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CONTENTS

CHAPTERS PAGES

PART–II

MODULE–IV

MODULE–V

MODULE–VI

MOCK TESTS

32.33.34.

Assertion-Reason & Column Matching Type Questions VI .....

.....

V AlgebraP

A-VI/1 A-VI/6

Mock Tests (For Revision)

(INVERSE TRIGONOMETRIC FUNCTION, DETERMINANTS & MATRICES)

(CALCULUS)

(THREE DIMENSIONAL GEOMETRY, VECTORS AND PROBABILITIES)

21.

Assertion-Reason & Column Matching Type Questions IV .....

24.25.26.27.28.29.30.31.

Assertion-Reason & Column Matching Type Questions V .....

Inverse Trigonometric Functions ..... 21/1–21/17––

A-IV/1–A-IV/5

Real Numbers ..... 24/1–24/6Limit and Continuity ..... 25/1–25/45Differentiability and Differentiation ..... 26/1–26/34Application of Derivatives ..... 27/1–27/39Indefinite Integrals ..... 28/1–28/29Definite Integrals ..... 29/1–29/48Areas under Curves ..... 30/1–30/23Differential Equations ..... 31/1–31/22

A-V/1–A-V/12

–––

–

..... MT/1–MT/15

Determinants ..... 22/1 22/37Matrices ..... 23/1 23/29

22.23.

Unit Test Paper No. 4 U-IV/1–U-IV/3

Unit Test Paper No. 5 U-V/1–U-V/3

Unit Test Paper No. 6 U-VI/1–U-VI/3

.....

Three Dimensional Geometry ..... 32/1 32/28ector ..... 33/1 33/38robability ..... 34/1 34/32

.....

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� IMPORTANT TERMS, FACTS AND FORMULAE

1. DOMAIN AND RANGE OF INVERSE FUNCTIONS

Function Domain Range

y = sin–1 x – 1 ≤ x ≤ 1 − ≤ ≤π π2 2

y

y = cos–1 x – 1≤ x ≤ 1 0 ≤ y ≤ π

y = tan–1 x R − < <π π2 2

y

y = cot–1 x R 0 < y < π

y = sec–1 x1

or 1xx

− ∞ < ≤ −⎧

⎨ ≤ < ∞⎩

02

< < ≠y yπ π,

y = cosec–1 x1

or 1xx

− ∞ < ≤ −⎧

⎨ ≤ < ∞⎩

− < < ≠π π2 2

0y y, .

2. PROPERTIES OF INVERSE FUNCTIONS

(A) sin–1 (sin x) = sin (sin–1 x) = x ; etc.

(B) 1 –1 1sin ( ) cosec ; etc.x

x− ⎛ ⎞=

⎜ ⎟

⎝ ⎠

(C) sin–1 ( – x) = – sin–1 x, cos–1 ( – x) = π – cos–1 x,tan–1 ( – x) = – tan–1 x ; etc.

(D) sin–1 x + cos–1 x = tan–1 x + cot–1 x = sec–1 x + cosec–1 x = .2

π

3. FORMULAE OF SUMS AND DIFFERENCES

(A) sin sin sin [ ]–1 –1− ± = − ± −1 2 21 1x y x y y x

(B) cos cos cos [ ]− − −± = + − −1 1 1 2 21 1x y xy x y

(C) tan tantan

tan

− −

−

−+ =

+−

<

++

−>

RS||

T||

1 1

1

1

11

11

x y

x y

xyxy

x y

xyxy

if

ifπ

(D) tan tan tan− − −− =−

+1 1 1

1x y

x y

xy

(E) tan tan tan .− − − −+ + =+ + −

− − −LNM

OQP

1 1 1

1x y z

x y z xyz

xy yz zxtan 1

cos–1

( – x) = π – cos–1

x

sin–1

x + cos–1

x =

π2

21/1

21 INVERSE TRIGONOMETRICFUNCTIONS

21/2 MODERN’S abc OF OBJECTIVE MATHEMATICS

4. MORE RESULTS

(A) (i) 2 2 11 1 2sin sin− −= −x x x (ii) 2 cos–1 x = cos–1 (2x2 – 1)

(iii) 22

1

1

1

2

11 1

21

2

21

2tan sin cos tan− − − −=

+=

−+

=−

xx

x

x

x

x

x(B) (i) 3 sin–1 x = sin–1 (3x – 4x3)

(ii) 3 cos–1 x = cos–1 (4x3 – 3x)

(iii) 33

1 31 1

3

2tan tan .− −=

−−

xx x

x

5. IMPORTANT SUBSTITUTIONS

Expression Substitution

a x2 2− x = a sin θ2 2

π π⎛ ⎞− < θ <⎜ ⎟

⎝ ⎠

a x a x2 2 2 2+ +or x = a tan θ2 2

π π⎛ ⎞− < θ <⎜ ⎟

⎝ ⎠

x a2 2− x = a sec θ (0 < θ < π)

a x a x2 2 2 2− +and x2 = a2 cos 2θ4 4

π π⎛ ⎞− < θ <⎜ ⎟

⎝ ⎠

1. sin–1 (sin x) = sin (sin–1 x) = x ; etc.2. sin–1 (– x) = – sin–1 x, cos–1 (– x) = π – cos–1 x ; etc.

3. sin sin sin ( )− − −+ = − + −1 1 1 2 21 1x y x y y x

4. cos cos cos ( )− − −+ = − − −1 1 1 2 21 1x y xy x y

5.

1

1 1

1

tan if 11

tan tantan if 1

1

x yxy

xyx y

x yxy

xy

−

− −

−

+⎧ <⎪ −⎪+ = ⎨ +⎪π + >⎪ −⎩

6. 22

1

1

1

2

11 1

21

2

21

2tan sin cos tan .− − − −=

+=

−+

=−

xx

x

x

x

x

x

Select the correct answer :

1. The value of 1 43sin cos

5−

⎛ ⎞⎛ ⎞π

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

is :

(A)3

5

π

(B)7

5

− π

(C)10

π

(D) .10

−π

(N.C.E.R.T. (Exemplar))

2. The value of 1 33

sin cos5

−

⎛ ⎞⎛ ⎞π

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

is :

(A)3

5

π

(B)7

5

− π

(C)10

π

(D) .10

−π


INVERSE TRIGONOMETRIC FUNCTIONS 21/3

3. The value of 1 7cot cos

25−

⎡ ⎤⎛ ⎞

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

is :

(A)25

24(B)

25

7

(C)24

25(D)

7.

24

(N.C.E.R.T. (Exemplar))4. The principal value of the expression

( )[ ]1cos cos 680−

− ° is :

(A)2

9

π

(B)2

9

− π

(C)34

9

π

(D) .9

π

(N.C.E.R.T. (Exemplar))5. If 3tan–1 x + cot–1 x = π, then x equals :

(A) 0 (B) 1

(C) –1 (D)1

.2


6. The domain of the function 1sin 1−

−x is :(A) [1, 2] (B) [–1, 1](C) [0, 1] (D) None of these.


7. If 1 12cos sin cos 0,

5− −

⎛ ⎞

⎜ ⎟+ =⎝ ⎠

x then x is equal to :

(A)1

5(B)

2

5

(C) 0 (D) 1.(N.C.E.R.T. (Exemplar))

8. The greatest and least values of (sin–1 x)2 + (cos–1 x)2

are respectively :

(A)2 25

and4 8

π π

(B) and2 2

π −π

(C)2 2

and4 4

π −π

(D)2

and 0.4

π

(N.C.E.R.T. (Exemplar))9. The domain of the function defined by :

f(x) = sin–1 x + cos x is :(A) [–1, 1] (B) [–1, π + 1](C) (–∞, ∞) (D) φ.

(N.C.E.R.T. (Exemplar))10. The value of sin (2 sin–1 (·6)) is :

(A) ·48 (B) ·96(C) 1·2 (D) sin 1·2.


11. If 1 1sin sin ,2

− −

π

+ =x y then the value of

cos–1 x + cos–1 y is :

(A)2

π

(B) π

(C) 0 (D)2

.3

π


12. If 1 1 4tan tan ,

5− −

π

+ =x y then cot–1 x + cot–1 y

equals :

(A)5

π

(B)2

5

π

(C)3

5

π

(D) π.


13. The value of 1 13 1

tan cos tan5 4

− −

⎛ ⎞

⎜ ⎟+⎝ ⎠

is :

(A)19

8(B)

8

19

(C)19

12(D)

3.

4


14. The value of the expression ( )( )1 1sin cot cos tan 1− −⎡ ⎤⎣ ⎦

is :

(A) 0 (B) 1

(C)1

3(D)

2.

3


15. The equation 1 1 1 1

tan cot tan3

− − −

⎛ ⎞

− = ⎜ ⎟

⎝ ⎠

x x has :

(A) no solution (B) unique solution(C) infinite number of solutions(D) two solutions.

(N.C.E.R.T. (Exemplar))16. If α ≤ 2 sin–1 x + cos–1 x ≤ β, then :

(A) ,2 2

−π π

α= β= (B) α = 0, β = π

(C)3

,2 2

−π π

α= β= (D) α = 0, β = 2π.



17. The value of tan2 (sec–1 2) + cot2(cosec–1 3) is :

(A) 5 (B) 11

(C) 13 (D) 15.(N.C.E.R.T. (Exemplar))

18. The value of the expression 1 1 12sec 2 sin

2− −

⎛ ⎞

⎜ ⎟+⎝ ⎠

is :

(A)6

π

(B)5

6

π

(C)7

6

π

(D) 1.


19. If 2

1 1 12 2 2

2 1 2sin cos tan ,

1 1 1− − −

⎛ ⎞⎛ ⎞ ⎛ ⎞−

⎜ ⎟+ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠+ + −

a a x

a a x

where a, x ∈ (0, 1), then the value of x is :

(A) 0 (B)2

a

(C) a (D) 22

.1−

a

a

(N.C.E.R.T. (Exemplar))20. If cos–1 x > sin–1 x, then :

(A)1

12< ≤x (B)

10

2≤ <x

(C)1

12

− ≤ <x (D) x > 0.

(N.C.E.R.T. (Exemplar))21. Considering only the principal values, if

1 1 1tan (cot ) sin cot ,

2x− −⎛ ⎞=

⎜ ⎟

⎝ ⎠

then x is :

(A)1

5(B)

2

5

(C)3

5(D) 5

3.

22. The value of sin (cot–1 x) is :

(A) 1 2+ x (B) x

(C) (1 + x2)–3/2 (D) (1 + x2)–1/2.

23. The principal value of 1 3sin

2− ⎛ ⎞

−⎜ ⎟⎜ ⎟

⎝ ⎠

is :

(A) – 2

3

π(B) − π

3

(C)4

3

π(D)

5

3

π.

24. If x = sin–1 k, y = cos–1 k, – 1 ≤ k ≤ 1, then the correctrelationship is :

(A) x + y = 2 (B) x – y = 2

(C) x y+ = π2

(D) x y− = π2

.

25. Two angles of a triangle are cot–1 2 and cot–1 3. Thenthe third angle is :

(A)3

4

π(B)

5

4

π

(C)π6

(D)π3

.

26. Solution set of cos–1 x – sin–1 x = sin–1 (1 – x) is :

(A) [ – 1, 1] (B) [ – 1, 0]

(C)1

0,2

⎡ ⎤

⎢ ⎥⎣ ⎦

(D) None of these.

27. If 0 ≤ x ≤ 1 and θ = sin–1 x + cos–1 x – tan–1 x, then :

(A) θ π=4

(B) θ π≥4

(C) θ π≤2

(D)π θ π4 2

≤ ≤ .

28. Principal value of :

1 13 7sin cos cos

2 6− −⎛ ⎞ ⎛ π ⎞⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎝ ⎠

is :

(A)π2

(B)3

2

π

(C)5

6

π(D) None of these.

29. Value of :

1 1 11 1 1cos 2 sin 3 cos

2 2 2− − −− ⎛ ⎞⎛ ⎞ ⎛ ⎞− + −

⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

– 4 tan–1 (– 1) equals :

(A)π

12(B)

25

12

π

(C)7

4

π(D)

11

4

π.

30. If sin sin cos− −+FHG

IKJ =1 11

51x , then x is equal to :

(A) 0 (B) 1

(C)1

5(D)

4

5.


31. The value of cos–15

cos3

π⎛ ⎞

⎜ ⎟

⎝ ⎠

+ sin–1 5

sin3

π⎛ ⎞

⎜ ⎟

⎝ ⎠

is :

(A)π2

(B)5

3

π

(C)10

3

π(D) 0. (Roorkee 2000)

32. The trigonometric equation sin–1 x = 2 sin–1 a has asolution for :

(A) all real values (B) | |a < 1

2

(C) | |a ≤ 1

2(D)

1

2

1

2< <| | .a

(A.I.E.E.E. 2003)

33. The value of x for which :sin (cot–1 (1 + x)) = cos (tan–1x) is

(A)1

2(B) 1

(C) 0 (D) – .1

2(I.I.T. (Screening) 2004)

34. If sin–1 5

x⎛ ⎞

⎜ ⎟⎝ ⎠

+ cosec–1 5

4⎛ ⎞

⎜ ⎟⎝ ⎠

= 2

π, then the value of x

is :(A) 3 (B) 4(C) 5 (D) 1. (A.I.E.E.E. 2007)

35. The value of − −

⎛ ⎞

+⎜ ⎟

⎝ ⎠

1 15 2cot cosec tan

3 3 is :

(A)5

17(B)

6

17

(C)3

17(D)

4

17.(A.I.E.E.E. 2008)

36. The value of 1 14 2tan cos tan

5 3− −⎡ ⎤⎛ ⎞ ⎛ ⎞+

⎢ ⎜ ⎟ ⎜ ⎟⎥

⎝ ⎠ ⎝ ⎠⎣ ⎦

is :

(A)6

17(B)

7

16

(C)16

7(D) None of these.

37. If cos–1 p + cos–1 q + cos–1 r = π, then :

p2 + q2 + r2 + 2pqr is :

(A) 3 (B) 1

(C) 2 (D) None of these.

38. If sin sin sin ,− − −+ + =1 1 1 3

2x y z

πthen :

x y zx y z

100 100 100101 101 101

9+ + −+ +

is :

(A) 1 (B) 2


39. The number of real solutions of :

tan ( ) sin− −+ + + + =1 1 21 12

x x x xπ

is :

(A) zero (B) one(C) two (D) infinite.

40. If cos cos ,− −+ =1 1x

a

y

bα then :

x

a

xy

ab

y

b

2

2

2

2

2− +cos α equals :

(A) cot2 α (B) cos2 α(C) tan2 α (D) sin2 α.

41. The principal value of :

1 1 9 9cos cos sin

10 102− ⎧ π π ⎫⎛ ⎞−⎨ ⎬⎜ ⎟

⎝ ⎠⎩ ⎭

is :

(A)3

20

π(B)

7

20

π

(C)7

10

π(D) None of these.

42. If i

n

ix n=

−∑ =

1

21sin ,π then

i

n

ix=∑

1

2is equal to :

(A) n (B) 2n

(C)n n( )+ 1

2(D) None of these.

43. sin cot–1 tan cos–1 x is equal to :

(A) x (B) 1 2− x

(C)1

x(D) None of these.

44. Value of 1 15cos tan tan

4−⎧ π ⎫⎛ ⎞

⎨ ⎬⎜ ⎟

⎝ ⎠⎩ ⎭

is :

(A) 1 (B) − 1

2

(C)1

2(D) None of these.


45. If cos–1 x + cos–1 y + cos–1 z = 3π, then :

xy + yz + zx equals :

(A) – 3 (B) – 1

(C) 0 (D) 3.

46. If i

n

ix n=

−∑ =

1

21sin π , then

i

n

ix=∑

1

2 equals :

(A)n n( )+ 1

2(B) 2n

(C) n (D) None of these.

47. The number of positive integral solutions of :

tan cos sin− − −++

=1 1

2

1

1

3

10x

y

yis :

(A) 0 (B) 1


48. Value of tan tan tan tan− − − −+ + +1 1 1 11

3

1

5

1

7

1

8is :

(A)π4

(B)3

4

π

(C) π (D) None of these.

49. If tan ,− + −=1

21 14

x

xthen x equals :

(A) tan 2 (B) tan 4

(C) tan 6 (D) tan 8.

50. The integral solution of :

1 1 11tan tan tan 3x

y− − −⎛ ⎞

+ =⎜ ⎟

⎝ ⎠

is :

(A) (1, 4) (B) (2, 1)

(C) (3, 13) (D) None of these.

51. If sin–1 x + cos–1 (1 – x) = sin–1 (– x), then x satisfies :

(A) 2x2 + 3x + 1 = 0 (B) 2x2 – 3x = 0

(C) 2x2 + x – 1 = 0 (D) 2x2 + x + 1 = 0.

52. If 1 11 1sec sin

1 1

x xy

x x− −⎛ ⎞ ⎛ ⎞+ −

= +⎜ ⎟ ⎜ ⎟− +⎝ ⎠ ⎝ ⎠

and

zx

x

x

x= cosec–1 2 3

3 2

3 2

2 31+

+FHG

IKJ +

++

FHG

IKJ

−cos , then y + z

equals :

(A) 0 (B)π2

(C) π (D) None of these.

53. If (tan ) (cot ) ,− −+ =1 2 1 225

8x x

πthen x equals :

(A) – 1 (B) 0

(C) 1 (D) 2.

54.m

n m

m m=

−∑

+ +1

14 2

2

2tan equals :

(A) tan–1 (n2 – n + 1) (B) tan–1 (n2 + n + 1)

(C) tan− ++ +

12

2 2

n n

n n(D) None of these.

55. If A tan 1=−

FHG

IKJ

− 3

2

x

k xand –1 2

B tan3

x k

k

−⎛ ⎞

= ⎜ ⎟

⎝ ⎠

,

then the value of (A – B) is :

(A) 0° (B) 30°

(C) 45° (D) 60°.

56. If sin sin ,− −+ =1 1

3 4 6

x y πthen the value of

x xy y2 2

9 4 3 16+ + is :

(A)1

4(B)

1

2

(C)3

4(D) None of these.

57. If 2 3

1sin .....2 4

x xx− ⎛ ⎞

− + −⎜ ⎟⎜ ⎟

⎝ ⎠

4 61 2cos .....

2 4 2

x xx− ⎛ ⎞ π

+ − + − =⎜ ⎟⎜ ⎟

⎝ ⎠

for 0 2< <| |x , then x equals :

(A) 1

2(B) 1

(C) − 1

2(D) – 1.

(I.I.T. (Screening) 2001)

58. cot ( cos ) tan ( cos ) ,− −− =1 1α α x then sin x =

(A) tan2

2

α(B) cot2

2

α

(C) tan α (D) cot .α2

(A.I.E.E.E. 2002)


59. The value of :

1 11 2 1

1 1 2

tan tan1

a x y a a

a y x a a− −⎛ ⎞ ⎛ ⎞− −

+⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

+−

+FHG

IKJ + +

−

+

FHG

IKJ

− − −

−tan ..... tan1 3 2

2 3

1 1

11 1

a a

a a

a a

a a

n n

n n

+ −tan 1 1

anis :

(A) 1tany

x− ⎛ ⎞

⎜ ⎟

⎝ ⎠

(B) 1tanx

y− ⎛ ⎞

⎜ ⎟

⎝ ⎠

(C) 1 (D) 0.

60. The value of :

21 1

2 2

1 2 1 1tan sin cos

2 21 1

x x

x x− −⎧ ⎫−⎪ ⎪+⎨ ⎬

+ +⎪ ⎪⎩ ⎭

is :

(A)2

11

2

x

xx

−≤ ≤if 0 (B)

2

11

2

x

xx

−<if

(C) not finite if x > 1 (D) None of these.

61. If tan tan( ) ( )

− −+

++

1 11

1 2

1

1 2 3

++

+ ++ +

− −tan( ) ( )

... tan( )

1 11

1 3 4

1

1 1n n

= tan–1 x, then x equals :

(A)n

n + 1(B)

n

n + 2

(C)n

n

−+

1

2(D)

n

n

++

1

2.

62. A.M. of the non-zero solutions of the equation :

tan tan tan− − −+

++

=1 1 12

1

2 1

1

4 1

2

x x xis :

(A)2

3(B)

5

3

(C)7

6(D)

11

3.

63. If x1, x2, x3, x4 are the roots of the equation :

x4 – x3 sin 2θ + x2 cos 2θ – x cos θ – sin θ = 0, thentan–1 x1 + tan–1 x2 + tan–1 x3 + tan–1 x4 equals :

(A)π θ2

− (B) θ

(C) – θ (D) π – θ.

64. If α < 1

32, then the number of solutions of

(sin–1 x)3 + (cos–1 x)3 = α π3 is :(A) 0 (B) 1(C) 2 (D) infinite.

65. If cos cos− −+ =1 1 5

12

x

a

y

b

πand

sin sin ,− −− =1 1

12

x

a

y

b

πthen the value of

x

a

y

b

2

2

2

2+ equals :

(A) 1 (B)3

4

(C)5

4(D) None of these.

66. If θ1, θ2, θ3 be the roots of x3 + mx2 + 3x + m = 0,then the general value of tan–1 θ1 + tan–1 θ2+ tan–1 θ3 is :

(A)nπ2

(B) nπ

(C) ( )2 12

n + π(D) None of these.

67. If cos cos ,− −− =1 1

2x

y α then 4x2 – 4xy cos α + y2

is equal to :(A) 4 (B) 2 sin2 α(C) – 4 sin2 α (D) 4 sin2 α.

(A.I.E.E.E. 2005)68. If 0 < x < 1, then :

21 x+ { }

1/ 221 1 1cos (cot ) sin (cot )x x x− −

⎡ ⎤−+⎣ ⎦

is equal to :

(A)21

x

x+

(B) x

(C) 21x x+(D) 21 x+

. (I.I.T. 2008)

1. The value of 23

1

1 1

cot cot 1 2n

n k

k−

= =

⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟+⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

∑ ∑ is :

(A)23

25(B)

25

23

(C)23

24(D)

24

23.

(I.I.T. (Advanced) 2013)


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

36. (D) 37. (B) 38. (C) 39. (C) 40. (D) 41. (D) 42. (B) 43. (A) 44. (C) 45. (D)46. (B) 47. (C) 48. (A) 49. (D) 50. (D) 51. (B) 52. (C) 53. (A) 54. (C) 55. (B)56. (A) 57. (B) 58. (A)

59. (B) 60. (A,C) 61. (B) 62. (C) 63. (A) 64. (A) 65. (C) 66. (B) 67. (D) 68. (C)

1. (D) 2. (A).

1. (D) 1 43sin cos

5−

⎛ ⎞⎛ ⎞π

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= 1 3sin cos 8

5−

⎛ ⎞⎛ ⎞π

⎜ ⎟π+⎜ ⎟⎝ ⎠⎝ ⎠

= 1 3sin cos

5−

⎛ ⎞π

⎜ ⎟

⎝ ⎠ = 1 3

sin sin2 5

−

⎛ ⎞⎛ ⎞π π

⎜ ⎟−⎜ ⎟⎝ ⎠⎝ ⎠

= 1sin sin10

−

⎛ ⎞⎛ ⎞−π

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= .10

−π

2. (D) 1 33sin cos

5−

⎛ ⎞⎛ ⎞π

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= 1 3sin cos 6

5−

⎛ ⎞⎛ ⎞π

⎜ ⎟π+⎜ ⎟⎝ ⎠⎝ ⎠

= 1 3sin cos

5−

⎛ ⎞⎛ ⎞π

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= 1 3sin sin

2 5−

⎛ ⎞⎛ ⎞π π

⎜ ⎟−⎜ ⎟⎝ ⎠⎝ ⎠

= 1sin sin10

−

⎛ ⎞⎛ ⎞−π

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= .10

−π

3. (D) 1 7cot cos

25−⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= 1 7cot cot

625 49−

⎡ ⎤

⎢ ⎥⎣ ⎦−

= 7 7

.24576

=

4. (A) cos–1[cos (– 680°)] = cos–1[cos 680°]= cos–1 [cos (720° – 40°)] = cos–1 [cos (–40°)]

= cos–1 [cos (40°)] = 40° = 2

.9

π

5. (B) 3 tan–1 x + cot–1 x = π ⇒ 2 tan–1 x + (tan–1 x + cot–1 x) = π

= 12 tan2

−

π

+ =πx ⇒ 12 tan2

−

π

=x

⇒ 1tan4

−

π

=x ⇒ tan–1 x = tan–1 1.

Hence, x = 1.

2. If tan–1 y = tan–1 x + tan–1 22

,1

x

x

⎛ ⎞

⎜ ⎟

⎝ ⎠−

when 1

.3

x <

Then a value of y is :

(A)3

23

1 3

x x

x

−

−

(B)3

23

1 3

x x

x

+

−

(C)3

23

1 3

x x

x

−

+

(D)3

23

.1 3

x x

x

+

+

(J.E.E (Main) 2015)

Moderns ABC Of ObjectiveMathematics JEE Main Part-2

Publisher : MBD GroupPublishers

ISBN : 9789383907663Author : J. P. Mohindru,Bharat Mohindru

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