CHAPTER--1
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1) tan(cos⁻¹x) is equal to
A) √(1-x²)/x B) x/(1+x²)
C) √(1+x²)/x D) √(1-x²)
2) If sin⁻¹x= π/5, for some x ∈ (-1,1), then the value of cos⁻¹x is
A) 3π/10 B) 5π/10
C) 7π/10 D) 9π/10
3) The domain of sin⁻¹x is..
A) (-π, π) B) [-1, 1]
C) (0, 2π) D)(-∞,∞)
4) The principal value of sin⁻¹(-√3/2) is...
A) -2π/3 B)-π/3 C) 4π/3 D) 5π/3
5) 4 tan⁻¹1/5 - tan⁻¹1/329 is
A) π B) π/2 C) π/3 D) π/4
6) If sin(sin⁻¹1/5+ cos⁻¹x)= 1, then x is equal to..
A) 1 B) 0 C) 4/5 D) 1/5
7) Value of tan⁻¹1/2 + tan⁻¹1/3 is.
A) 0 B) π/3 C) π/6 D) π/4
8) If A= tan⁻¹x then sin 2A is..
A) 2x/(1-x²) B) 2x/√(1-x²)
C) 2x/(1+x²) D) none
9) The value of sin(2sin⁻¹(0.8)) is
A) sin 1.2° B) sin 1.6°
C) 0.48 C) 0.96
10) cos⁻¹(cos(5π/4)) is given by
A) 5π/4 B) 3π/4 C) -π/4 D) none
11) cos⁻¹1/2 + 2 sin⁻¹1/2 =
A) π/4 B) π/6 C) π/3 D) 2π/3
12) If sin⁻¹x + sin⁻¹y =2π/3, then cos⁻¹x + cos⁻¹y =
A) 2π/3 B) π/3 C) π/6 D) π
13) tan⁻¹1/4 + tan⁻¹2/9=
A) 1/2 cos⁻¹3/5 B) 1/2 sin⁻¹3/5
C) 1/2 tan⁻¹3/5 D) tan⁻¹1/2
14) tan⁻¹x/y - tan⁻¹{(x-y)/(x+y)} =
A) π/2 B) π/3 C) π/4 D) π/4 or -3π/4
15) The solution of sin⁻¹{2a/(1-a²)} - cos⁻¹{(1-b²)/(1+b²)}= tan⁻¹{2x/(1-x²)} is...
A) (a-b)/(1-ab) B) (1+ab)/(a-b)
C) (ab -1)/(a+b) D) (a-b)/(1+ab)
16) If sin⁻¹x/5 + cosec⁻¹5/4=π/2, then x=
A) 4 B) 5 C) 1 D) 3
17) 2 tan⁻¹1/3 + tan⁻¹1/7=
A) tan⁻¹49/29 B) π/2 C) 0 D) π/4
18) cos⁻¹15/17 + sin⁻¹1/5=
A) π/2 B)cos⁻¹171/221 C)π/4 D)N
19) cot [cos⁻¹(7/25)]=
A) 25/24 B) 25/7 C) 24/25 D) n
20) sin⁻¹3/5 + tan⁻¹1/7=
A) π/4 B) π/2 C)cos⁻¹4/5 D) π
21) A solution of the Equation tan⁻¹(1+x) + tan⁻¹(1-x)=π/2 is
A) x=1 B) x=-1 C) x=0 D) x=π
22) If x²+ y²+ z²= r², then tan⁻¹(xy/zr) + tan⁻¹(yz/xr) + tan⁻¹(xz/yr)=
A) π B) π/2 C) 0 D) n
23) If x+y+z= xyz, then tan⁻¹x + tan⁻¹y + tan⁻¹z=
A) π B) π/2 C) 1 D) N
24) If xy+ yz+ zx= 1, then tan⁻¹x + tan⁻¹y+ tan⁻¹z =
A) π B) π/2 C) 1 D) none
25) If x₁ , x₂ , x₃ , x₄ are roots of the Equation x⁴ - x³ sin 2b + x² cos 2b - x cos b - sinb = 0. Then tan⁻¹x₁ + tan⁻¹x₂ + tann⁻¹x₃ + tan⁻¹x₄=
A) b B) π/2 - b C) π- b D) - b
26) The principal value of sin⁻¹(sin(2π/3)) is..
A) 2π/3 B) -2π/3 C) π/3 D) 4π/3
27) cos⁻¹1/2 + 2 sin⁻¹1/2 =
A) π/4 B) π/6 C) π/3 D) 2π/3
28) The value of cos⁻¹{cos(7π/6)}
A) 7π/6 B) 5π/3 C) 5π/6 D) 13π/6
29) The value of cos(2cos⁻¹0.8)
A) 0.48 B) 0.96 C) 0.6 D) none
30) The value of sin(cot⁻¹x) is..
A) 1/(1+x²) B) 1/√(1+x²)
C) x/√(1+x²) D) x/(1+x²)
31) The value of tan{cos⁻¹4/5 + tan⁻¹2/3}
A) 6/17 B) 7/16 C) 17/6 D) n
32) The value of tan(1/2 cos⁻¹√5/3) is...
A) (3+√5)/2 B) (3+√5)
C) (3-√5)/2. D) None
33) If sin⁻¹{2a/(1+a²)} + sin⁻¹{2b/(1+b²)} = 2 tan⁻¹x, then x is equal
A) (a-b)/(1+ab) B) b/(1+ab)
C) b /(1-ab) D) (a+b)/(1-ab)
34) The value of
Cot⁻¹[{√(1-sinx) + √(1+ sinx)}/{√(1-sinx) - √(1+sinx)}] is (0<x<2π)
A) π - x/2 B) 2π - x C) x/2 D) 2π - x/2
35) sin[cot⁻¹{cos(tan⁻¹x)}] is..
A) √{(x²+2)/(x²+1)}
B) √{(x²+1)/(x²+2)}
C) x/√(x²+1) D) 1/√(x²+1)
36) If x≥ 1 then 2tan⁻¹x + sin⁻¹{2x/(1+x²)} is equal to
A) 4 tan⁻¹x B) 0 C) π/2 D) π
37) If A= tan⁻¹{x√3/(2k -x)} and B= tan⁻¹{(2x- k)/k√3}, then the value of A- B is..
A) 0 B) 45° C) 60° D) 30°
38) The Equation sin⁻¹x - cos⁻¹x = cos⁻¹√3/2 has
A) no solution
B) Unique solution
C) infinite number of solutions
D) none of these
39) If sin⁻¹x + sin⁻¹(1-x) = cos⁻¹x, then x equals
A) 1, -1 B) 1,0 C) 0, 1/2 D) none
40) If π/2 ≤ x ≤ 3π/2, then sin⁻¹(sinx) is equal to
A) x B) -x C) π + x D) π - x
41) If π≤ x ≤ 2π, then cos⁻¹(cosx) is equal to
A) x B) - x C) 2π + x D) 2π - x
42) The value of sin⁻¹(sin 10) is
A) 10 B)10 -3π C) 3π - 10 D) none
43) The value of tan⁻¹1 + tan⁻¹2+ tan⁻¹3 is
A) 0 B) 1 C) π D) -π
44) The value of sin⁻¹(cos(33π/5)) is
A) 3π/5 B) 7π/5 C) π/10 D) -π/10
45) If k≤ sin⁻¹x + cos⁻¹x+ tan⁻¹x ≤ K, then
A) k=0, K=π B) k=0, K=π/2
C) k= π/2, K=π D) none
46) The smallest and the largest values of tan⁻¹{1-x)/(1+x)} , 0≤x≤1 are
A) 0, π B) 0, π/4
C) -π/4, π/4 D) π/4, π/2
47) The greatest and least values of (sin⁻¹x)³ + (cos⁻¹x)³ are
A) -π/2, π/2 B) -π³/8, π³/8
C) π³/32, 7π³/8 D) none
48) If a< 1/32, then the number of solutions of (sin⁻¹x)³ +(cos⁻¹x)³= aπ³ is
A) 0 B) 1 C) 2 D) infinite
49) If x takes negative permissible value, then sin⁻¹x =
A) cos⁻¹√(1-x²) B)- cos⁻¹√(1-x²)
C)cos⁻¹√(x²- 1) D)π- cos⁻¹√(1-x²)
50) If 4 sin⁻¹x + cos⁻¹x= π, then x is
A) 1/2 B) √3/2. C) -1/2 D) none
51) sin⁻¹(1-x)- 2 sin⁻¹x= π/2, then x equal to
A) 0, 1/2 B) 0, 1/2 C) 0 D) none
52) If (tan⁻¹x)² + (cot⁻¹x)²= 5π²/8 then x is
A) -1 B) 1 C) 0 D) none
53) If tan x + tan(π/3 + x)+ tan(-π/3 +x)= K tan 3x, then K is
A) 1 B) 1/3 . C) 3 D) none
54) If x+ 1/x = 2, the principal value of sin⁻¹x is
A) π/4 B) π/2. C) π D) 3π/2
55) The numerical value of tan(2 tan⁻¹1/5 - π/4) is
A) 1 B) 0 C) 7/17 D) -7/17
56) If tan⁻¹(x +y)=33, and x= tan⁻¹3 , then y is
A) 0.3 B) tan⁻¹1.3 C) tan⁻¹0.3 D) tan⁻¹1/18
57) Two angles of a triangle are cot⁻¹2 and cot⁻¹3 then the third angles is
A) π/4 B) 3π/4 C) π/6 D) π/3
58) If A= 2tan⁻¹(2√2 - 1) and B= 3 sin⁻¹1/3 + sin⁻¹3/5, then
A) A= B B) A< B C) A> B D) none
59) tan⁻¹√{a(a+b+c)/abc} + tan⁻¹√{b(a+b+c)/cba} + tan⁻¹√{c(a+b+c)/abc} is..
A) π/4 B) π/2 C) π. D) 0
60) If sin⁻¹x + sin⁻¹y + sin⁻¹z= 3π/2, the value of x¹⁰⁰ + y¹⁰⁰ + z¹⁰⁰ - 9/(x¹⁰¹ + y¹⁰¹ + z¹⁰¹) is
A) 0. B) 1. C) 2 D) 3
61) a³/2 cosec²(1/2 tan⁻¹a/b) + b²/2 sec²(1/2 tan⁻¹b/a) is ..
A) (a-b)(a²+b²) B) (a+b)(a²-b²)
C) (a+b)(a²+b²) D) none
62) If a, b are positive quantities and if a₁= (a+b)/2, b₁=√(ab), a₂=(a₁+b₁)/2, b₂= √(a₂b₁) and so on, then
A) a∞= √(b²-a²)/cos⁻¹(a/b)
B) b∞= √(b²-a²)/cos⁻¹(a/b)
C) b∞= √(b²+a²)/cos⁻¹(b/a) d) n
63) tan 2π/5 - tan π/15 - √3 tan2π/5 tan π/15 is equal to
A) -√3 B) 1/√3 C) 1 D) √3
64) IF a₁ , a₂ , a₃, ..., aₙ is an AP with common difference d, then
tan[tan⁻¹{d/(1+ a₁a₂)} + tan⁻¹{d/(1+ a₂a₃)} +.....+ tan⁻¹{d/(1+a ₙ₋₁aₙ)} is equal to
A) (n-1)d/(a₁ + aₙ)
B) (n-1)d/(1+ a₁aₙ)
C) nd/(1+ a₁ aₙ)
D) (aₙ-a₁)/(a₁+ aₙ)
65) The number of real solutions of
tan⁻¹√(x(x+1)) + sin⁻¹√(x²+x+1) = π/2 is
A) zero B) one C) two D) infinite
CHAPTER -- 2
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FILL IN THE BLANKS
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1) The value of
tan⁻¹1/3+ tan⁻¹1/5+ tan⁻¹1/7 + tan⁻¹1/8 is____
2) If a, b, c are positive real positive real numbers and x= tan⁻¹√A + tan⁻¹√B + tan⁻¹√C, where A= a(a+b+c)/bc ,
B= b(a+b+c)/ac, C= c(a+b+c)/ab then tan x is equal to_____
3) ⁿₘ₌₁∑ tan⁻¹{2m/(m⁴+m²+2)}=_
4) The sum of first n terms of the series cot⁻¹3 + ⁻¹cot⁻¹7 + cot⁻¹13 + cot⁻¹21+ .... is given by _____
5) tan⁻¹(yz/xr) + tan⁻¹(xz/yr) + tan⁻¹(xy/zr) is equal to____ where r²= x²+ y²+z²
6) The value of sin[cos⁻¹√5/4 + tan⁻¹√(5/11) is ____
7) The number of positive integeral pairs satisfying the Equation tan⁻¹a + tan⁻¹b = tan⁻¹3 is _____
8) The value of sin(1/4 sin⁻¹√63/8) is _____
9) The value of cot(cos⁻¹(-1/3)=_
10) tan[1/2 sin⁻¹{2x/(1+x²)} + 1/2 cos⁻¹{(1-y²)/(1+y²)}]=____
11) If cot⁻¹x + cot⁻¹y+ cot⁻¹z= π/2 then x+y+z equal to_____
12) If cos⁻¹x/a + cos⁻¹y/b = K, then x²/a² + y²/b²= ____
13) sin[ cot⁻¹{tan(cos⁻¹x)}]=___
14) The greatest of the two angles A= tan⁻¹(2√2 - 1) and B= 3sin⁻¹1/3 + sin⁻¹3/5= ____
15) cot(π/4 - 2 cot⁻¹3)= ___
16) tan(cos⁻¹4/5 + tan⁻¹2/3= __
17) If sin⁻¹3/5 +sin⁻¹5/13 = sin⁻¹x, then x is ______
18) sin(sin⁻¹1/2 + cos⁻¹1/2)=____
19) tan⁻¹{(c₁x -y)/(c₁y +x)} +tan⁻¹{(c ₂ - c₁)/(1+c₂c₁)} + tan⁻¹ {c₃ - c₂)/(1+c₃c₂)} + ...+tan⁻¹1/cₙ = __
20) If cos⁻¹x +cos⁻¹y + cos⁻¹z=π, then x²+y²+z²+2xyz= _____
21) If tan⁻¹a/x + tan⁻¹b/x = π/2, then x= _____
22) If 3 sin⁻¹{2x/(1+x²)} - 4 cos⁻¹{(1-x²)/(1+x²)} + 2tan⁻¹{2x/(1-x²)} =π/3 then x = _____
23) If tan⁻¹(x-1) +tan⁻¹(x+1) + tan⁻¹x + 1= tan⁻¹3x then x=___
24) 2tan⁻¹1/3 +tan⁻¹1/7 = ___
25) cos(tan⁻¹3/4)= _____
CHAPTER --3
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1) If cos⁻¹x +cos⁻¹y + cos⁻¹z+ cos⁻¹u= 2π, then
A) x¹⁹⁹⁹+ y²⁰⁰⁰+ z²⁰⁰1+ u²⁰⁰²=
A) 1 B) -1 C) 0 D) none
2) If sin⁻¹x₁+ sin⁻¹x₂ +....+sin⁻¹x₂ₙ = nπ, then x₁ + x₂+ ...x₂ₙ=
A) n B) 0. C) 2n D) none
3) If sin⁻¹x₁+ sin⁻¹x₂+....+sin⁻¹x₂ₙ = nπ, then ²ⁿᵢⱼ₌₁∑ xᵢ xⱼ equal to
ᵢ≠ ⱼ
A) 2n B)n(2n-1) C) n(n-1) D) none
4) If cos⁻¹x₁ + cos⁻¹x₂ +...+cos⁻¹xₙ = nπ, then x₁ + x₂² + x₃² + ....+ xₙⁿ =
A) -1 B) 1 C) 0 D) none
1) if x satisfies the inequation x² - x - 2> 0 than a value exists for
A) sin⁻¹x B) sec⁻¹x= π/5, C) cos⁻¹x D) none
2) if x,y are roots of the Equation 6x² + 11x + 3= 0 then
A) both cos⁻¹x and cos⁻¹y are real
B) Both cosec⁻¹x and cosec⁻¹y are real
C) Both cot⁻¹x and cot⁻¹y are real
D) none
3) Let f(x)= sec⁻¹+ tan⁻¹x. Then f(x) is real
A) x ∈ [-1, 1] B) x∈R
C) x∈ (- ∞, -1] ∪[1,∞) D) none
4) If cos⁻¹ - sin⁻¹x= 0 then x is..
A) ±1/√2 B) 1C) √2 D) 1/√2
5) cosec⁻¹(cos x) is real if
A) A) x ∈ [-1, 1] B) x∈R
C) x is an odd multiple of π/2
D) x is a multiple of π
6) The principal value of sin⁻¹{sin5π/6} is
A) π/6 B) 5π/6 C) 7π/6 D) none
7) The principal value of cos⁻¹(- sin 7π/6) is
A) 5π/6 B) 7π/6 C) π/3 D) none
8) The principal value of sin⁻¹[ cos(sin⁻¹√3/2)) is
A) π/6 B) π/3 C) -π/3 D) none
9) The principal value of cos⁻¹{1/√2(cos 9π/10 - sin 9π/10)} is
A) 3π/20 B) 7π/20 C) 7π/10 D) n
10) The value of cos(tan⁻¹(tan 15π/4)) is
A) 1/√2 B) -1/√2 C) 1 D) none
11) -2π/5 is the principal value of
A) cos⁻¹(cos 7π/5)
B) sin⁻¹(sin 7π/5)
C) sec⁻¹(sec 7π/5). D) none
12)If cos⁻¹x + cos⁻¹y + cos⁻¹z = 3π, then xy + yz + zx is equal to
A) -3 B) 0 C) 3 D) -1
13) IF ¹⁰ᵢ₌₁∑ cos⁻¹xᵢ = 0 then. ¹⁰ᵢ₌₁ ∑ xᵢ is
A) 0 B) 10 C) 5 D) none
14) If ²ⁿᵢ₌₁∑ sin⁻¹xᵢ = nπ then ²ⁿᵢ₌₁∑ xᵢ is equal to
A) n B) 2n C) n(n-1)/2 d) none
15) The value of cos⁻¹(-1/2) + sin⁻¹(-√3/2) is...
A) π/3 B) 0 C) 2π/3 D) none
16) The value of tan(2tan⁻¹1/5 - π/4) is
A) 0. B) 1. C) 7/17 D) none
17) The Formula cos⁻¹{(1-x²)/(1+x²)}= 2 tan⁻¹x holds only for
A) x∈R B) |x|≤ 1 C) x ∈ (-1, 1]
D) x∈ [-1, +∞,)
18) tan⁻¹a + tan⁻¹b, where a> 0, b> 0, ab > 1, is equal to
A) tan⁻¹{(a+b)/(1-ab)}
B) tan⁻¹{(a+b)/(1-ab)}- π
C) π+ tan⁻¹{(a+b)/(1- ab)} D) none
19) The set of values of x for which tan⁻¹{x/√(1-x²)}= sin⁻¹x holds is..
A) R B) [-1,1] C)[0,1] D) [-1,0]
20) cos⁻¹[x²/2 + √(1-x²). √(2 - x²/4)= cos⁻¹x/2 - cos⁻¹x holds for
A) |x|≤1 B) x∈R C) 0≤x ≤ 1 D) 1-≤x≤0
21) If f(x)= sin⁻¹{√3x/2 - 1/2 √(1-x²)}, -1/2 ≤ x ≤ 1, then f(x) is equal to
A) sin⁻¹1/2 - sin⁻¹x
B) sin⁻¹x - π/6 C) sin⁻¹x + π/6 D) none
22) The Formula 2sin⁻¹x= sin⁻¹{2x √(1-x²)} holds for
A) x∈ [0,1]. B) x∈ [-1/√2, 1/√2]
C) x∈(-1,0) D) x∈[-√3/2, √3/2]
23) cos⁻¹(cosx)= x is satisfied by
A) x∈R B) x∈[0,π] C) x∈[-1,1] D) n
24) 2 tan⁻¹x + sin⁻¹{2x/(1+x²)} is independent of x then
A) x∈[1, +∞) B) x∈ (-1, 1]
C) (-∞, -1] D) none
25) If tan⁻¹2, tan⁻¹3 are two angles then the third angle is
A) π/4 B) 3π/4 C) π/2 D) none
26) ³ₙ₌₁∑ tan⁻¹1/n =
A) 0 B) π C) π/2 D) n
27) The value of tan{cos⁻¹4/5 + sin⁻¹2/√13 is
A) 7/16 B)17/6 C) 6/17 D) n
28) If sin⁻¹x + cos⁻¹x = π/6 then x is
A) 1/2 B) √3/2 C) -1/2 D) N
29) The value of 2 tan⁻¹[{√(1+x²) - 1}/x] is equal to
A) cot⁻¹x B) sec⁻¹x C) tan⁻¹x D) n
A) π B) π/2 C) 1 D) n
30) The value of 2 tan⁻¹1/3 + tan⁻¹1/7 is
A) π/4 B) π/2 C) π D) none
31) The value of cot⁻¹3 + cosec⁻¹√5 is
A) π/3 B) π/2 C) π/4 D) none
32) sin cot⁻¹tan cos⁻¹x is..
A) x B) √(1-x²) C) 1/x D) none
33) The value of tan⁻¹1/3 + sin
tan⁻¹1/5 + tan⁻¹1/7+ tan⁻¹1/8 is
A) π B) π/4 C) 3π/4 D) none
34) The value of tan²(sec⁻¹2) + cot²(cosec⁻¹3) is..
A) 13 B) 15 C) 11 D) none
35) tan(π/4 + 1/2 cos⁻¹x) + tan(π/4 - 1/2 cos⁻¹x), x≠ 0, is equal to
A) x B) 2x C) 2/x D) n
36) The number of real solutions of the Equation √(1+cos2x) = √2 sin⁻¹(sinx), -π≤ x ≤π is
A) 0 B) 1 C) 2 D) infinite
37) The Number of real solutions of tan⁻¹√x(x+1)+ sin⁻¹√(x²+x+1)= π/2
A) zero B) one C) two D) infinite
38) The Number of positive integeral solutions of the Equation tan⁻¹x + cos⁻¹{y/(1+y²)} = sin⁻¹3/√10 is
A) 0ne B) two C) zero D) none
39) Considering principal values, the number of solutions of tan⁻¹2x + tan⁻¹3x=π/4 is
A) two B) three C) one D) none
40) The number of real solutions of (x,y), where |y|= sinx, y=
Cos⁻¹(cosx), -2π≤ x ≤ 2π, is
A) 2 B) 1 C) 3 D) 4
41) If cos⁻¹x > sin⁻¹x then
A) x<0 B) -1<x <0
C) 0≤ x < 1/√2 D) 1≤ x< 1/√2
42) If If cot⁻¹n/π > π/6, n x∈ N, then the maximum value of n is
A) 1 B) 5 C) 9 D) none
43) The set of values of k for which x² - Kx + sin⁻¹(sin 4) > 0 for all real x is
A) ¢ B) (-2,2) C) 8 d) none
44) Let f(x)= sin⁻¹x + cos⁻¹x. then π/2 is equal to
A) f(-1/2) B) f(k²-2k+3), k ∈ R
C) f{1/(1+k²)} k ∈ R D) f(-2)
45) At x= 3/2, the value is real for
A) tan⁻¹x B) cosec⁻¹x
C) cos⁻¹2x D) none
46) If 1/2 < |x|< 1 then which of the following are real ?
A) sin⁻¹x B) tan⁻¹x c) sec⁻¹x,
D) sin⁻¹x
47) Let tan⁻¹(tan 5π/4)= a, tan⁻¹(- tan 2π/3) = b. Then
A) a> b B) 4a - 3b= 0
C) a+ b= 7π/11 D) none
48) Let f(x) = ₑ cos⁻¹sin(x+π/3). then
A) f(8π/9) = ₑ 5π/18
B) f(8π/9) ₑ13π/18
C) f(-7π/4)ₑπ/12
D) f(-7π/4)ₑ11π/12
49) If f(x) = cos⁻¹x+ cos⁻¹{x/2 + 1/2 √(3- 3x²)} then
A) f(2/3)=π/3
B) f(2/3)= 2cos⁻¹2/3 - π/3
C) f(1/3)= π/3
D) f(1/3) = 2cos⁻¹1/3 - π/3
50) The value of tan{1/2 sin⁻¹{2x/(1+x²)} + 1/2 cos⁻¹{(1-x²)/(1+x²)} is
A) 2x/(1-x²) if 0≤x ≤ 1
B) 2x/(1-x²) if x < 1
C) not finite x> 1 D) none
51) a, b and z are three angles given by a= 2tan⁻¹(√2 -1), b= 3 sin⁻¹1/√2 + sin⁻¹(-1/2) ≤ K and z= cos⁻¹1/3. Then
A) a> b B) b<z
C) a< z D) none
52) if 0< x < 1 then tan⁻¹{√(1-x²)/(1+x)} is equal to
A) 1/2 cos⁻¹x B) cos⁻¹√{(1+x)/2} C) sin⁻¹√{(1-x)/2} d) none
53) one of the values of x satisfying tan(sec⁻¹x) = sin cos⁻¹1/√5 is
A) √5/3 B) 3/√5 C) √5/3 d) -3/√5
54) If sin⁻¹x+ sin⁻¹y.cos⁻¹x - cos⁻¹y = π/3 then the number of values of (x,y) is..
A) two B) four C) zero D) none
55) The solution set of the Equation cos⁻¹x- sin⁻¹x= sin⁻¹(1-x) is
A) [-1,1] B) [0, 1/2] C) [-1,0] d) N
CHAPTER-- 4
1) the principal value of sin⁻¹(sin 2π/3) is
A) 2π/3 B) - π/3 C) π/3 D) none
2) The value of tan⁻¹1/3 + cos⁻¹ 2/√5 is
A) π/4 B) 5π/4 C) -3π/4 D) none
3) 2 tan⁻¹{cosec(tan⁻¹x) - tan (cot⁻¹x) is equal to
A) tan⁻¹1/x B) π- tan⁻¹x
C) tan⁻¹x
D) π - cot⁻¹x
4) 2(sin⁻¹cos sin⁻¹x + cos⁻¹sin cos⁻¹x) =
A) π/2 B) 0 C) 2x D) π
5) sin cos⁻¹tan sec⁻¹x (where |x|< √2) is equal to
A) √(x² -2) B) √(2-x²)
C) 1/√(x² -2). D) 1/√(2-x²)
6) tan(π/4+ 1/2 cos⁻¹a/b + tan(π/4 - 1/2 cos⁻¹a/b is equal to
A) 2b/a B) b/2a C) b/a D) a/2b
7) The most general solution of the Equation tan⁻¹(1/2 secx) + cot⁻¹(2 cosx) =π/3 are
A) nπ+ π/3 B) nπ - π/3
C) nπ ± π/6 D) 2nπ ± π/6
8) The most general solution of the Equation sin[2 cos⁻¹cot(2 tan⁻¹x)]= 0 are
A) ±1, ±(1+√2) B) 1, ±(1- √2)
C) ±1, ±(1±√2) D) -1, ±(1- √2)
9) For x=√3/2 and y= 1/√2,
tan⁻¹{x√(1- y²)} + y√(1- x²)] =
A) 7π/12 B) π/12 C) 5π/12 D) n
10) tan⁻¹1+ tan⁻¹2+ tan⁻¹3 is equal to
A) 0 B) π C) π/2 D) none
11) If cot⁻¹(√cosa) - tan⁻¹√(cosa) = x, then sinx is equal to
A) tan²a/2 B) cot²a/2 C) tana D) cota
12) If sin⁻¹(x - x²/2 +x³/4 ...) + Cos⁻¹(x² - x⁴/2+ x⁶/4 - ...)=π/2, 0<|x|< √2, then x is equal to
A) 1/2 B) 1 C) -1/2 D) -1
13) The value of cos[cos⁻¹(-√3/2) + π/6] is
A) 1. B) 0. C) -1. D) none
14) The value of cos⁻¹(-1/2) + sin ⁻¹ (-√3/2) is
A) π B) 0 C) 2π/3 D) none
15) ³⁰ᵢ₌₁∑ sin⁻¹xᵢ = 15π, then ³⁰ᵢ₌₂∑xᵢ is equal to
A)120 B) 60 C) 30 D) 15
16) If ᵢ₌₀ᵏ∑cos⁻¹xᵢ= 0 then ᵢ₌₀ᵏ∑xᵢ is equal to
A) k(k+1)/2 B) 2k
C) k(k-1)/2. D) k
17) The value of sin[cos⁻¹(-1/2)+ π/6] is equal to
A) 1/2 B) √3/2 ) -1/2 D) -√3/2
18) a sin⁻¹x - b cos⁻¹x = c, then a sin⁻¹x+ cos⁻¹x is equal to
A) {πab+c(a+b)}/(a+b)
B) {πab- c(a+b)}/(a+b)
C) {πab+c(a-b)}/(a+b)
D) {πba+c(a-b)}/(a-b)
19) If 3 sin⁻¹x - 4 cos⁻¹x= 4, then 4 sin⁻¹x + 3 cos⁻¹x is equal to
A) (4-25π)/7. B) (25π - 4)/7
C) (4 + 25π)/7 D) none
20) If sin⁻¹x + sin⁻¹y+ cos⁻¹z = 2π
Then x+y+z is equal to
A) 0. B) 3 c) 2 d) 1
21) The value of sin [cot⁻¹{tan(cos⁻¹x)}] (where x> 0) is
A) x/√(1-x²). B) √(1-x²)/x
C) x D) none
22) If (sin⁻¹x)²+ (cos⁻¹x)²= 5π²/8, x< 0 then x is equal to
A) -1/2 B) -1 C) -1/√2 D) none
23)If tan⁻¹√(x(x+1)) + sin⁻¹√(x²+x+1)= π/2, then x is
A) 0. B) 0, -1 c) 0,1 d) -1
24) Cos tan⁻¹sin cot⁻¹x (where x>0) is equal to
A) √{(x²+1)/(x²+2)}
B) √{(x²+2)/(x²+1)}
C) √{(x²+1)/(x²- 1)}
D) √{(x²-1)/(x²+1)}
25) cot⁻¹1/2 - 1/2 cot⁻¹4/3 =
A) -π/4 B) π/4 C) π/2 D) -π/2
26) The value of tan⁻¹{x cosa/(1- x cosa)} - cot⁻¹{cosa/(x- sina) is
A) independent of a
B) independent of x
C) π/2. D) 0
27) If 0≤ A < π/4, then tan⁻¹(1/2 tan 2A) + tan⁻¹(cotA) + tan⁻¹(cot³A) is equal to
A) π/4 B) 0 C) π/2 D) π
28) If a= sin⁻¹{1-x²)/(1+x²)} and b= 2 tan⁻¹{1+x)/(1-x)}, 0< x <1, then (a+b)/2 is equal to
A) π/4 B) π C) π/2. D) 0
29) cos⁻¹√[{1+√(1+x²)}/{2√(1+x²)}] (where x ≥ 0) is equal
A) 1/2 tan⁻¹x B) tan⁻¹x
C) 1/2 cot⁻¹x D) cot⁻¹x
30) The number of real solutions of the Equation √(1+ cos 2x) = √2 sin⁻¹(sinx), -π≤ x ≤π is
A) 0 b) 2. C) 2. D) none
31) If 0≤ a, b < 1 and sin⁻¹{2a/(1+a²)} + sin⁻¹{2b/(1+b²)} = 2 tan⁻¹x, then x is equal to
A) ab/(1-ab). B) (a+b)/(1-ab)
C) (a+b)/(ab -1) D) ab/(ab -1)
32) The value of tan[sin⁻¹3/5 + cot⁻¹3/2 is..
A) 7/16 B) 16/7 C) 17/6 D) 6/17
33) sin⁻¹[log₃x/3 ] is defined for
A) 1≤ x≤ 9 B) 1/9 ≤ x≤ 1
C) 1/9 ≤ x≤ 9 D) 0 < x≤ 9
34) If 2tan⁻¹(cosx) = tan⁻¹(2 cosec x), then x is equal to
A) nπ - π/4 B) 2nπ - π/4
C) nπ + π/4 D) 2π + π/4
35) The value of cos⁻¹x +cos⁻¹(x/2 + 1/2 √(3-3x²)), 1/2, ≤ x ≤1, is
A) 2π/3 B) π/6 C) π/3 D) none
36) The solution of the Equation sin⁻¹6x + sin⁻¹6√3 x= -π/2 is
A) 1/12 B) -1/12 C) ±1/12 D) n
37) The number of solutions of the Equation 2sin⁻¹√(x² - x+1)+ cos⁻¹(x² -x)= 3π/2 is
A) 0 B) 2. C) 3. D) 1
38) If a, b, c be positive real numbers, then tan⁻¹√{a(a+b+c)/bc} + tan⁻¹√{b(a+b+c)/ca} + +
tan⁻¹y √{c(a+b+c)/ab} is equal to
A) 0. B) π/2. C) π/4. D) π
39) a= sin⁻¹x + cos⁻¹x - tan⁻¹x, x≥ 0, then the smallest Interval in which a lies is given by
A) π/2 ≤a ≤3π/4 B) -π/4 ≤ a ≤ 0
B) 0 ≤ a ≤π/4 D) π/4 ≤ a≤π/2
40) a, b, c are three angles given by a= 2 tan⁻¹(√2 - 1), b= 3sin⁻¹1/√2 + sin⁻¹(1/√2) and c= cos⁻¹1/3, then
A) b> c B) a> c c) a> b d) none
41) If cos⁻¹x + cos⁻¹y + cos⁻¹z = 3π then xy + yz + zx is equal to
A) 3. B) 0 c) -3 d) -1
42) The value of cos⁻¹(cos 5π/3) + sin⁻¹(sin 5π/3) is equal to
A) 10π/3 B) π/2 C) 5π/2 D) 0
43) If sinx⁻¹+ sin⁻¹y + sin⁻¹z= 3π/2, then the value of x³ + y³ + z³ - 3xyz is equal to
A) -6 b) 6. C) 0. D) none
44) cot+cos⁻¹a+ cos⁻¹b) + cot(sin⁻¹a + sin⁻¹b) is equal to
A) π/2 B) π/4 C) - π/4 D) 0
45) If ∆ ABC is a right angled triangle with ang C =π/2, then tan⁻¹{a/(b+c)} + tan⁻¹{b/(c+a)} is equal to
A) π/2 B) π/4 C) 3π/4 D) 0
46) The number of solution (s) of the Equation sin⁻¹5/x + sin⁻¹2/x = π/2 is..
A) 0. .b) 1. C) 2 d) infinitely many
47) ∞ₖ₌₁∑ cot⁻¹(k² + 3/4) =
A) π/2 B) -π/2 C) cot⁻¹1/2 d) cot⁻¹2
48) cot (sin⁻¹√(13/17)= sin(tan⁻¹a), then a is equal to
A)4/17. B) √{17²-13²)/(17x13)}
C) √{17²-13²)/(17²+13³)} D) 2/3
49) If the three positive real numbers x,y,z (y<1) are in GP and tan⁻¹x, tan⁻¹y, tan⁻¹z are in AP, then
A) x= y= z B) z= 1/x
C) x= y= z = 0. D) none
50) The value of tan⁻¹{(sin 2 -1)/cos 2} is equal to
A) π/2 -1 B) 2 - π/2 C) 1 - π/4
D) π/4 -1
51) cot⁻¹9/2+ cot⁻¹25/2 + cot⁻¹49/2+ .....n terms is equal to
A) tan⁻¹{2n/(4n+5)}
B) cot⁻¹{2n/(4n+5)}
C) tan⁻¹{2n/(4n+3)}
D) cot⁻¹{2n/(4n+3)}
52) ∞ₖ₌₁∑ tan⁻¹(1/2r²)= t, then tant is equal to
A) 0. B) 1. C) π/4 d) none
CHAPTER -- 5
**************
1) The value of tan⁻¹1+ cos⁻¹-1/2 + sin⁻¹-1/2 =
A) π/4 B)5π/12 C) 3π/4 D)13π/12
2) cos(1/2 cos⁻¹1/8)=
A) 3/4 B) -3/4 C) 1/16 D) 1/4
3) 6 sin⁻¹(x²-6x+8.5)=π, if
A) x=1 B) x=2 C) x=3 D) x=4
4) The inequality sin⁻¹(sin 5)> x² - 4x holds if
A) x= 2 -√(9-2π) B) x= 2+ √(9-2π)
C) x∈2± √(9-2π) D) x>2 + √(9-2π)
5) If 1< x <√2, the number of solution of the Equation tan⁻¹(x-1) + tan⁻¹x+ tan⁻¹(x+1)= tan⁻¹3x is
A) 0 B) 1 C) 2 D) 3
6)sin(2 tan⁻¹1/3)+ cos(tan⁻¹2√2)
A) 12/15 B) 13/15 C) 14/15 D) 1
7) If 4sin⁻¹x+ cos⁻¹x=π then x is.
A) 1/2 B) 3/2 C) 5/2 D) none
8) The value of cos⁻¹(cos(8π/7)) =
A) 3π/7 B) 4π/7 C) 5π/7 D) 6π/7
9) The greater of tan 1 and tan⁻¹ 1 is ____
10) If sin⁻¹x+ sin⁻¹y=π, then x= Ky, where K= ____
11) The sum of the infinite series cot⁻¹ 2 +cot⁻¹8 + cot⁻¹18 + cot⁻¹32 + .... is equal to___
12) The value of sin⁻¹[cot{sin⁻¹√{(2- √3)/4} + cos⁻¹√12/4 + sec⁻¹√2}] =___
13) The numerical value of tan(cos⁻¹4/5 + tan⁻¹2/3) is___
14) sin⁻¹(-√2/2)+ cos⁻¹(-1/2) - tan⁻¹(-√3) + cot⁻¹(-1/√3)=____
15) The only integer satisfying the Equation sin⁻¹x + sin⁻¹(1-x) = cos⁻¹x is zero. T/F
16) tan⁻¹5 + tan⁻¹3 - cot⁻¹4/7 = ____
17) If 2 tan⁻¹(cosx)= tan⁻¹(2cosecx), then x= nπ - π/5, where n is any integer. T/F
18) If A= tan⁻¹1/7 and B= tan⁻¹1/3 , prove that cos 2A = sin 4B.
19) Prove that cos⁻¹x= 2sin⁻¹√{(1-x)/2} = 2 cos⁻¹{(1+x)/2}
20) If U= cot⁻¹√cosa - tan⁻¹√cosa prove that sin U = tan²(a/2)
21) cos⁻¹x = cos⁻¹{x/2 + 1/2 √(3-3x²)}
22) Prove
tan⁻¹x+ tan⁻¹(1/x) = π/2 if x> 0
- π/2 if x < 0
23) show 2tan⁻¹{tana/2 tan(π/2 - b/2)} = tan⁻¹{(sina cos b)/(cos a+ sin b)}
24) Solve sin⁻¹6x +sin⁻¹6√3 x= -π/2
25) If sin[2cos⁻¹{cot(2tan⁻¹x)}]=0
26) If sin⁻¹x + sin⁻¹y + sin⁻¹z=π, prove x⁴+y⁴+z⁴+4x²y²z³= 2(x²y² + y²z²+ z²x²)
27) a= tan⁻¹(2tan²x) - 1/2 sin⁻¹{3sinx/(5+4 cos 2x)} then x =
28) sin⁻¹x > cos⁻¹x holds for
A) all values of x. B) x∈ (0,1/√2)
C) x∈ (1/√2, 1) D) x= .75
29) tan(π/4 + 1/2 cos⁻¹a/b) + tan(π/4 - 1/2 cos⁻¹a/b) is..
A) a/b B) 2b/a C) a/2b D) b/2a
30) If cos⁻¹p/a + cos⁻¹q/b = x, then p²/a² - (2pqcosx)/ab +q²/b² =
A) sin²x B) cos²x C) tanx d) cot²x
31) The number of solution of the Equation tan⁻¹{1/(2x+1)} + tan⁻¹{1/(4x+1)}= tan⁻¹2/x² is .
A) 1. B) 2. C) 3. D) 4
32) The Equation sin⁻¹x = 2sin⁻¹a has a solution for
A) all real values of a
B) a< 1 C) -1/√2 ≤ a ≤ 1/√2
D) -1< a < 1
33) If tan⁻¹[x+(2x)] - tan⁻¹4/x - tan⁻¹[x -(2/x)]= 0, then x= ___ or __
34) The numerical value of tan(2tan⁻¹1/5 - π/4) is _____
35) If sin⁻¹{2a/(1+a²) + sin⁻¹{2b/(1+b²)} = 2 tan⁻¹x, then ____
36) If y= 2tan⁻¹x + sin⁻¹[2x/(1+x²)] for allx, then___ < y < _____
37) If a= sin⁻¹x + cos⁻¹x - tan⁻¹x (0≤ x ≤ 1), then the interval in which a lies is given by π/4≤ a≤π/2. T/F
38) If a= cot⁻¹7 + cot⁻¹8 + tan⁻¹18 then cota= 3. T/F
39) cos⁻¹√{(a-x)/(a-b)} = sin⁻¹√{x -b)/(a-b)} if a> x > b or a < x< b. T/F
40) Prove:
A) cos{tan⁻¹(sin(cot⁻¹x))}= √{(x²+1)/(x²+2)}
B) tan⁻¹x = 2tan⁻¹(cosec tan⁻¹x - tan cot⁻¹x ).
41) simplify : cos(2 cos⁻¹x + sin⁻¹x) at x= 1/5.
42) prove
A) tan⁻¹1 + tan⁻¹2 + tan⁻¹3= 2(tan⁻¹1 + tan⁻¹1/2+ tan⁻¹1/3=π
B) tan⁻¹√{a(a+b+c)/bc} + tan⁻¹√{b(a+b+c)/ca} + tan⁻¹√{c(a+b+c)/ab} =π
43) Find the simplest value of
tan⁻¹√[{(1 +x²)-1}/x] and tan[1/2 sin⁻¹{2x/(1+x²)} + 1/2 cos⁻¹{(1-y²)/(1+y²)}]
44) Express the Equation
Cot⁻¹[y/√(1-x² - y²)} =2 tan⁻¹{(3-4x²)/4x²} - tan⁻¹{(3-4x²)/x²} as a rational integeral Equation in x and y.
45) Prove:
A) If cos⁻¹x + cos⁻¹y + cos⁻¹z=π, then x²+y²+z²+2xyz= 1
B**) tan⁻¹{1/2(cos 2a sec 2b + cos 2b sec 2a)}= tan⁻¹[tan²(a+b) tan²(a-b) ]
C) tan⁻¹(yz/rz) + tan⁻¹(zx/ry) + tan⁻¹(xy/rz)= π/2 where r²= x²+ y²+ z²
46) If cos⁻¹x/2 + cos⁻¹y/3 = a, Prove that 9x²- 12xy cos a + 4y²= 36sin²a.
47)A) Solve the equation tan⁻¹(x+1) + tan⁻¹(x -1)= tan⁻¹31 and prove that sec²(tan⁻¹2) + cosec²(cot⁻¹3) = 15.
B) 2 cos⁻¹x= sin⁻¹(2x√(1-x²)).
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