Let $P=\begin{bmatrix}1&0&0\\[2pt]3&1&0\\[2pt]9&3&1\end{bmatrix}$ and $Q=[q_{ij}]$ be two $3\times 3$ matrices such that $Q-P^{5}=I_{3}$. Then $\displaystyle \frac{2q_{11}+q_{31}}{q_{32}}$ is equal to:

1

Consider three boxes, each containing $10$ balls labelled $1,2,\ldots,10$. Suppose one ball is randomly drawn from each…

Topic: JEE Main 2019 (12 January Morning Shift)

2

Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^{2}=4x$, and let $X$ be any point on the arc $POQ$ of this …

Topic: JEE Main 2019 (12 January Morning Shift)

3

If the vertices of a hyperbola are at $(-2,0)$ and $(2,0)$ and one of its foci is at $(-3,0)$, then which one of the fo…

Topic: JEE Main 2019 (12 January Morning Shift)

4

The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabo…

Topic: JEE Main 2019 (12 January Morning Shift)

5

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the exper…

Topic: JEE Main 2019 (12 January Morning Shift)

6

Let $S=\{1,2,3,\ldots,100\}$. The number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is ev…

Topic: JEE Main 2019 (12 January Morning Shift)

7

If $\displaystyle \frac{z-\alpha}{z+\alpha}\ (\alpha\in\mathbb{R})$ is a purely imaginary number and $|z|=2$, then a va…

Topic: JEE Main 2019 (12 January Morning Shift)

8

Let $f$ and $g$ be continuous functions on $[0,a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$. Then $\displaystyle \in…

Topic: JEE Main 2019 (12 January Morning Shift)

9

A ratio of the $5^{\text{th}}$ term from the beginning to the $5^{\text{th}}$ term from the end in the binomial expansi…

Topic: JEE Main 2019 (12 January Morning Shift)

10

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:

Topic: JEE Main 2019 (12 January Morning Shift)

1

If $\lambda$ be the ratio of the roots of the quadratic equation in $x$, \[ 3m^{2}x^{2}+m(m-4)x+2=0, \] then the least …

Topic: JEE Main 2019 (12 January Morning Shift)

2

The maximum value of $3\cos\theta+5\sin\!\left(\theta-\dfrac{\pi}{6}\right)$ for any real value of $\theta$ is:

Topic: JEE Main 2019 (12 January Morning Shift)

3

An ordered pair ($\alpha $, $\beta $) for which the system of linear equations (1 + $\alpha $) x + $\beta $y + z = 2 …

Topic: JEE Main 2019 (12 January Morning Shift)

4

Let $S$ be the set of all points in $(-\pi,\pi)$ at which the function $f(x)=\min\{\sin x,\cos x\}$ is not differentiab…

Topic: JEE Main 2019 (12 January Morning Shift)

5

The integral $\displaystyle \int \cos(\log_e x)\,dx$ is equal to (where $C$ is a constant of integration):

Topic: JEE Main 2019 (12 January Morning Shift)

6

$\displaystyle \lim_{x\to \pi/4}\frac{\cot^{3}x-\tan x}{\cos\!\left(x+\frac{\pi}{4}\right)}$ is:

Topic: JEE Main 2019 (12 January Morning Shift)

7

The area (in sq. units) of the region bounded by the parabola $y=x^{2}+2$ and the lines $y=x+1$, $x=0$ and $x=3$, is:

Topic: JEE Main 2019 (12 January Morning Shift)

8

For x > 1, if $(2x)^{2y}=4e^{2x-2y}$, then $\,(1+\log_e 2x)^2\,\dfrac{dy}{dx}$ is equal to :

Topic: JEE Main 2019 (12 January Morning Shift)

9

If the straight line $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta)$, th…

Topic: JEE Main 2019 (12 January Morning Shift)

10

Considering only the principal values of inverse functions, the set $A = \{x \ge 0 : \tan^{-1}(2x) + \tan^{-1}(3x) = …

Topic: JEE Main 2019 (12 January Morning Shift)