The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is:

1

Let $ P = [p_{ij}] $ and $ Q = [q_{ij}] $ be two square matrices of order $ 3 $ such that $ q_{ij} = 2^{(i + j - 1)} p_…

Topic: JEE Main 2026 (24 January Evening Shift)

2

$ \left( \frac{1}{3} + \frac{4}{7} \right) + \left( \frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2} \r…

Topic: JEE Main 2026 (24 January Evening Shift)

3

Let the image of parabola $ x^2 = 4y $, in the line $ x - y = 1 $ be $ (y + \alpha)^2 = b(x - c) $, $ a, b, c \in \math…

Topic: JEE Main 2026 (24 January Evening Shift)

4

Let $ \alpha_1, \alpha_2, \alpha_3, \alpha_4 $ be an A.P. of four terms such that each term of the A.P. and its common …

Topic: JEE Main 2026 (24 January Evening Shift)

5

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

Topic: JEE Main 2026 (24 January Evening Shift)

6

Let the angles made with the positive $x$-axis by two straight lines drawn from the point $P(2, 3)$ and meeting the lin…

Topic: JEE Main 2026 (24 January Evening Shift)

7

Let the length of the latus rectum of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, $ (a > b) $, be $30$. If i…

Topic: JEE Main 2026 (24 January Evening Shift)

8

Let $ f(x) = \int \frac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2} ,dx $, $ x > 0 $, $ \lim_{x \to 0} f(x) = 0 $ and $ f(1) =…

Topic: JEE Main 2026 (24 January Evening Shift)

1

The largest value of $ n $, for which $ 40^n $ divides $ 60! $, is:

Topic: JEE Main 2026 (24 January Evening Shift)

2

The sum of all values of $ \alpha $, for which the shortest distance between the lines $ \frac{x+1}{\alpha} = \frac{y-…

Topic: JEE Main 2026 (24 January Evening Shift)

3

Let $ f(\alpha) $ denote the area of the region in the first quadrant bounded by $ x = 0, x = 1, y^2 = x $ and $ y = | …

Topic: JEE Main 2026 (24 January Evening Shift)

4

If the domain of the function $ f(x) = \sin^{-1} \left( \frac{1}{x^2 - 2x - 2} \right) $, is $ (-\infty, \alpha] \cup […

Topic: JEE Main 2026 (24 January Evening Shift)

5

Let $ X = { x \in \mathbb{N} : 1 \le x \le 19 } $ and for some $ a, b \in \mathbb{R} $, $ Y = { ax + b : x \in X } $. I…

Topic: JEE Main 2026 (24 January Evening Shift)

6

Consider the following three statements for the function $ f : (0, \infty) \to \mathbb{R} $ defined by$ f(x) = |\log_e …

Topic: JEE Main 2026 (24 January Evening Shift)

7

Let $ y = y(x) $ be a differentiable function in the interval $ (0, \infty) $ such that $ y(1) = 2 $.and $ \lim_{t \to …

Topic: JEE Main 2026 (24 January Evening Shift)

8

Let $ f $ be a function such that $ 3f(x) + 2f\left( \frac{m}{19x} \right) = 5x $, $ x \ne 0 $, where $ m = \sum_{i=1}^…

Topic: JEE Main 2026 (24 January Evening Shift)

9

The smallest positive integral value of $ a $, for which all the roots of $ x^4 - ax^2 + 9 = 0 $ are real and distinct,…

Topic: JEE Main 2026 (24 January Evening Shift)

10

Let $ \vec{a} = 2\hat{i} - 5\hat{j} + 5\hat{k} $ and $ \vec{b} = \hat{i} - \hat{j} + 3\hat{k} $. If $ \vec{c} $ is a ve…

Topic: JEE Main 2026 (24 January Evening Shift)