JEE Mains · Maths · STD 11 - 12. limits
\(\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)\) is equal to:
- A \(\frac{9 \pi^2}{8}\)
- B \(\frac{11 \pi^2}{10}\)
- C \(\frac{3 \pi^2}{2}\)
- D \(\frac{5 \pi^2}{9}\)
Answer & Solution
Correct Answer
(A) \(\frac{9 \pi^2}{8}\)
Step-by-step Solution
Detailed explanation
\( \lim _{x \rightarrow \frac{\pi}{2}} \frac{0-\{\sin (2 x)+\cos (x)\} \cdot 3 x^2}{2\left(x-\frac{\pi}{2}\right)} \) \( =\lim _{x \rightarrow \frac{\pi}{2}} \frac{-\{2 \sin x \cos x+\cos x\} 3 x^2}{2\left(x-\frac{\pi}{2}\right)} \)…
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 mean of the data
be \(5.\) If \(m\) and \(\sigma^2\) are respectively the mean deviation about the mean and the variance of the data, then \(\frac{3 \alpha}{m+\sigma^2}\) is equal to \(..........\).\(X\) \(1\) \(3\) \(5\) \(7\) \(9\) \((f)\) \(4\) \(24\) \(28\) \(\alpha\) \(8\) JEE Mains 2023 Hard - Let \(\mathrm{ABC}\) be a triangle with \(\mathrm{A}(-3,1)\) and \(\angle \mathrm{ACB}=\theta, 0<\theta<\frac{\pi}{2} .\) If the equation of the median through \(\mathrm{B}\) is \(2 \mathrm{x}+\mathrm{y}-3=0\) and the equation of angle bisector of \(\mathrm{C}\) is \(7 \mathrm{x}-4 \mathrm{y}-1=0\) then \(\tan\, \theta\) is equal to:JEE Mains 2021 Hard
- Let \(x=2\) be a root of the equation \(x^2+p x+q=0\) and \(f(x)=\left\{\begin{array}{cc}\frac{1-\cos \left(x^2-4 p x+q^2+8 q+16\right)}{(x-2 p)^4}, & x \neq 2 p \\ 0, & x=2 p\end{array}\right.\) Then \(\lim _{x \rightarrow 22^{+}}[f(x)]\) where [. ] denotes greatest integer function, is \(........\)JEE Mains 2023 Hard
- Let \(x=(8 \sqrt{3}+13)^{13}\) and \(y=(7 \sqrt{2}+9)^9\). If \([t]\) denotes the greatest integer \(\leq t\), thenJEE Mains 2023 Hard
- Let \(P Q\) be a chord of the parabola \(y^2=12 x\) and the midpoint of \(PQ\) be at \((4,1)\). Then, which of the following point lies on the line passing through the points \(\mathrm{P}\) and \(\mathrm{Q}\) ?JEE Mains 2024 Medium
- Let the position vectors of the vertices \(A, B\) and \(C\) of a triangle be \(2 \hat{i}+2 \hat{j}+\hat{k}, \quad \hat{i}+2 \hat{j}+2 \hat{k}\) and \(2 \hat{i}+\hat{j}+2 \hat{k}\) respectively. Let \(l_1, l_2\) and \(l_3\) be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) respectively, then \(l_1^2+l_2^2+l_3^2\) equals :JEE Mains 2024 Hard
More PYQs from JEE Mains
- Let \(A=\left[a_{i j}\right]\) be a square matrix of order \(3\) such that \(a_{i j}=2^{j-i}\), for all \(i, j=1,2,3\). Then, the matrix \(A ^{2}+ A ^{3}+\ldots+ A ^{10}\) is equal toJEE Mains 2022 Hard
- The sum and product of the mean and variance of a binomial distribution are \(82.5\) and \(1350\) respectively. They the number of trials in the binomial distribution is.JEE Mains 2022 Hard
- Let \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\) and \(A =\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]\). For \(k \in N\), if \(X ^{\prime} A ^{ k } X =33\), then \(k\) is equal to.JEE Mains 2022 Hard
- In an examination,\(5\) students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is \(..........\).JEE Mains 2023 Hard
- Let \(a_{n}=\int_{-1}^{n}\left(1+\frac{x}{2}+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\ldots \ldots .+\frac{x^{n-1}}{n}\right) d x\) for \(n \in N\). Then the sum of all the elements of the set \(\left\{n \in N: a_{n} \in(2,30)\right\}\) is \(...........\)JEE Mains 2022 Hard
- If \(\lim _{\mathrm{t} \rightarrow 0}\left(\int_0^1(3 x+5)^{\mathrm{t}} \mathrm{d} x\right)^{\frac{1}{t}}=\frac{\alpha}{5 \mathrm{e}}\left(\frac{8}{5}\right)^{\frac{2}{3}}\), then \(\alpha\) is equal to ________JEE Mains 2025 Hard