Inverse Trignometry DPP

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Fresher Course for IIT JEE (Main & Advanced)–2017 Course : Fresher(XL) Batch

DAILY PRACTICE PROBLEM SHEET Subject : Mathematics

DPPS 1

Topic : Inverse Trigonometric Function Q.1

Q.2

Q.3

Q.4

Q.5

3π  x  –1  1  The number of real values of x satisfying tan–1  is :  + tan  3  = 2 4 1− x  x  (A) 0 (B) 1 (C) 2 (D) infinitely many

If the equation sin–1(4x – x2 – 5) + cos–1(4x – x2 – 5) + λx = 0 has a real solution. Then the value of [λ] is: (where [⋅] denotes greatest integer function) (A) –2 (B) –1 (C) 0 (D) Infinitely many

1 1 1 tan–1   + tan–1   + tan–1   + ……. ∞ =  3 7  13  π π π (B) (C) (A) 4 2 3

13π   –1 –1 tan–1  – tan  + cot (9) + cosec 8   7π (A) π (B) 8 If

∑ tan r =1

(A) – Q.7

1 3

–1

Q.9

 41    is equal to  4    5π (C) 8

4r + 4 π   + cot–1 m, then m is :  3  =– 2 4 4 r + 4 r + 3 r + 3   1 (B) (C) 3 3

(D)

3π 4

(D) – 3

The value of Lim (sin–1 [sin x] + cos–1 [cosx] – 2tan–1 [tan x]) is equal to [Note : [k] denotes largest integer x →0

function less than or equal to k.] π (A) π (B) 2 Q.8

π 6

    π θ3 θ5 θ 7 θ2 θ4 If tan–1  − α + θ − then maximum value of α + − + .....∞  + cot–1  α − 1 + − + .....∞  = 3! 5! 7 ! 2! 4!     2 equals to 1 1 1 (B) 1 (C) (D) (A) – 2 2 2

∞

Q.6

(D)

(C)

3π 2

(D) non-existent

Number of values of x satisfying the equation cos–13x + sin–12x = π is (A) 0 (B) 1 (C) 2 (D) 3 2  2x  –1  x − 1   , then the value of (f(10) – g(100)) is equal to Let f(x) = sin–1   and g(x) = cos  2 2  1+ x   x +1 (A) π – 2(tan–1(10) + tan–1(100)) (B) 0 (C) 2(tan–1(100) – tan–1(10)) (D) 2(tan–1(10) – tan–1(100))

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Q.10

Q.11

 x2 − c  π π   . Then the possible values of 'c' for which g is Let g : R →  ,  is defined by g(x) = sin–1  2  6 2   1+ x  surjective function, is 1 1   1   1  (B)  − 1, −  (D)  − , 1 (A)   (C) −  2 2 2        2  1 1   > sec–1 Statement-1 : cosec–1  + 2 2 

1 1   +  2 2

Statement-2 : cosec–1 x > sec–1 x if 1 ≤ x < 2 (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1. (C) Statement-1 is True, Statement-2 is False. (D) Statement-1 is False, Statement-2 is True. Passage # 1 (Q.12 & 13)

Q.12 Q.13

Q.14

Consider a real-valued function f(x) =

sin −1 x + 2 + 1 − sin −1 x

The domain of definition of f(x) is (A) [–1, 1] (B) [sin 1, 1]

(C) [–1, sin 1]

(D) [–1, 0]

The range of f(x) is (A) [0, 3 ]

(C) [1,

(D) [ 3 , 6 ]

(B) [1,

3]

6]

Match the following Column – I with Column - II (A) (B) (C)

(D)

Column – I If the equation x2 + 4 + 3 sin(ax + b) – 2x = 0 has atleast one real solution, where a, b ∈ [0, 2π], then sin(a + b) can be equal to sin–1 x ≤ cos–1 x, then x can be equal to The number of the ordered pairs (x, y) satisfying |y| = cos x and y = sin–1(sin x), where –2π ≤ x ≤ 3π, is equal to nπ 2 and If n ∈ N and the set of equations cos–1 x + (sin–1 y)2 = 4 π2 (sin–1 y) 2 − x = is consistent, then n can be equal to 16

Q.15

 π 1 π 1 1 −1  1   −1  1   Find the value of  tan  + cos   + tan  − cos   . 2 n =1   4 2  n   n   4 2

Q.16

If x = α satisfies the equation

Column – II

(P)

–1

(Q)

0

(R)

1

(S)

4

(T)

5

100

∑

sin −1 x 2 + cos −1 x = – 3, then find the value of (α2 + 2α + 3). −1 2 −1 cos x + sin x ANSWERS :

1. (A) 2. (B) 3. (A) 8. (A) 9. (C) 10. (C) 14. A → P; B → P,Q; C → T; D → R

4. (D) 11. (A) 15. 5050

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5. (C) 12. (C) 16. 2

6. (A) 13. (D)

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7. (B)

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