Let $A = \{x \in \mathbb{R} : [x + 3] + [x + 4] \le 3\}$, and $B = \left\{ x \in \mathbb{R} : 3^x \left( \sum_{r=1}^{\infty} \frac{3}{10^r} \right)^{x - 3} < 3^{-3x} \right\}$, where $[\,]$ denotes the greatest integer function. Then,

1

If the system of equations $x + y + a z = b$ $2x + 5y + 2z = 6$ $x + 2y + 3z = 3$ has infinitely many solut…

Topic: JEE Main 2023 (6 April Morning Shift)

2

The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L : 9x + 5y = 45…

Topic: JEE Main 2023 (6 April Morning Shift)

3

One vertex of a rectangular parallelepiped is at the origin $O$ and the lengths of its edges along the $x$, $y$ and $z$…

Topic: JEE Main 2023 (6 April Morning Shift)

4

The mean and variance of a set of $15$ numbers are $12$ and $14$ respectively. The mean and variance of another set o…

Topic: JEE Main 2023 (6 April Morning Shift)

5

Let $A = [a_{ij}]_{2\times 2}$, where $a_{ij}\ne 0$ for all $i,j$ and $A^{2}=I$. Let $a$ be the sum of all diagonal e…

Topic: JEE Main 2023 (6 April Morning Shift)

6

Let $5f(x)+4f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+3,\; x>0.$ Then $18\displaystyle\int_{1}^{2} f(x)\,dx$ is equal…

Topic: JEE Main 2023 (6 April Morning Shift)

7

Let $I(x)=\displaystyle \int \frac{x^{2}\big(x\sec^{2}x+\tan x\big)}{(x\tan x+1)^{2}}\,dx.$ If $I(0)=0$, then $I\!\le…

Topic: JEE Main 2023 (6 April Morning Shift)

8

Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}$.…

Topic: JEE Main 2023 (6 April Morning Shift)

9

Let $a_{1},a_{2},a_{3},\ldots,a_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its …

Topic: JEE Main 2023 (6 April Morning Shift)

10

The sum of all the roots of the equation $\lvert x^{2}-8x+15\rvert-2x+7=0$ is:

Topic: JEE Main 2023 (6 April Morning Shift)