If the distance of the point $ P(43, \alpha, \beta) $ from the line $ \vec{r} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k}),; \mu \in \mathbb{R} $ along a line with direction ratios $ 3, -1, 0 $ is $ 13\sqrt{10} $, then $ \alpha^2 + \beta^2 $ is equal to ______

1

Let $ f $ be a differentiable function satisfying $ f(x) = 1 - 2x + e^{-x} \int_0^x e^t f(t),dt,; x \in \mathbb{R} $ …

Topic: JEE Main 2026 (28 January Evening Shift)

2

Three persons enter in a lift at the ground floor. The lift will go upto $ 10^{th} $ floor. The number of ways, in whic…

Topic: JEE Main 2026 (28 January Evening Shift)

3

$ \sum_{r=1}^{25} \frac{r}{r^2 + r + 1} = \frac{p}{q} $, where $ p $ and $ q $ are integers such that $ \gcd(p, q) = 1 …

Topic: JEE Main 2026 (28 January Evening Shift)

4

$ \frac{6}{3^{26}} + \frac{10.1}{3^{25}} + \frac{10.2}{3^{24}} + \frac{10.2^2}{3^{23}} + \ldots + \frac{10.2^{24}}{3} $…

Topic: JEE Main 2026 (28 January Evening Shift)

5

Let $ f(x) = \int \frac{dx}{x^{2/3} + 2x^{1/2}} $ be such that $ f(0) = -26 + 24\log_e 2 $. If $ f(1) = a + b\log_e 3 $…

Topic: JEE Main 2026 (28 January Evening Shift)

6

Let $ A = { z \in \mathbb{C} : |z - 2| \le 4 } $ and $ B = { z \in \mathbb{C} : |z - 2| + |z + 2| = 5 } $. Then the…

Topic: JEE Main 2026 (28 January Evening Shift)

7

Let $ P $ be a point in the plane of the vector$ \overrightarrow{AB} = 3\hat{i} + \hat{j} - \hat{k} $ and $ \overrighta…

Topic: JEE Main 2026 (28 January Evening Shift)

8

Let $ Q(a, b, c) $ be the image of the point $ P(3, 2, 1) $ in the line$ \frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1…

Topic: JEE Main 2026 (28 January Evening Shift)

9

The sum of all the elements in the range of$ f(x) = \text{Sgn}(\sin x) + \text{Sgn}(\cos x) + \text{Sgn}(\tan x) + \tex…

Topic: JEE Main 2026 (28 January Evening Shift)

10

Let $ y = y(x) $ be the solution of the differential equation$ x \frac{dy}{dx} - y = x^2 \cot x,; x \in (0,\pi) $.If $ …

Topic: JEE Main 2026 (28 January Evening Shift)

1

Let $ A = \left[ \matrix{ 3 & -4 \cr 1 & -1 } \right] $ and $ B $ be two matrices such that $ A^{100} = 100B + I $. The…

Topic: JEE Main 2026 (28 January Evening Shift)