Inverse Trigonometric Functions18 questions

Inverse Trigonometric Functions — JEE Maths Practice Questions & Solutions

18 questions on Inverse Trigonometric Functions with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

hard
Two functions f(x)f(x) and g(x)g(x) are defined as f(x)=log⁡3∣x−2x2−10x+24∣f(x)=\log_{3}\left|\dfrac{x-2}{x^{2}-10x+24}\right| and g(x)=sin⁡−1(2[x]−315)g(x)=\sin^{-1}\left(\dfrac{2[x]-3}{15}\right), then find the number of even integers for which (f(x)+g(x))(f(x)+g(x)) is defined ([⋅][\cdot] is GIF).
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hard
If sin⁡(30°+arctan⁡x)=1314\sin(30°+\arctan x)=\dfrac{13}{14} and 0<x<10<x<1, the value of xx is a3b\dfrac{a\sqrt{3}}{b}, where a,ba,b are positive integers with no common factors. Find the value of (a+b2)\left(\dfrac{a+b}{2}\right).
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medium
Let SS be the set of domain of f(x)=π2−tan⁡−1−x2+5x−6f(x)=\sqrt{\dfrac{\pi}{2}-\tan^{-1}\sqrt{-x^{2}+5x-6}}. If λ=α+1α\lambda=\alpha+\dfrac{1}{\alpha} where α∈S\alpha\in S and λ\lambda is an integer, then find the value of λ2\lambda^{2}.
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medium
If x=sin⁡−1(a6+1)+cos⁡−1(a4+1)−tan⁡−1(a2+1)x=\sin^{-1}(a^{6}+1)+\cos^{-1}(a^{4}+1)-\tan^{-1}(a^{2}+1), a∈Ra\in\mathbb{R}, then find the value of sec⁡2x\sec^{2}x.
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medium
If the roots of the equation x3−10x+11=0x^{3}-10x+11=0 are u,v,wu,v,w, then find the value of 3csc⁡2(tan⁡−1u+tan⁡−1v+tan⁡−1w)3\csc^{2}(\tan^{-1}u+\tan^{-1}v+\tan^{-1}w).
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medium
If the domain of the function f(x)=3cos⁡−1(4x)−πf(x)=\sqrt{3\cos^{-1}(4x)-\pi} is [a,b][a,b], then find the value of (4a+64b)(4a+64b).
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medium
If cot⁡−1(4+24)+cot⁡−1(4+64)+cot⁡−1(4+124)+…∞=tan⁡−1(ab)\cot^{-1}\left(4+\dfrac{2}{4}\right)+\cot^{-1}\left(4+\dfrac{6}{4}\right)+\cot^{-1}\left(4+\dfrac{12}{4}\right)+\ldots\infty=\tan^{-1}\left(\dfrac{a}{b}\right) where aa and bb are coprime, then find the value of (a3+b3)(a^{3}+b^{3}).
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medium
If lim⁡n→∞∑k=1ncot⁡−1(1+k+k2)=cot⁡−1(α)+cot⁡−1(β)\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\cot^{-1}(1+k+k^{2})=\cot^{-1}(\alpha)+\cot^{-1}(\beta), where α,β\alpha,\beta are prime numbers, then find (α+β)(\alpha+\beta).
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medium
If all the roots of the equation x3−3x=0x^{3}-3x=0 satisfy the equation (α−sin⁡−1(sin⁡2))x2−(β−tan⁡−1(tan⁡1))x+γ2−2γ+1=0(\alpha-\sin^{-1}(\sin 2))x^{2}-(\beta-\tan^{-1}(\tan 1))x+\gamma^{2}-2\gamma+1=0, then find the value of ∣cot⁡(β+γ)+cot⁡α∣|\cot(\beta+\gamma)+\cot\alpha|.
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medium
Find the number of values of xx satisfying simultaneously sin⁡−1x=2tan⁡−1x\sin^{-1}x=2\tan^{-1}x and tan⁡−1x(x−1)+csc⁡−11+x−x2=π2\tan^{-1}\sqrt{x(x-1)}+\csc^{-1}\sqrt{1+x-x^{2}}=\dfrac{\pi}{2}.
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medium
If ∑r=0∞2 arccot(r2+r+42)=kπ\displaystyle\sum_{r=0}^{\infty}2\,\text{arccot}\left(\dfrac{r^{2}+r+4}{2}\right)=k\pi, then find the value of kk.
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medium
Let f(α)=sin⁡−1(sin⁡α)+cos⁡−1(cos⁡α)f(\alpha)=\sin^{-1}(\sin\alpha)+\cos^{-1}(\cos\alpha) and g(β)=sin⁡−1(sin⁡β)+tan⁡−1(tan⁡β)g(\beta)=\sin^{-1}(\sin\beta)+\tan^{-1}(\tan\beta). Find the value of f(100)+g(8)f(100)+g(8).
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medium
Find the number of solutions of the equation sin⁡−1(4sin⁡2θ+sin⁡θ)+cos⁡−1(−1+6sin⁡θ)=π2\sin^{-1}(4\sin^{2}\theta+\sin\theta)+\cos^{-1}(-1+6\sin\theta)=\dfrac{\pi}{2} in θ∈[0,5π]\theta\in[0,5\pi].
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medium
Let g:R→Rg:\mathbb{R}\to\mathbb{R} be defined as g(x)=sgn(x2−5x+6)g(x)=\text{sgn}(x^{2}-5x+6). Find the number of solutions of sin⁡x=cos⁡−1(g(sin⁡−1x))\sin x=\cos^{-1}(g(\sin^{-1}x)) lying in [0,314][0,314].
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medium
If A=cot⁡−1(1)+12cot⁡−1(12)+13cot⁡−1(13)A=\cot^{-1}(1)+\dfrac{1}{2}\cot^{-1}\left(\dfrac{1}{2}\right)+\dfrac{1}{3}\cot^{-1}\left(\dfrac{1}{3}\right) and B=cot⁡−1(1)+2cot⁡−1(2)+3cot⁡−1(3)B=\cot^{-1}(1)+2\cot^{-1}(2)+3\cot^{-1}(3), then ∣B−A∣=aπb+cdcot⁡−1(3)|B-A|=\dfrac{a\pi}{b}+\dfrac{c}{d}\cot^{-1}(3) where a,b,c,d∈Na,b,c,d\in\mathbb{N} in lowest form. Find (a+b+c+d)(a+b+c+d).
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easy
Let f(x)=tan⁡−1(x2+kx+9−x)f(x)=\tan^{-1}(x^{2}+kx+9-x). If the range of f(x)f(x) lies in the interval (0,π2)\left(0,\dfrac{\pi}{2}\right) for all values of x∈Rx\in\mathbb{R}, then find the maximum integral value of kk.
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easy
The number of points x∈[−π2,3π2]x\in\left[-\dfrac{\pi}{2},\dfrac{3\pi}{2}\right] satisfying the equation 1+sin⁡−1(sin⁡x)=π31+\sin^{-1}(\sin x)=\dfrac{\pi}{3}.
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easy
If f(x)=tan⁡−11x2+x+1+tan⁡−11x2+3x+3+tan⁡−11x2+5x+7+⋯f(x) = \tan^{-1}\dfrac{1}{x^2+x+1}+\tan^{-1}\dfrac{1}{x^2+3x+3}+\tan^{-1}\dfrac{1}{x^2+5x+7}+\cdots to ∞\infty terms, then the value of ∣f′(0)∣|f'(0)| is.
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