Embibe Experts Solutions for Chapter: Definite Integration, Exercise 1: JEE Advanced Paper 2 - 2019

Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathrm{ℝ}$ be a continuous function such that $f\left(0\right)=1$ and ${\int }_0^{\frac{\pi }{3}}f\left(t\right)dt=0$

Then which of the following statements is (are) TRUE?

For any real number $x$, let $\left[x\right]$ denotes the largest integer less than or equal to $x$. If $I={\int }_0^{10}\left[\sqrt{\frac{10x}{x+1}}\right]dx$, then the value of $9I$ is ____.

For $a\in R, \left|a\right|>1,$ let $\underset{n\to \infty }{\lim }\left(\frac{1+\sqrt[3]{2}+\dots +\sqrt[3]{n}}{n^{7/3}\left(\frac{1}{{\left(an+1\right)}^2}+\frac{1}{{\left(an+2\right)}^2}+\dots +\frac{1}{{\left(an+n\right)}^2}\right)}\right)=54.$ Then the possible value(s) of $a$ is/are:

The value of the integral $\underset{0}{\overset{\pi /2}{\int }}\frac{3\sqrt{\cos \theta }}{{\left(\sqrt{\cos \theta }+\sqrt{\sin \theta }\right)}^5}d\theta$ equals

The value of the integral $\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{1+\sqrt{3}}{{\left({\left(x+1\right)}^2{\left(1-x\right)}^6\right)}^{\frac{1}{4}}} dx$ is

Which of the following inequalities is/are TRUE?

If $I=\frac{2}{\pi }\underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}\frac{dx}{\left(1+e^{\mathrm{s}\mathrm{i}\mathrm{n}x}\right)\left(2-\mathrm{c}\mathrm{o}\mathrm{s}2x\right)}$ then $27I^2$ equals____

For each positive integer $n$, let $y_n={\frac{1}{n}\left((n+1)(n+2)\mathrm{...}(n+n)\right)}^{\frac{1}{n}}$. If $\underset{n\to \infty }{\lim }{y}_n=L,$ then the value of $\left[L\right]$ (where $\left[x\right]$ is the greatest integer less than or equal to $x$) is ____