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JEE • MATHEMATICS • INVERSE TRIGONOMETRY

Inverse trigonometry practice · Mixed difficulty

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Showing 31-60 of 605 questions

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31 Easy
31 of 605
Analyzing +4 | -1
The maximum value of sec⁡−1(7−5(x2+3)2(x2+2))\sec ^{-1}\left(\frac{7-5\left(x^{2}+3\right)}{2\left(x^{2}+2\right)}\right) is:
View answer and explanation
Correct option: D
2π3\frac{2 \pi}{3}
Option D (2π/3) is correct because it represents the maximum value of the inverse secant function after simplifying the expression inside.
32 Easy
32 of 605
Analyzing +4 | -1
If x=sin⁡−1(sin⁡10)x=\sin ^{-1}(\sin 10) and y=cos⁡−1(cos⁡10)y=\cos ^{-1}(\cos 10), then y−xy-x is equal to:
View answer and explanation
Correct option: D
π\pi
Option D (π\pi) is correct because, after finding the correct ranges for the inverse trigonometric functions, the difference y-x simplifies to π\pi.
33 Easy
33 of 605
Application +4 | -1
If cos⁡−1x+cos⁡−1(y/2)=α\cos ^{-1} x+\cos ^{-1}(y / 2)=\alpha, then 4x2−4xycos⁡α+y24 x^2-4 x y \cos \alpha+y^2 is equal to
View answer and explanation
Correct option: A
4sin⁡2α4 \sin ^2 \alpha
Option A (4sin⁡2α4 \sin ^2 \alpha) is correct because by applying the cosine subtraction formula and simplifying, the expression 4x2−4xycos⁡α+y24 x^2-4 x y \cos \alpha+y^2 is equal to 4sin⁡2α4 \sin ^2 \alpha.
34 Medium
34 of 605
Application +4 | -1
cos⁡−1[cos⁡{2cot⁡−1(2−1)}]\cos ^{-1}\left[\cos \left\{2 \cot ^{-1}(\sqrt{2}-1)\right\}\right] is equal to
View answer and explanation
Correct option: D
3π4\frac{3 \pi}{4}
Option B (π212\frac{{\pi^2}}{{12}}) is correct because the integral is solved using properties of even functions and trigonometric substitutions.
35 Easy
35 of 605
Analyzing +4 | -1
If sin⁡−1x+sin⁡−1y+sin⁡−1z=π,  then  x4+y4+z4+4x2y2z2=k(x2y2+y2z2+z2x2){\sin ^{ - 1}}x + {\sin ^{ - 1}}y + {\sin ^{ - 1}}z = \pi ,\;\mathrm{then}\;{x^4} + {y^4} + {z^4} + 4{x^2}{y^2}{z^2} = k\left( {{x^2}{y^2} + {y^2}{z^2} + {z^2}{x^2}} \right) Where kk is equal to
View answer and explanation
Correct option: B
2
Option B (2) is correct because after applying trigonometric identities and simplifying the given equation, the value of k is found to be 2.
36 Medium
36 of 605
Analyzing +4 | -1
If x1,x2,  x3,  x4{x_1},{x_2},\;{x_3},\;{x_4} are the roots of the equation x4−x3sin⁡2β−xcos⁡β−sin⁡β=0,  {x^4} - {x^3}\sin 2\mathrm{\beta } - x\cos \mathrm{\beta } - \sin \mathrm{\beta } = 0,\; then tan⁡−1+tan⁡−1x2+tan⁡−1x3+tan⁡−1x4{\tan ^{ - 1}} + {\tan ^{ - 1}}{x_2} + {\tan ^{ - 1}}{x_3} + {\tan ^{ - 1}}{x_4} is equal to
View answer and explanation
Correct option: B
π2−β\frac{\pi }{2} - \mathrm{\beta }
Option D (xx) is correct because simplifying the given expression using trigonometric identities leads to the result xx.
37 Easy
37 of 605
Application +4 | -1
If 0≤x<∞0 \le x < \infty, then cos⁡−1(1−x21+x2){\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right) equals
View answer and explanation
Correct option: A
2tan⁡−1x2{\tan ^{ - 1}}x
Option A (2tan⁡−1x2{\tan ^{ - 1}}x) is correct because it directly applies the trigonometric identity cos⁡−1(1−x21+x2)=2tan⁡−1x{\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right) = 2{\tan ^{ - 1}}x for 0≤x<∞0 \le x < \infty.
38 Medium
38 of 605
Application +4 | -1
Number of solutions to the equation sin⁡−1x−cos⁡−1x=cos⁡−1(32)\sin ^{-1} x-\cos ^{-1} x=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right) is
View answer and explanation
Correct option: A
One
Option A (One) is correct because solving the trigonometric equation yields only one valid solution for x.
39 Medium
39 of 605
Application +4 | -1
[tan⁡−1x]2−2[tan⁡−1x]+1≤0\left[\tan ^{-1} x\right]^2-2\left[\tan ^{-1} x\right]+1 \leq 0, where [x][x] denotes greatest interger ≤x\leq x, iff x∈[tan⁡k,tan⁡2)x \in[\tan k, \tan 2) then evaluate kk.
View answer and explanation
Correct option: A
1
Option A (1) is correct because solving the inequality leads to the condition that x belongs to the interval [tan 1, tan 2), thus k = 1.
40 Medium
40 of 605
Application +4 | -1
If sin⁡−1x+sin⁡−1y+sin⁡−1z=3π/2\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=3 \pi / 2, then the value of x100+y100+z100−9x101+y101+z101x^{100}+y^{100}+z^{100}-\frac{9}{x^{101}+y^{101}+z^{101}} is
View answer and explanation
Correct option: B
0
Option B (0) is correct because, given the condition sin⁡−1x+sin⁡−1y+sin⁡−1z=3π/2\sin^{-1}x + \sin^{-1}y + \sin^{-1}z = 3\pi/2, it implies x=y=z=1, which simplifies the expression to 0.
41 Medium
41 of 605
Analyzing +4 | -1
If sin⁡−1(x2−x42+x64……∞)+cos⁡−1(x−x22+x34….∞)=π2\sin ^{-1}\left(x^2-\frac{x^4}{2}+\frac{x^6}{4} \ldots \ldots \infty\right)+\cos ^{-1}\left(x-\frac{x^2}{2}+\frac{x^3}{4} \ldots . \infty\right)=\frac{\pi}{2} for 0<∣x∣<20 < |x| < \sqrt{2}, then cot⁡−1x\cot ^{-1} x equals to
View answer and explanation
Correct option: D
π4\frac{\pi}{4}
Option D (π4\frac{\pi}{4}) is correct because solving the given equation after simplifying the infinite geometric series leads to x=1x=1, and thus cot⁡−1(1)=π4\cot^{-1}(1) = \frac{\pi}{4}.
42 Easy
42 of 605
Application +4 | -1
The solution of tan⁡−12x+tan⁡−13x=π4{\tan ^{ - 1}}2x + {\tan ^{ - 1}}3x = \frac{\pi }{4} is
View answer and explanation
Correct option: A
16\frac{1}{6}
Option A (16\frac{1}{6}) is correct because it is the only solution that satisfies the original equation after checking for extraneous roots.
43 Easy
43 of 605
Application +4 | -1
tan⁡−1c1x−yc1y+x+tan⁡−1c2−c11+c2c1+tan⁡−1c3−c21+c3c2+...+tan⁡−11cn{\tan ^{ - 1}}\frac{{{c_1}x - y}}{{{c_1}y + x}} + {\tan ^{ - 1}}\frac{{{c_2} - {c_1}}}{{1 + {c_2}{c_1}}} + {\tan ^{ - 1}}\frac{{{c_3} - {c_2}}}{{1 + {c_3}{c_2}}} + ... + {\tan ^{ - 1}}\frac{1}{{{c_n}}} is equal to
View answer and explanation
Correct option: C
tan⁡−1xy{\tan ^{ - 1}}\frac{x}{y}
Option C is correct because the given expression simplifies to tan⁡−1xy{\tan ^{ - 1}}\frac{x}{y} due to the telescoping nature of the series.
44 Medium
44 of 605
Application +4 | -1
If tan⁡−1a+tan⁡−1b=sin⁡−11−tan⁡−1c{\tan ^{ - 1}}a + {\tan ^{ - 1}}b = {\sin ^{ - 1}}1 - {\tan ^{ - 1}}c, then
View answer and explanation
Correct option: C
1a+1b+1c−1abc=0\frac{1}{a} + \frac{1}{b} + \frac{1}{c} - \frac{1}{{abc}} = 0
Option C (1a+1b+1c−1abc=0\frac{1}{a} + \frac{1}{b} + \frac{1}{c} - \frac{1}{{abc}} = 0) is correct because it is the result of simplifying the given equation using the tangent addition formula and the fact that sin⁡−11=π2{\sin ^{ - 1}}1 = \frac{\pi}{2}.
45 Easy
45 of 605
Application +4 | -1
The value of cot⁡(∑n=119cot⁡−1(1+∑p=1n2p))\cot \left(\sum_{n=1}^{19} \cot ^{-1}\left(1+\sum_{p=1}^n 2 p\right)\right) is
View answer and explanation
Correct option: D
2119\frac{21}{19}
Option D (2119\frac{21}{19}) is correct because the summation telescopes, simplifying the expression inside the cotangent function.

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46 Easy
46 of 605
Analyzing +4 | -1
The number of solutions of the equation, sin⁡−1x=2tan⁡−1x\sin ^{-1} x=2 \tan ^{-1} x (in principal values) is :
View answer and explanation
Correct option: A
1
Option A (1) is correct because the equation sin⁡−1x=2tan⁡−1x\sin^{-1}x = 2 \tan^{-1}x has only one solution, which is x=0.
47 Medium
47 of 605
Application +4 | -1
If tan⁡−1a+xa+tan⁡−1a−xa=π6\tan ^{-1} \frac{a+x}{a}+\tan ^{-1} \frac{a-x}{a}=\frac{\pi}{6}, then x2x^2 is equal to
View answer and explanation
Correct option: C
23a22 \sqrt{3} a^2
Option C (23a22 \sqrt{3} a^2) is correct because it is the solution obtained after simplifying the given trigonometric equation using the tangent addition formula.
48 Easy
48 of 605
Understanding +4 | -1
cos⁡(cos⁡−1(725))\cos \left(\cos ^{-1}\left(\frac{7}{25}\right)\right) is equal to
View answer and explanation
Correct option: D
None of these
None of these is correct. The correct answer is 725\frac{7}{25} because cos⁡(cos⁡−1(x))=x\cos(\cos^{-1}(x)) = x for xx in the domain of cos⁡−1(x)\cos^{-1}(x), which is [−1,1][-1, 1].
49 Medium
49 of 605
Application +4 | -1
If xr=cos⁡π2r+isin⁡π2rx_r=\cos \frac{\pi}{2^r}+i \sin \frac{\pi}{2^r}, then x1x2x3…∞=x_1 x_2 x_3 \ldots \infty=
View answer and explanation
Correct option: D
-1
Option D is correct because the infinite product simplifies to -1 using Euler's formula and properties of geometric series.
50 Easy
50 of 605
Application +4 | -1
A value of xx satisfying tan⁡−1(x+3)−tan⁡−1(x−3)=sin⁡−1(3/5)\tan ^{-1}(x+3)-\tan ^{-1}(x-3)=\sin ^{-1}(3 / 5) is
View answer and explanation
Correct option: C
4
Option C (4) is the correct answer because it satisfies the given equation after applying the tangent subtraction formula and solving the resulting quadratic equation.
51 Easy
51 of 605
Analyzing +4 | -1
If α=sin⁡−132+sin⁡−113\alpha = {\sin ^{ - 1}}\frac{{\sqrt 3 }}{2} + {\sin ^{ - 1}}\frac{1}{3} and β=cos⁡−132+cos⁡−113\beta = {\cos ^{ - 1}}\frac{{\sqrt 3 }}{2} + {\cos ^{ - 1}}\frac{1}{3}, then
View answer and explanation
Correct option: C
α<β\alpha < \beta
Option C (α<β\alpha < \beta) is correct because by adding the two equations, we find α+β=π\alpha + \beta = \pi, and since α<π2\alpha < \frac{\pi}{2}, it follows that α<β\alpha < \beta.
52 Easy
52 of 605
Application +4 | -1
If tan⁡−1(ax)+tan⁡−1(bx)=π2,{\tan ^{ - 1}}\left( {\frac{a}{x}} \right) + {\tan ^{ - 1}}\left( {\frac{b}{x}} \right) = \frac{\pi }{2}, then xx is equal to
View answer and explanation
Correct option: A
ab\sqrt {ab}
Option A (ab\sqrt {ab}) is correct because solving the equation tan⁡−1(ax)+tan⁡−1(bx)=π2{\tan ^{ - 1}}\left( {\frac{a}{x}} \right) + {\tan ^{ - 1}}\left( {\frac{b}{x}} \right) = \frac{\pi }{2} leads to x=abx = \sqrt{ab}.
53 Medium
53 of 605
Application +4 | -1
If sin⁡−1x+sin⁡−1y+sin⁡−1z=3π2\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2} then the value of x100+y100+z100−3x101+y101+z101{x}^{100}+{y}^{100}+{z}^{100}-\frac{3}{{x}^{101}+{y}^{101}+{z}^{101}} is
View answer and explanation
Correct option: C
2
Option C (2) is correct because the value of the expression is 2, derived from the given condition on inverse sines.
54 Easy
54 of 605
Application +4 | -1
If 4cos⁡−1x+sin⁡−1x=π4 \cos ^{-1} x+\sin ^{-1} x=\pi, then xx equals to
View answer and explanation
Correct option: D
32\frac{\sqrt{3}}{2}
Option D (32\frac{\sqrt{3}}{2}) is correct because using the identity sin⁻¹x + cos⁻¹x = π/2 simplifies the equation, leading to x = √3/2.
55 Easy
55 of 605
Application +4 | -1
If tan⁡−11+x2−1−x21+x2+1−x2=α\tan ^{-1} \frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}=\alpha, then x2x^2 is equal to
View answer and explanation
Correct option: D
sin⁡2α\sin 2 \alpha
Option D (sin⁡2α\sin 2 \alpha) is correct because after applying componendo and dividendo and simplifying, x2x^2 equals sin⁡2α\sin 2 \alpha.
56 Easy
56 of 605
Application +4 | -1
If 12≤x≤1\frac{1}{{\sqrt 2 }} \le x \le 1, then sin⁡−1(2x1−x2){\sin ^{ - 1}}\left( {2x\sqrt {1 - {x^2}} } \right) equals
View answer and explanation
Correct option: B
π−2sin⁡−1x\pi - 2{\sin ^{ - 1}}x
Option B (π−2sin⁡−1x\pi - 2{\sin ^{ - 1}}x) is correct because it accounts for the range of the inverse sine function when 12≤x≤1\frac{1}{{\sqrt 2 }} \le x \le 1.
57 Easy
57 of 605
Application +4 | -1
sin⁡−1(cos⁡(sin⁡−1x))+cos⁡−1(sin⁡(cos⁡−1x))\sin ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)+\cos ^{-1}\left(\sin \left(\cos ^{-1} x\right)\right) is equal to
View answer and explanation
Correct option: B
π2\frac{\pi}{2}
Option B (π2\frac{{\pi}}{{2}}) is correct because using trigonometric identities simplifies the expression to π2\frac{{\pi}}{{2}}.
58 Easy
58 of 605
Application +4 | -1
Let cos⁡(2tan⁡−1x)=12\cos \left(2 \tan ^{-1} x\right)=\frac{1}{2}, then the value of xx will be
View answer and explanation
Correct option: B
13\frac{1}{\sqrt{3}}
Option B (13\frac{{1}}{{\sqrt{3}}}) is correct because it is the value of x that satisfies the given equation.
59 Easy
59 of 605
Application +4 | -1
If tan⁡−1x+tan⁡−1y=π4{\tan ^{ - 1}}x + {\tan ^{ - 1}}y = \frac{\pi }{4}, then
View answer and explanation
Correct option: A
x+y+xy=1x + y + xy = 1
Option A (x+y+xy=1x + y + xy = 1) is correct because it is derived by applying the tangent addition formula to the given equation and simplifying.
60 Medium
60 of 605
Analyzing +4 | -1
The value of cot⁡(∑n=119cot⁡−1(1+∑p=1n2p))\cot \left(\sum_{n=1}^{19} \cot ^{-1}\left(1+\sum_{p=1}^{n} 2 p\right)\right) is:
View answer and explanation
Correct option: A
2119\frac{21}{19}
Option A (2119\frac{{21}}{{19}}) is correct because by simplifying the summation and using the arctangent subtraction formula, the expression reduces to 2119\frac{{21}}{{19}}.
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