (c) .40 (d) .68

17. (a)1

2 (b)

3

4 (c)

3

26 (d)

5

3618. A 19. B

20. C 21. C 22. D 23. A

24. A 25. C 26. B 27. C

28. C 29. B 30. False 31. False

32. False 33. True 34. True 35. False

36. True 37. 0.15 38. 0.3 39. { }2,4,6E =

40. 0.2 41. 0.2

42. (a) ↔ (iv) (b) ↔ (v) (c) ↔ (i) (d) ↔ (iii) (e) ↔ (ii)43. (a) ↔ (iv) (b) ↔ (iii) (c) ↔ (ii) (d) ↔ (i)

The weightage of marks over different dimensions of the question paper shall be asfollows:

1. Weigtage of Type of Questions Marks(i) Objective Type Questions : (10) 10 × 1 = 10(ii) Short Answer Type questions : (12) 12 × 4 = 48

(viii) Long Answer Type Questions : (7) 7 × 6 = 42Total Questions : (29) 100

2. Weightage to Different Topics

S.No. Topic Objective Type S.A. Type L.A. Type TotalQuestions Questions Questions

1. Sets - 1(4) - 4(1)2. Relations and Functions - - 1(6) 6(1)3. Trigonometric Functions 2(2) 1(4) 1(6) 12(4)4. Principle of Mathematical - 1(4) - 4(1)

Induction5. Complex Numbers and 2(2) 1(4) - 6(3)

Quadratic Equations -6. Linear Inequalities 1(1) 1(4) - 5(2)7. Permutations and

Combinations - 1(4) - 4(1)8. Binomial Theorem - - 1(6) 6(1)9. Sequences and Series - 1(4) - 4(1)10. Straight Lines 2(2) 1(4) 1(6) 12(4)11. Conic Section - - 1(6) 6(1)12. Introduction to three - 1(4) - 4(1)

dimensional geometry13. Limits and Derivatives 1(1) 1(4) - 5(2)14. Mathematical Reasoning 1(1) 1(4) - 5(2)15. Statistics - 1(4) 1(6) 10(2)16. Probability 1(1) - 1(6) 7(2)

Total 10(10) 48(12) 42(7) 100(29)

MATHEMATICS - CLASS XITime : 3 Hours

Max. Marks : 100

DESIGN OF THE QUESTIONPAPER


SAMPLE QUESTION PAPER

Mathematics Class XI

General Instructions

(i) The question paper consists of three parts A, B and C. Each question ofeach part is compulsory.

(ii) Part A (Objective Type) consists of 10 questions of 1 mark each.

(iii) Part B (Short Answer Type) consists of 12 questions of 4 marks each.

(iv) Part C (Long Answer Type) consists of 7 questions of 6 marks each.

PART - A

1. If tan θ = 1

2 and tan φ =

1

3, then what is the value of (θ + φ)?

2. For a complex number z, what is the value of arg. z + arg. z , z ≠ 0?

3. Three identical dice are rolled. What is the probability that the same number willappear an each of them?

Fill in the blanks in questions number 4 and 5.

4. The intercept of the line 2x + 3y – 6 = 0 on the x-axis is ................. .

5. 20

1 coslimx

x

x→

− is equal to ................. .

In Questions 6 and 7, state whether the given statements are True or False:

6.1

2, 0x xx

+ ≥ ∀ >

7. The lines 3x + 4y + 7 = 0 and 4x + 3y + 5 = 0 are perpendicular to each other.

In Question 8 to 9, choose the correct option from the given 4 options, out of whichonly one is correct.

8. The solution of the equation cos2θ + sinθ + 1 = 0, lies in the interval

(A) ,4 4

π π⎛ ⎞−⎜ ⎟⎝ ⎠(B) 3

,4 4

π π⎛ ⎞⎜ ⎟⎝ ⎠

(C) 3 5,

4 4

π π⎛ ⎞⎜ ⎟⎝ ⎠

(D) 5 7,

4 4

π π⎛ ⎞⎜ ⎟⎝ ⎠

DESIGN OF THE QUESTION PAPER 323

9. If z = 2 + 3i , the value of z z⋅ is

(A) 7 (B) 8 (C) 2 3i− (D) 1

10. What is the contrapositive of the statement? “If a number is divisible by 6, then itis divisible by 3.

PART - B

11. If A′ ∪ B = U, show by using laws of algebra of sets that A ⊂ B, where A′denotes the complement of A and U is the universal set.

12. If cos x = 1

7 and cos y =

13

14, x, y being acute angles, prove that x – y = 60°.

13. Using the principle of mathematical induction, show that 23n – 1 is divisible by 7for all n ∈ N.

14. Write z = – 4 + i 4 3 in the polar form.

15. Solve the system of linear inequations and represent the solution on the numberline:

3x – 7 > 2 (x – 6) and 6 – x > 11 – 2x

16. If a + b + c ≠ 0 and , ,b c c a a b

a b c

+ + + are in A.P., prove that

1 1 1, ,

a b c are

also in A.P.

17. A mathematics question paper consists of 10 questions divided into two parts Iand II, each containing 5 questions. A student is required to attempt 6 questionsin all, taking at least 2 questions from each part. In how many ways can thestudent select the questions?

18. Find the equation of the line which passes through the point (–3, –2) and cuts offintercepts on x and y axes which are in the ratio 4 : 3.

19. Find the coordinates of the point R which divides the join of the points P(0, 0, 0)and Q(4, –1, –2) in the ratio 1 : 2 externally and verify that P is the mid point ofRQ.

20. Differentiate f(x) = 3

3 4

x

x

−+ with respect to x, by first principle.


21. Verify by method of contradiction that p = 3 is irrational.

22. Find the mean deviation about the mean for the following data:

xi

10 30 50 70 90

fi

4 24 28 16 8

PART C

23. Let f(x) = x2 and g(x) = x be two functions defined over the set of non-

negative real numbers. Find:

(i) (f + g) (4) (ii) (f – g) (9) (iii) (fg) (4) (iv) (9)f

g⎛ ⎞⎜ ⎟⎝ ⎠

24. Prove that: (sin 7 sin 5 ) (sin 9 sin 3 )

tan 6(cos 7 cos5 ) (cos9 cos3 )

x x x xx

x x x x

+ + +=

+ + +

25. Find the fourth term from the beginning and the 5th term from the end in the

expansion of

103

2

3

3

x

x

⎛ ⎞−⎜ ⎟⎝ ⎠ .

26. A line is such that its segment between the lines 5x – y + 4 = 0 and 3x + 4y – 4 = 0is bisected at the point (1, 5). Find the equation of this line.

27. Find the lengths of the major and minor axes, the coordinates of foci, the verti-

ces, the ecentricity and the length of the latus rectum of the ellipse 2 2

1169 144

x y+ = .

28. Find the mean, variance and standard deviation for the following data:

Class interval: 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100

Frequency: 3 7 12 15 8 3 2

29. What is the probability that(i) a non-leap year have 53 Sundays.

(ii) a leap year have 53 Fridays (iii) a leap year have 53 Sundays and 53 Mondays.


MARKING SCHEME

MATHEMATICS CLASS XI

PART - A

Q. No. Answer Marks

1.4

π1

2. Zero 1

3.1

361

4. 3 1

5.1

21

6. True 1

7. False 1

8. D 1

9. A 1

10. If a number is not divisible by 3, 1then it is not divisible by 6.

PART - B

11. B = B ∪ φ = B ∪ (A ∩ A′) 1

= (B ∪ A) ∩ (B ∪ A′) 1

= (B ∪ A) ∩ (A′ ∪ B) = (B ∪ A) ∩ U (Given) 1

= B ∪ A1

2

⇒ A ⊂ B.1

2


12. cos x =1

7 ⇒ sin x = 2 1 4 3

1 cos 149 7

x− = − =

1

cos y =13

14 ⇒ sin y =

169 3 31

196 14− = 1

cos(x – y) = cosx cosy + sinx siny1

2

=1 13 4 3 3 3 1

7 14 7 14 2⎛ ⎞ ⎛ ⎞ + ⋅ =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ 1

⇒ x – y = 3

π 1

2

13. Let P(n) : “23n – 1 is divisble by 7”1

2

P(1) = 23 – 1 = 8 – 1 = 7 is divisible by 7 ⇒ P(1) is true.1

2

Let P(k) be true, i.e, “23k – 1 is divisible by 7”, ∴ 23k – 1 = 7a, a ∈ Z1

We have : 23(k + 1) – 1 = 23k . 23 – 11

= (23k – 1) 8 + 7 = 7a . 8 + 7 = 7(8a + 1)1

2

⇒ P(k + 1) is true, hence P(n) is true n∀ ∈ N1

2

14. Let – 4 + i 4 3 = r (cosθ + i sinθ)1

2

⇒r cosθ =– 4, r sinθ = 4 3 ⇒ r2 = 16 + 48 = 64 ⇒ r = 8. 11

2


tanθ = – 3 ⇒ θ = π – 2

3 3

π π= 1

1

2

∴ z = – 4 + i 4 3 = 8 2 2

cos sin3 3

iπ π⎛ ⎞+⎜ ⎟⎝ ⎠

1

2

15. The given in equations are :

3x – 7 > 2(x – 6) ... (i) and 6 – x > 11 – 2x ... (ii)

(i) ⇒ 3x – 2x > – 12 + 7 or x > –5 ... (A) 1

(ii) ⇒ –x + 2x > 11 – 6 or x > 5 ... (B) 1

From A and B, the solutions of the given system are x > 5 1

Graphical representation is as under:

1

16. Given , ,b c c a a b

a b c

+ + + are in A.P.

∴1 ,1 ,1b c c a a b

a b c

+ + ++ + + will also be in A.P. 1

1

2

⇒ , ,a b c a b c a b c

a b c

+ + + + + + will be in A.P.

1

Since,a + b + c ≠ 0

⇒1 1 1

, ,a b c

will also be in A.P. 11

2

17. Following are possible choices:

Choice Part I Part II(i) 2 4(ii) 3 3 1(iii) 4 2

}


∴Total number of ways of selecting the questions are:

= ( )5 5 5 5 5 52 4 3 3 4 2C C C C C C× + × + × 1

1

2

=10 × 5 + 10 × 10 + 5 × 10 = 200 11

2

18. Let the intercepts on x-axis and y-axis be 4a, 3a respectively1

2

∴Equation of line is : 14 3

x y

a a+ = 1

1

2or 3x + 4y = 12a

(–3, –2) lies on it ⇒ 12a = –17 11

2Hence, the equation of the line is

3x + 4y + 17 = 01

219. Let the coordinates of R be (x, y, z)

∴x = 1(4) 2(0)

41 2

−= −

− 1

y = 1( 1) 2(0)

11 2

− −=

− 1

z = 1( 2) 2(0)

21 2

− −=

− ∴ R is (– 4, 1, 2) 1

Mid point of QR is 4 4 1 1 2 2

, ,2 2 2

− + − −⎛ ⎞⎜ ⎟⎝ ⎠ i.e., (0, 0, 0) 1

Hence verified.

20. f (x) =3

3 4

x

x

−+ ∴ f (x + Δx) =

3 ( )

3 4( )

x x

x x

− + Δ+ + Δ

1

2

f ′(x) =0

0

3 3lim

( ) ( ) 3 4 4 3 4lim

x

x

x x xf x x f x x x x

x x

Δ →

Δ →

− − Δ −−

+ Δ − + + Δ +=

Δ Δ 1


=0

(3 ) (3 4 ) (3 4 4 ) (3 )lim

( ) (3 4 4 ) (3 4 )x

x x x x x x

x x x xΔ →

− − Δ + − + + Δ −Δ + + Δ +

1

2

= 2 2

0

9 12 3 4 3 4 9 3 12 4 12 4lim

( ) (3 4 4 ) (3 4 )x

x x x x x x x x x x x x

x x x xΔ →

+ − − − Δ − Δ − + − + − Δ + Δ=

Δ + + Δ +

1

= 20

15 15lim

( ) (3 4 4 ) (3 4 ) (3 4 )x

x

x x x x xΔ →

− Δ −= =

Δ + + Δ + + 1

21. Assume that p is false, i.e., ~p is true

i.e., 3 is rational1

2∴ There exist two positive integers a and b such that

3a

b= , a and b are coprime

1

2⇒ a2 = 3b2 ⇒ 3 divides a2 ⇒ 3 divides a 1∴ a = 3c, c is a positive integer,∴ 9c2 = 3b2 ⇒ b2 = 3c2 ⇒ 3 divides b also 1∴ 3 is a common factor of a and b which is a contradictionas a, b are coprimes. 1

Hence p : 3 is irrational is true.

22. xi: 10 30 50 70 90

fi: 4 24 28 16 8 ∴ 80if =∑

1

2

fi x

i: 40 720 1400 1120 720 ∴ 4000i if x =∑ 1

| | | |i id x x= − : 40 20 0 20 40 ∴ Mean = 501

2

fi | d

i |: 160 480 0 320 320 ∴ | | 1280i if d =∑ 1

∴ Mean deviation = 1280

1680

= 1


PART C

23. (f + g) (4) = f(4) + g(4) = (4)2 + 4 = 16 + 2 = 18 11

2

(f – g) (9) = f(9) – g(9) = (9)2 – 9 = 81 – 3 = 78 11

2

(f . g) (4) = f(4) . g(4) = (4)2 . (4) = (16) (2) = 32 11

2

f

g⎛ ⎞⎜ ⎟⎝ ⎠ (9) =

2(9) (9) 8127

(9) 39

f

g= = = 1

1

2

24. sin 7x + sin 5x = 2 sin 6x cosx 1

sin 9x + sin 3x = 2 sin 6x cos 3x 1

cos 7x + cos 5x = 2 cos 6x cosx 1

cos 9x + cos 3x = 2 cos 6x cos 3x 1

∴ L.H.S =2sin 6 cos 2sin 6 cos3

2cos 6 cos 2cos 6 cos3

x x x x

x x x x

++

1

2

=sin 6 (cos3 cos ) sin 6

cos 6 (cos3 cos ) cos 6

x x x x

x x x x

+=

+ 1

= tan 6x1

2

25. Using 1Tr + = Cn n r rr x y− ⋅

we have 1

T4 =

7 33

3 2

310

3

xC

x

⎛ ⎞ −⎛ ⎞⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ 1

= 15 154

10.9.8 1 40

3.2.1 273x x− ⋅ ⋅ = − 1

5th term from end = (11 – 5 + 1) = 7th term from beginning 1


∴ T7 =

4 63

6 2

310

3

xC

x

⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ 1

=210.9.8.7 3

18904.3.2.1 1

⋅ = 1

26. Let the required line intersects theline 5x – y + 4 = 0 at (x

1, y

1) and

the line 3x + 4y – 4 = 0 at (x2, y

2).

∴ 5x1 – y

1 + 4 = 0 ⇒

y1 = 5x

1 + 4

3x2 + 4y

2 – 4 = 0 ⇒y

2 = 24 3

4

x−

∴ Points of inter section are (x1, 5x

1 + 4), 2

2

4 3,

4

xx

−⎛ ⎞⎜ ⎟⎝ ⎠

1

2

∴ 1 2 12

x x+= and

21

4 35 4

4 52

xx

−+ +

= 1

⇒ x1 + x

2 = 2 and20x

1 – 3x

2 = 20

1

2

Solving to get 1

26

23x = , 2

20

23x = 1

∴ y1 =

222

23, y

2 =

8

23

1

2

∴ Equation of line is y – 5 =

2225

23 ( 1)26

123

x−

−−

1

or 107x – 3y – 92 = 01

2


27. Here a2 = 169 and b2 = 144 ⇒ a = 13, b = 12 1

∴ Length of major axis = 26

Length of minor axis = 24

Since e2 = 2

2

144 25 51 1

169 169 13

be

a− = − = ∴ = 1

foci are (± ae, 0) = 5

13 ,013

⎛ ⎞± ⋅⎜ ⎟⎝ ⎠ = (± 5, 0) 1

vertices are (± a, 0) = (± 13, 0) 1

latus rectum = 22 2(144) 288

13 13

b

a= = 1

28. Classes: 30-40 40-50 50-60 60-70 70-80 80-90 90-100

f: 3 7 12 15 8 3 2∴ 50f =∑1

2

xi: 35 45 55 65 75 85 95

di: =

65

10ix −

–3 –2 –1 0 1 2 3

fi d

i: –9 –14 –12 0 8 6 6 15i if d = −∑ 1

2i if d : +27 28 12 0 8 12 18, 2 105i if d =∑ 1

Mean x = 15

65 10 65 3 6250

− × = − = 1

Variance σ2 =

22105 15

10 20150 50

⎡ ⎤−⎛ ⎞− ⋅ =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦1

1

2

S.D. σ = 201 14.17= 1

29. (i) Total number of days in a non leap year = 365

= 52 weeks + 1 day 1


∴ P(53 sun days) = 1

71

(ii) Total number of days in a leap year = 366

= 52 weeks + 2 days 1

∴ These two days can be Monday and Tuesday, Tuesday and Wednes-day, Wednesday and Thursday, Thursday and Friday, Friday and Satur-day, Saturday and Sunday, Sunday and Monday

∴ P(53 Fridays) = 2

7

1

2

(iii) P(53 Sunday and 53 Mondays) = 1

7 (from ii) 1

1

2

answers...answers 305 (iii) ( f g) x = 2x3 + x3 + 2x + 1 (iv)f g ⎛⎞ ⎜⎟ ⎝⎠ x = 221 1 x x...

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