Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202522 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
22 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

JEE Main 2025 · 22 January · Morning Shift · Mathematics · Inverse Trigonometric Functions

Using the principal values of the inverse trigonometric functions,...

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of 1616\left(\left(\sec ^{-1} x\right)^2+\left(\operatorname{cosec}^{-1} x\right)^2\right) is :

Options

  1. A24π224 \pi^2
  2. B22π222 \pi^2
  3. C31π231 \pi^2
  4. D18π218 \pi^2

Correct answer

B. 22π222 \pi^2

Step-by-step solution

16(sec⁡−1x)2+(cosec⁡−1x)2Sec⁡−1x=a∈[0,π]−{π2}cosec⁡−1x=π2−a=16[a2+(π2−a)2]=16[2a2−πa+π24]max⁡]a=π=16[2π2−π2+π24]=20π2min⁡]a=π4=16[2×π216−π24+π24]=2π2Sum⁡=22π2\begin{aligned} & 16\left(\sec ^{-1} x\right)^2+\left(\operatorname{cosec}^{-1} x\right)^2 \\ & \operatorname{Sec}^{-1} x=a \in[0, \pi]-\left\{\frac{\pi}{2}\right\} \\ & \operatorname{cosec}^{-1} x=\frac{\pi}{2}-a \\ & =16\left[a^2+\left(\frac{\pi}{2}-a\right)^2\right]=16\left[2 a^2-\pi a+\frac{\pi^2}{4}\right] \\ & \max ]_{a=\pi}=16\left[2 \pi^2-\pi^2+\pi \frac{2}{4}\right]=20 \pi^2 \\ & \min ]_{a=\frac{\pi}{4}}=16\left[\frac{2 \times \pi^2}{16}-\frac{\pi^2}{4}+\frac{\pi^2}{4}\right]=2 \pi^2 \\ & \operatorname{Sum}=22 \pi^2\end{aligned}
Practice Inverse Trigonometric Functions on Quantrex Academy →

More from Inverse Trigonometric Functions

Considering only the principal values of the inverse trigonometric functions, the value… JEE Advanced 2026Let α = 3 ^ -1 ((6)/(11)) and β = 3 ^ -1 ((4)/(9)) , where inverse trigonometric… JEE Main 2026If ( ^ -1 (x 2 )) = ( ^ -1 1-x^2 ) , x ∈ (0,1) , then the value of x is : JEE Main 2026Let [·] denote the greatest integer function. If the domain of the function f(x) = ^ -1… JEE Main 2026Let 0 < α < 1 , β = (1)/(3α) and ^ -1 (1-α) + ^ -1 (1-β) = (π)/(4) . Then 6(α + β) is… JEE Main 2026If (π)/(4) + ∑_ p=1 ^ 11 ^ -1 ( 2^ p-1 1 + 2^ 2p-1 ) = α , then α is equal to __________. JEE Main 2026If y = ^ -1 ((3 x - 4 x)/(4 x + 3 x)) + 2 ^ -1 ( x 1+ 1-x^2 ) , then (dy)/(dx) at x = 3 2… JEE Main 2026Considering the principal values of inverse trigonometric functions, the value of the… JEE Main 2026If k= ((π)/(4)+(1)/(2) ^ -1 ((2)/(3)))+ ((1)/(2) ^ -1 ((2)/(3))) , then the number of… JEE Main 2026If the domain of the function f(x)= ^ -1 ( 1 x^ 2 -2 x-2 ) , is (-∞, α] [β, γ] [δ, ∞)… JEE Main 2026The number of solutions of ^ -1 4x + ^ -1 6x = (π)/(6) , where - 1 2 6 < x < 1 2 6 , is… JEE Main 2026Let the maximum value of ( ^ -1 x)^ 2 +( ^ -1 x)^ 2 for x ∈[- 3 2 , 1 2 ] be m n π^ 2… JEE Main 2026If the domain of the function f(x)= ^ -1 ((2 x-5)/(11-3 x))+ ^ -1 (2 x^ 2 -3 x+1) is the… JEE Main 2026The total number of real solutions of the equation θ= ^ -1 (2 θ)-(1)/(2) ^ -1 ((6 θ)/(9+… JEE Advanced 2025The value of ^ -1 ( 1+ ^2(2) -1 (2) )- ^ -1 ( 1+ ^2((1)/(2)) +1 ((1)/(2)) ) is equal to JEE Main 2025The sum of the infinite series ^ -1 ((7)/(4))+ ^ -1 ((19)/(4))+ ^ -1 ((39)/(4))+ ^ -1… JEE Main 2025 Full Inverse Trigonometric Functions list Inverse Trigonometric Functions JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs