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Mere Teacher · Previous Year Questions
JEE Main Mathematics · Inverse Trigonometric Functions · Set 1 (Q1-30)
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1.
The sum of the infinite series cot−1(47)+cot−1(419)+cot−1(439) is:
(1)2π+cot−1(21)
(2)2π−cot−1(21)
(3)2π−tan−1(21)
(4)2π+tan−1(21)
2.
For non-negative integer n, let f(n)=∑k=0nsin2(n+2k+1π)∑k=0nsin(n+2k+1π)sin(n+2k+2π) Assuming cos−1x takes values in [0,π] which of the following options is/are correct? (JEE Advanced 2019)
(1)f(4)=23
(2)If α=tan(cos−1f(6)), then α2+2α−1=0
(3)sin(7cos−1f(5))=0
(4)limn→∞f(n)=21
3.
If cot(cos−1x)=sec(tan−1(b2−a2a))b>a, then x=
(1)2b2−a2b
(2)abb2−a2
(3)2b2−a2a
(4)ab2−a2
4.
Given 0≤x≤21 then the value of tan(sin−1(2x+21−x2)−sin−1x) is
(1)1
(2)3
(3)-1
(4)31
5.
Domain of cos−1[x] is, where [ ] denotes a greatest integer function
(1)(−1,2]
(2)(−1,2)
(3)[−1,2]
(4)[−1,2)
6.
If 2sinh−1(1−a2a)=log(1−x1+x), then x is equal to
(1)a
(2)a1
(3)1−a2
(4)1−a21
7.
The value of x which satisfies sin(cot−1x)=cos(tan−1(1+x)) is
(1)−21
(2)21
(3)-1
(4)1
8.
The value of sin−1(cos553π) is
(1)53π
(2)5−3π
(3)10π
(4)10−π
9.
If ∑r=150tan−12r21=p, then the value of tanp is :
(1)102101
(2)5150
(3)100
(4)5051
10.
Considering the principal values of the inverse trigonometric functions, sin−1(23x+211−x2),−21<x<21 is equal to
(1)6−5π−sin−1x
(2)65π−sin−1x
(3)6π+sin−1x
(4)4π+sin−1x
11.
If cos−1x+cos−1y+cos−1z=3π, then
(1)x+y+z−3=0
(2)x+y+z+3=0
(3)x+2y+3z−5=0
(4)x−y−z=0
12.
If α=3sin−1(116) and β=3cos−1(94), where the inverse trigonometric functions take only the principal values, then the correct option(s) is (are) (JEE Advanced 2015)
(1)cosβ>0
(2)sinβ<0
(3)cos(α+β)>0
(4)cosα<0
13.
The number of solutions of the equationsin−1(x2+31)+cos−1(x2−32)=x2 for x∈[−1,1] and [x]denotes the greatest integer less than or equal to x, is:
(1)2
(2)0
(3)4
(4)Infinite
14.
If tan−1(1+1.21)+tan−1(1+2.31)+…+tan−1(1+n(n+1)1)=tan−1[x], then x is equal to
(1)n+11
(2)n+1n
(3)n+21
(4)n+2n
15.
For any y∈R, let cot−1(y)∈(0,π) and tan−1(y)∈(−2π,2π). Then, the sum of all solutions of the equation tan−19−y26y+cot−16y9−y2=32π for 0<|y|<3, is equal to [2023 Adv.]
(1)23−3
(2)3−23
(3)43−6
(4)6−43
16.
If sin−1x+cos−1y=52π, then cos−1x+sin−1y is
(1)52π
(2)53π
(3)54π
(4)103π
17.
cot−1(2⋅12)+cot−1(2⋅22)+cot−1(2⋅32)+… up to ∞=
(1)π/2
(2)π/5
(3)π/4
(4)π/3
18.
If x=log(y+y2+1) then y=
(1)tanhx
(2)cothx
(3)sinhx
(4)coshx
19.
If sin−1x<cos−1x, then
(1)−1≤x<21
(2)−3≤x<−1
(3)21<x≤1
(4)1<x<3
20.
Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos−1(x)-2sin−l(x) = cos−1(2x) is equal to :
(1)0
(2)l
(3)21
(4)−21
21.
Let x∈R and |x|<1. Then tanh−1x=
(1)21log(1−x1+x)
(2)21log(1+x1−x)
(3)21log(x+1−x2)
(4)21log(x−1−x2)
22.
If α and β are the least and the greatest values of f(x)=(sin−1x)2+(cos−1x)2 for all x∈R respectively, then 8(α+β)=
(1)π2
(2)11π2
(3)9π2
(4)25π2
23.
The real valued function f(x)=x−[x]cosec−1x, where [x] denotes the greatest integer less than or equal to x, isdefined for all x belonging to:
(1)all reals except integers
(2)all non-integers except the interval [−1,1]
(3)all integers except 0,−1,1
(4)all reals except the Interval [−1,1]
24.
The domain of the function defined by f(x)=cos−1x−1 is
(1)[0,2]
(2)[−1,1]
(3)[0,1]
(4)[1,2]
25.
The value of cot(∑n=119cot−1(1+∑p=1n2p)) is:
(1)1921
(2)2119
(3)2322
(4)2223
26.
IfS=(x∈R:sin(x2+2x+2x+1)−sin−1(x2+1x)=4π)then ∑x∈S(sin((x2+x+5)2π)−cos((x2+x+5)π)) is equal to_________.
27.
50tan(3tan−1(21)+2cos−1(51))+42tan(21tan(22)) is equal to_____.
28.
If y=cos(3π+cos−12x), then (x−y)2++3y2 is equal to
29.
If k=tan(4π+21cos−1(32))+tan(21sin−1(32)), then the number of solutions of the equation sin−1(kx−1)=sin−1x−cos−1x is ____
30.
Let S=(x:cos−1x=π+sin−1x+sin−1(2x+1)). Then ∑(2x−1)2 is equal to .
