List of top Questions asked in JEE Main

\[ \sum_{k=1}^{n} \left( \alpha^k + \frac{1}{\alpha^k} \right)^2 = 20, \quad \alpha \text{ is one of the roots of } x^2 + x + 1 = 0, \text{ then } n = ? \]

If the system of equations: $$ \begin{aligned} 3x + y + \beta z &= 3 \\2x + \alpha y + z &= 2 \\x + 2y + z &= 4 \end{aligned} $$ has infinitely many solutions, then the value of $ 22\beta - 9\alpha $ is:

Let $ y = y(x) $ be the solution curve of the differential equation $ x(x^2 + e^x) \, dy + \left( e^x(x - 2) y - x^3 \right) \, dx = 0, \quad x>0, $ passing through the point $ (1, 0) $. Then $ y(2) $ is equal to:

If the image of the point $ (4, 4, 3) $ in the line $ \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} $ is $ (a, \beta, \gamma) $, then $ a + \beta + \gamma $ is equal to:

Let P be the parabola, whose focus is $ (-2, 1) $ and directrix is $ 2x + y + 2 = 0 $. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:

Let $ A = [a_{ij}] $ be a square matrix of order 2 with entries either 0 or 1. Let $ E $ be the event that $ A $ is an invertible matrix. Then the probability $ P(E) $ is:

Two equal sides of an isosceles triangle are along $ -x + 2y = 4 $ and $ x + y = 4 $. If $ m $ is the slope of its third side, then the sum of all possible distinct values of $ m $ is:

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability that A wins if A makes the first throw is:

Let {an}ⁿ⁼⁰_∞be a sequence such that a₀=a₁=0 and a_n+2=3a_n+1−2a_n+1,∀ n≥0. Then a₂₅a₂₃−2a₂₅a₂₂−2a₂₃a₂₄+4a₂₂a₂₄is equal to

Let $ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} $ and $ \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k} $. Let $ L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in {R} $ and $ L_2 : \vec{r} = (\hat{j} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} $ be two lines. If the line $ L_3 $ passes through the point of intersection of $ L_1 $ and $ L_2 $, and is parallel to $ \vec{a} + \vec{b} $, then $ L_3 $ passes through the point:

The area of the region, inside the circle $(x-2\sqrt{3})^2 + y^2 = 12$ and outside the parabola $y^2 = 2\sqrt{3}x$ is:

Let $ A = \{1, 2, 3, \dots, 10\} $ and $ B = \left\{ \frac{m}{n} : m, n \in A, m <n \text{ and } \gcd(m, n) = 1 \right\} $. Then $ n(B) $ is equal to :

Let $f: [0, 3] \to A$ be defined by $f(x) = 2x^3 - 15x^2 + 36x + 7$ and $g: [0, \infty) \to B$ be defined by $g(x) = \frac{x}{x^{2025} + 1}$. If both functions are onto and $S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \}$, then $n(S)$ is equal to:

In a group of 3 girls and 4 boys, there are two boys $ B_1 $ and $ B_2 $. The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $ B_1 $ and $ B_2 $ are not adjacent to each other, is:

If \[ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6 \], then f(1) is equal to:

Let P be the foot of the perpendicular from the point $ (1, 2, 2) $ on the line \[ \frac{x-1}{1} = \frac{y + 1}{-1} = \frac{z - 2}{2} \] Let the line $ \mathbf{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k})$, $ \lambda \in \mathbb{R} $, intersect the line $L$ at $Q$. Then $ 2(PQ)^2 $ is equal to:

The number of integral terms in the binomial expansion of \[ \left(11^{\frac{1}{2}} + 17^{\frac{1}{8}}\right)^{1024} \] is:

If $ z_1, z_2, z_3 \in \mathbb{C} $ are the vertices of an equilateral triangle, whose centroid is $ z_0 $, then $ \sum_{k=1}^{3} (z_k - z_0)^2 $ is equal to

Let the line passing through the points $ (-1, 2, 1) $ and parallel to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \] intersect the line \[ \frac{x + 2}{3} = \frac{y - 3}{2} = \frac{z - 4}{1} \] at the point P. Then the distance of P from the point Q(4, -5, 1) is:

Evaluate the following limit: \[ \lim_{x \to \infty} \frac{(2x^2 - 3x + 5) \left( 3x - 1 \right)^{x/2}}{(3x^2 + 5x + 4) \sqrt{(3x + 2)^x}}. \] The value of the limit is:

The sum of all rational numbers in $ (2 + \sqrt{3})^8 $ is:

For a statistical data $ x_1, x_2, \dots, x_{10} $ of 10 values, a student obtained the mean as 5.5 and \[ \sum_{i=1}^{10} x_i^2 = 371. \] He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively.
The variance of the corrected data is:

Let a curve $ y = f(x) $ pass through the points $ (0,5) $ and $ (\log 2, k) $. If the curve satisfies the differential equation: \[ 2(3+y)e^{2x}dx - (7+e^{2x})dy = 0, \] then $ k $ is equal to:

The sum 1 + 3 + 11 + 25 + 45 + 71 + ... upto 20 terms, is equal to

Let R = {(1, 2), (2, 3), (3, 3)}} be a relation defined on the set $ \{1, 2, 3, 4\} $. Then the minimum number of elements needed to be added in $ R $ so that $ R $ becomes an equivalence relation, is: