Let $\mathbb{N}$ denote the set of all natural numbers. Define two binary relations on $\mathbb{N}$ as $R_1 = {(x,y) \in \mathbb{N} \times \mathbb{N} : 2x + y = 10}$ and $R_2 = {(x,y) \in \mathbb{N} \times \mathbb{N} : x + 2y = 10}$. Then:

1

The least positive integer $n$ for which $\left(\dfrac{1 + i\sqrt{3}}{1 - i\sqrt{3}}\right)^{n} = 1$ is:

Topic: JEE Main 2018 (16 April Morning Shift)

2

Let $p, q$ and $r$ be real numbers $(p \ne q,, r \ne 0)$, such that the roots of the equation $\dfrac{1}{x+p} + \dfrac{…

Topic: JEE Main 2018 (16 April Morning Shift)

3

If the function $f$ defined as $f(x)=\dfrac{1}{x}-\dfrac{kx-1}{e^{2x}-1},\ x\ne0$, is continuous at $x=0$, then the ord…

Topic: JEE Main 2018 (16 April Morning Shift)

4

Let $M$ and $m$ be respectively the absolute maximum and the absolute minimum values of the function $f(x)=2x^{3}-9x^{2…

Topic: JEE Main 2018 (16 April Morning Shift)

5

If $\displaystyle \int \frac{\tan x}{1+\tan x+\tan^2 x},dx = x - \frac{K}{\sqrt{A}}\tan^{-1}!\left(\frac{K\tan x + 1}{\…

Topic: JEE Main 2018 (16 April Morning Shift)

6

The number of numbers between $2000$ and $5000$ that can be formed with the digits $0,1,2,3,4$ (repetition of digits is…

Topic: JEE Main 2018 (16 April Morning Shift)

7

$\displaystyle\lim_{x\to0} \dfrac{(27+x)^{1/3}-3}{9-(27+x)^{2/3}}$ equals :

Topic: JEE Main 2018 (16 April Morning Shift)

8

Let $\dfrac{1}{x_1},\dfrac{1}{x_2},\ldots,\dfrac{1}{x_n}$ $(x_i\ne0\text{ for }i=1,2,\ldots,n)$ be in A.P. such that $x…

Topic: JEE Main 2018 (16 April Morning Shift)

9

Let $M$ and $m$ be respectively the absolute maximum and the absolute minimum values of the function $f(x) = 2x^{3} - 9…

Topic: JEE Main 2018 (16 April Morning Shift)

10

If the function $f$ defined as $f(x) = \dfrac{1}{x} - \dfrac{kx - 1}{e^{2x} - 1}, ; x \ne 0$, is continuous at $x = 0$,…

Topic: JEE Main 2018 (16 April Morning Shift)

1

Two different families $A$ and $B$ are blessed with equal number of children. There are $3$ tickets to be distributed a…

Topic: JEE Main 2018 (16 April Morning Shift)

2

If an angle $A$ of $\triangle ABC$ satisfies $5\cos A + 3 = 0$, then the roots of the quadratic equation $9x^{2} + 27x …

Topic: JEE Main 2018 (16 April Morning Shift)

3

The mean and the standard deviation (s.d.) of five observations are $9$ and $0$, respectively. If one of the observatio…

Topic: JEE Main 2018 (16 April Morning Shift)

4

Let A, B and C be three events, which are pair-wise independent and $\overrightarrow E $ denotes the completement of an…

Topic: JEE Main 2018 (16 April Morning Shift)

5

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{c} = \hat{j} - \hat{k}$, and a vector $\vec{b}$ be such that $\vec{a…

Topic: JEE Main 2018 (16 April Morning Shift)

6

If the angle between the lines $\dfrac{x}{2}=\dfrac{y}{2}=\dfrac{z}{1}$ and $\dfrac{5-x}{-2}=\dfrac{7y-14}{p}=\dfrac{z-…

Topic: JEE Main 2018 (16 April Morning Shift)

7

If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus an its nearest vertex on …

Topic: JEE Main 2018 (16 April Morning Shift)

8

The locus of the point of intersection of the lines $ \sqrt{2}x - y + 4\sqrt{2}k = 0$ and $\sqrt{2}kx + ky - 4\sqrt{2} …

Topic: JEE Main 2018 (16 April Morning Shift)

9

If a circle $C$, whose radius is $3$, touches externally the circle $x^{2}+y^{2}+2x-4y-4=0$ at the point $(2,2)$, then …

Topic: JEE Main 2018 (16 April Morning Shift)