For each $t \in \mathbb{R}$, let $[t]$ be the greatest integer less than or equal to $t$. Then $\displaystyle \lim_{x \to 0^{+}} x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \cdots + \left[\frac{15}{x}\right]\right)$

1

Let $f(x) = x^{2} + \dfrac{1}{x^{2}}$ and $g(x) = x - \dfrac{1}{x}$, $x \in \mathbb{R} - {-1,0,1}$. If $h(x) = \dfrac{f…

Topic: JEE Main 2018 (Offline)

2

Let $S={t\in\mathbb{R}: f(x)=|x-\pi|,(e^{|x|}-1)\sin|x|\ \text{is not differentiable at }t}$. Then the set $S$ is equal…

Topic: JEE Main 2018 (Offline)

3

Let $g(x)=\cos x^{2}$, $f(x)=\sqrt{x}$ and $\alpha,\beta\ (\alpha

Topic: JEE Main 2018 (Offline)

4

The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \frac{\sin^{2}x}{1+2^{x}},dx$ is :

Topic: JEE Main 2018 (Offline)

5

The integral $\displaystyle \int \frac{\sin^{2}x \cos^{2}x}{\left(\sin^{5}x + \cos^{3}x \sin^{2}x + \sin^{3}x \cos^{2}x…

Topic: JEE Main 2018 (Offline)

6

Let $y = y(x)$ be the solution of the differential equation $\sin x \dfrac{dy}{dx} + y \cos x = 4x,\ x \in (0,\pi).$ If…

Topic: JEE Main 2018 (Offline)

7

A straight line through a fixed point $(2,3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is …

Topic: JEE Main 2018 (Offline)

8

Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$. …

Topic: JEE Main 2018 (Offline)

9

If $\displaystyle \sum_{i=1}^{9}(x_i-5)=9$ and $\displaystyle \sum_{i=1}^{9}(x_i-5)^{2}=45$, then the standard deviatio…

Topic: JEE Main 2018 (Offline)

10

A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its colour is observed and this bal…

Topic: JEE Main 2018 (Offline)