The value of the integral \[ \int_{-\log_e 2}^{\log_e 2} e^x \left( \log_e\!\left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx \] is equal to:

1

Let $y = y(x)$ be a solution curve of the differential equation \[ (1 - x^2 y^2)\,dx = y\,dx + x\,dy. \] If the line …

Topic: JEE Main 2023 (11 April Morning Shift)

2

Let $w_1$ be the point obtained by the rotation of $z_1 = 5 + 4i$ about the origin through a right angle in the anticlo…

Topic: JEE Main 2023 (11 April Morning Shift)

3

Let $x_1, x_2, \ldots, x_{100}$ be in an arithmetic progression, with $x_1 = 2$ and their mean equal to $200$. If $y_…

Topic: JEE Main 2023 (11 April Morning Shift)

4

Let $S = \{ M = [a_{ij}], \; a_{ij} \in \{0, 1, 2\}, \; 1 \le i, j \le 2 \}$ be a sample space and $A = \{ M \in S : …

Topic: JEE Main 2023 (11 April Morning Shift)

5

Let $R$ be a rectangle given by the lines $x = 0$, $x = 2$, $y = 0$ and $y = 5$. Let $A(\alpha, 0)$ and $B(0, \beta)$…

Topic: JEE Main 2023 (11 April Morning Shift)

1

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the var…

Topic: JEE Main 2023 (11 April Morning Shift)

2

For any vector $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$, with $10|a_i| < 1, \; i = 1, 2, 3$, consider the …

Topic: JEE Main 2023 (11 April Morning Shift)

3

Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that $\mathbf{A}' = \alpha \mathbf{A} + \mathbf{I}$,…

Topic: JEE Main 2023 (11 April Morning Shift)

4

Let $f(x) = \lfloor x^2 - x \rfloor + | -x + \lfloor x \rfloor |$, where $x \in \mathbb{R}$ and $\lfloor t \rfloor$ d…

Topic: JEE Main 2023 (11 April Morning Shift)

5

Consider ellipses $\mathbf{E_k} : kx^2 + k^2y^2 = 1, \; k = 1, 2, \ldots, 20$. Let $\mathbf{C_k}$ be the circle which…

Topic: JEE Main 2023 (11 April Morning Shift)

6

Area of the region \[ \{(x, y) : x^2 + (y - 2)^2 \le 4, \; x^2 \ge 2y\} \] is:

Topic: JEE Main 2023 (11 April Morning Shift)

7

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i} + \ha…

Topic: JEE Main 2023 (11 April Morning Shift)

8

The number of triplets&nbsp;, where&nbsp;&nbsp;are distinct non negative integers satisfying&nbsp;, is :

Topic: JEE Main 2023 (11 April Morning Shift)

9

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 m…

Topic: JEE Main 2023 (11 April Morning Shift)

10

The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^{2} \geq 0$ i…

Topic: JEE Main 2023 (11 April Morning Shift)