JEE Mains · Maths · STD 12 - 2. inverse trigonometric function
Let \(0 < \alpha < 1\), \(\beta = \dfrac{1}{3\alpha}\) and \(\tan^{-1}(1-\alpha) + \tan^{-1}(1-\beta) = \dfrac{\pi}{4}\). Then \(6(\alpha + \beta)\) is equal to:
- A \(6\)
- B \(7\)
- C \(8\)
- D \(9\)
Answer & Solution
Correct Answer
(B) \(7\)
Step-by-step Solution
Detailed explanation
Given \(\tan^{-1}(1-\alpha) + \tan^{-1}(1-\beta) = \dfrac{\pi}{4}\) Applying the formula \(\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\dfrac{x+y}{1-xy}\right)\), we get: \(\dfrac{(1-\alpha) + (1-\beta)}{1 - (1-\alpha)(1-\beta)} = \tan\left(\dfrac{\pi}{4}\right) = 1\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If the term independent of \(x\) in the expansion of \(\left(\sqrt{\mathrm{ax}}{ }^2+\frac{1}{2 \mathrm{x}^3}\right)^{10}\) is 105 , then \(\mathrm{a}^2\) is equal to :JEE Mains 2024 Medium
- Let the mean and the variance of 6 observation \(a, b\), \(68,44,48,60\) be \(55\) and \(194 \), respectively if \(a>b\), then \(a+3 b\) isJEE Mains 2024 Hard
- Consider the triangles with vertices \(A (2,1) B (0,0)\) and \(C ( t , 4), t \in[0,4]\). It the maximum and the minimum perimeters of such triangles are obtained at \(t=\alpha\) and \(t=\beta\) respectively, then \(6 \alpha+21 \beta\) is equal to \(.........\).JEE Mains 2023 Hard
- For the system of linear equations \(2 x+4 y+2 a z=b\) \(x+2 y+3 z=4\) \(2 x-5 y+2 z=8\) which of the following is NOT correct?JEE Mains 2023 Hard
- The sum of all the solutions of the equation \((8)^{2 x}-16 \cdot(8)^x+48=0\) is :JEE Mains 2024 Hard
- If \(\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}\) are in an \(A.P.\) and \(\log _e \mathrm{a}-\) \(\log _e 2 b, \log _e 2 b-\log _e 3 c, \log _e 3 c-\log _e a\) are also in an \(A.P,\) then \(a: b: c\) is equal toJEE Mains 2024 Hard
More PYQs from JEE Mains
- If a tangent to the ellipse \(x^{2}+4 y^{2}=4\) meets the tangents at the extremities of its major axis at \(\mathrm{B}\) and \(\mathrm{C}\), then the circle with \(\mathrm{BC}\) as diameter passes through the point:JEE Mains 2021 Hard
- Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability that the ball drawn is white is \( p/q \) (where \( gcd(p,q)=1 \)), then \( p+q \) is equal to:JEE Mains 2026 Easy
- Consider a matrix \(A =\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]\), where \(\alpha, \beta, \gamma\) are three distinct natural numbers. If \(\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32} \times 3^{16}\), then the number of such \(3 -\) tuples \((\alpha, \beta, \gamma)\) is \(.....\)JEE Mains 2022 Hard
- If \(\lim _{x \rightarrow 0} \frac{\alpha x e^{x}-\beta \log _{e}(1+x)+\gamma x^{2} e^{-x}}{x \sin ^{2} x}=10, \alpha, \beta, \gamma \in R\), then the value of \(\alpha+\beta+\gamma\) is \(......\)JEE Mains 2021 Hard
- Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be defined as \(f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 .\) Then, the value of \(\sum_{\mathrm{k}=1}^{20} \frac{1}{\sin (\mathrm{k}) \sin (\mathrm{k}+\mathrm{f}(\mathrm{k}))}\) is equal to:JEE Mains 2021 Hard
- Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a thrice differentiable odd function satisfying \(f^{\prime}(\mathrm{x}) \geq 0, f^{\prime}(\mathrm{x})=f(\mathrm{x}), f(0)=0, f^{\prime}(0)=3\). Then \(9 f\left(\log _{\mathrm{c}} 3\right)\) is equal to _______.JEE Mains 2025 Hard