JEE Mains · Maths · STD 12 - 5. continuity and differentiation
If \(y^{2}+\log _{e}\left(\cos ^{2} x\right)=y, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right),\) then
- A \(\operatorname{|y}^{\prime \prime}(0) \mid=2\)
- B \(\left|y^{\prime}(0)\right|+\left|y^{\prime \prime}(0)\right|=3\)
- C \(\operatorname{|y}^{\prime}(0)\left|+\operatorname{|y}^{\prime \prime}(0)\right|=1\)
- D \(y^{\prime \prime}(0)=0\)
Answer & Solution
Correct Answer
(A) \(\operatorname{|y}^{\prime \prime}(0) \mid=2\)
Step-by-step Solution
Detailed explanation
\(y ^{2}+\ln \left(\cos ^{2} x \right)= y \quad x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) for \(x=0 \quad y=0\) or 1 Differentiating wrt x \(\Rightarrow 2 y y^{\prime}-2 \tan x=y^{\prime}\) \(\text { At }(0,0) y^{\prime}=0\) At \((0,1) y^{\prime}=0\) Differentiating wrt…
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
- Let the vertex \(A\) of a triangle \(ABC\) be \((1, 2)\), and the mid-point of the side \(AB\) be \((5, -1)\). If the centroid of this triangle is \((3, 4)\) and its circumcenter is \((\alpha, \beta)\), then \(21(\alpha + \beta)\) is equal to:JEE Mains 2026 Medium
- If a circle \(C\) passing through \((4, 0)\) touches the circle \(x^2 + y^2 + 4x - 6y - 12 = 0\) externally at a point \((1, -1),\) then the radius of the circle \(C\) isJEE Mains 2013 Hard
- Let \(\left\{a_{n}\right\}_{n-1}^{\infty}\) be a sequence such that \(a_{1}=1, a_{2}=1\) and \(a_{n+2}=2 a_{n+1}+a_{n}\) for all \(n \geq 1 .\) Then tha value of \(47 \sum_{n=1}^{\infty} \frac{a_{n}}{2^{3 n}}\) is equal to \(.....\)JEE Mains 2021 Hard
- Let \(z_0\) be a root of the quadratic equation, \(x^2 + x + 1= 0.\) If \(z = 3 + \,6iz_0^{81}\, - 3iz_0^{93}, \) then arg \(z\) is equal toJEE Mains 2019 Hard
- Let \(f\left( x \right) = \left\{ \begin{array}{l}
\max \left\{ {\left| x \right|,{x^2}} \right\},\,\,\,\,\left| x \right| \le 2\\
8 - 2\left| x \right|,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2 < \left| x \right| \le 4\,\,\,\,
\end{array} \right.\). Let \(S\) be the set of points in the interval \((-4, 4)\) at which \(f\) is not differentiable. Then \(S\)JEE Mains 2019 Hard - Let \(f(x)=\int x^3 \sqrt{3-x^2} d x\). If \(5 f(\sqrt{2})=-4\), then \(f(1)\) is equal toJEE Mains 2025 Medium
More PYQs from JEE Mains
- Let \(K\) be the set of all real values of \(x\) where the function \(f\left( x \right) = \sin \,\left| x \right| - \left| x \right| + 2\,\left( {x - \pi } \right)\,\cos \,\left| x \right|\) is not differentiable. Then the set \(K\) is equal toJEE Mains 2019 Hard
- Sum of squares of modulus of all the complex numbers \(z\) satisfying \(\bar{z}=i z^{2}+z^{2}-z\) is equal toJEE Mains 2022 Hard
- Let \(A\) and \(B\) be independent events such that \(\mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=2 \mathrm{p} .\) The largest value of \(\mathrm{p}\), for which \(\mathrm{P}\) (exactly one of \(\mathrm{A}, \mathrm{B}\) occurs \()=\frac{5}{9}\), is :JEE Mains 2021 Hard
- The number of solutions of the equation \(\cos \left(x+\frac{\pi}{3}\right) \cos \left(\frac{\pi}{3}-x\right)=\frac{1}{4} \cos ^{2} 2 x, x \in[-3 \pi\) \(3 \pi]\) isJEE Mains 2022 Hard
- If \(\left\{a_{i}\right\}_{i=1}^{n}\) where \(n\) is an even integer, is an arithmetic progression with common difference \(1\) , and \(\sum \limits_{ i =1}^{ n } a _{ i }=192, \sum \limits_{ i =1}^{ n / 2} a _{2 i }=120\), then \(n\) is equal toJEE Mains 2022 Hard
- Let \(f(x)\) be a polynomial of degree \(5\), and have extrema at \(x = 1\) and \(x = -1\). If \(\displaystyle\lim_{x \to 0} \left(\dfrac{f(x)}{x^3}\right) = -5\), then \(f(2) - f(-2)\) is equal to:JEE Mains 2026 Hard