Let $f$ be a twice differentiable non-negative function such that

$(f(x))^2 = 25 + \int_0^x \big((f(t))^2 + (f'(t))^2\big),dt$.

Then the mean of $f(\log 1),; f(\log 2),; \ldots,; f(\log 625)$ is equal to ______.

1

Let the area of the region bounded by the curve $y = \max{\sin x, \cos x}$, lines $x = 0,; x = \frac{3\pi}{2}$, and th…

Topic: JEE Main 2026 (23 January Morning Shift)

2

Let $|A| = 6$, where $A$ is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot…

Topic: JEE Main 2026 (23 January Morning Shift)

3

Let the direction cosines of two lines satisfy the equations: $4\ell + m - n = 0$ and $2mn + 10n\ell + 3\ell m = 0$. …

Topic: JEE Main 2026 (23 January Morning Shift)

4

Let the mean and variance of $8$ numbers $-10,; -7,; -1,; x,; y,; 9,; 2,; 16$ be $\dfrac{7}{2}$ and $\dfrac{293}{4}$,…

Topic: JEE Main 2026 (23 January Morning Shift)

5

The value of $\dfrac{{}^{100}C_{50}}{51} + \dfrac{{}^{100}C_{51}}{52} + \cdots + \dfrac{{}^{100}C_{100}}{101}$ is:

Topic: JEE Main 2026 (23 January Morning Shift)

6

Among the statements: I : \( \begin{vmatrix} 1 & \cos\alpha & \cos\beta \\ \cos\alpha & 1 & \cos\gamma \\ \cos\beta & …

Topic: JEE Main 2026 (23 January Morning Shift)

7

The value of the integral $\displaystyle \int_{\pi/24}^{5\pi/24} \frac{dx}{1 + \sqrt{3}\tan 2x}$ is:

Topic: JEE Main 2026 (23 January Morning Shift)

8

Let \( f(x) = \begin{cases} \dfrac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \ne -\frac{3}{2}, \frac{1}{2} \\ b, & x = -\frac…

Topic: JEE Main 2026 (23 January Morning Shift)

9

A building construction work can be completed by two masons $A$ and $B$ together in $22.5$ days. Mason $A$ alone can co…

Topic: JEE Main 2026 (23 January Morning Shift)

10

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function$f(\theta) = 4\left(\sin^4\l…

Topic: JEE Main 2026 (23 January Morning Shift)

1

From the first $100$ natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If …

Topic: JEE Main 2026 (23 January Morning Shift)

2

The number of $4$-letter words, with or without meaning, which can be formed using the letters $PQRPRSTUVP$, is ______.

Topic: JEE Main 2026 (23 January Morning Shift)

3

If $f(x) = \begin{cases} \dfrac{a|x| + x^2 - 2(\sin x)(\cos x)}{x}, & x \ne 0 \ b, & x = 0 \end{cases}$ is cont…

Topic: JEE Main 2026 (23 January Morning Shift)

4

Let $\frac{\pi}{2} < \theta < \pi$ and $\cot\theta = -\frac{1}{2\sqrt{2}}$. Then the value of$\sin\left(\frac{150…

Topic: JEE Main 2026 (23 January Morning Shift)

5

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

Topic: JEE Main 2026 (23 January Morning Shift)