JEE Main 2025 (Jan 22nd, Shift 2) Practice Questions & Answers

Let $\alpha, \beta, \gamma$α,β,γ and $\delta$δ be the coefficients of $x^7, x^6, x^5$x7,x6,x5 and x respectively in the expansion of $(x+\sqrt{x^3-1})^5 + (x-\sqrt{x^3-1})^5$(x+x3−1​)5+(x−x3−1​)5, x > 1. If u and v satisfy the equations $\alpha u + \beta v = 18$αu+βv=18, $\gamma u + \delta v = 20$γu+δv=20, then u + v equals :

In a group of 3 girls and 4 boys, there are two boys $B_1$B1​ and $B_2$B2​. 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 $B_1$B1​ and $B_2$B2​ are not adjacent to each other, is :

Let P($4,4\sqrt{3}$4,43​) be a point on the parabola $y^2 = 4ax$y2=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

For a 3 x 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3 x 3 matrix such that $|A| = \frac{1}{2}$∣A∣=21​ and trace (A) = 3. If B = adj(adj(2A)), then the value of |B| + trace (B) equals:

Suppose that the number of terms in an A.P. is 2k, k∈N. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to

Let a line pass through two distinct points P(-2, -1, 3) and Q, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$3i^+2j^​+2k^. If the distance of the point Q from the point R(1, 3, 3) is 5, then the square of the area of ΔPQR is equal to:

If $\lim_{x \to 0} \left( \left( \frac{e}{1 - e} \right) \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha$limx→0​((1−ee​)(e1​−1+xx​))x=α, then the value of $\frac{\log_e \alpha}{1 + \log_e \alpha}$1+loge​αloge​α​ equals:

Let $f(x) = \int_0^{x^2} \frac{t^2-8t+15}{e^t} dt, x \in \mathbb{R}$f(x)=∫0x2​ett2−8t+15​dt,x∈R. Then the numbers of local maximum and local minimum points of f, respectively, are :

The perpendicular distance, of the line $\frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2}$2x−1​=−1y+2​=2z+3​ from the point P(2, -10, 1), is:

If $x = f(y)$x=f(y) is the solution of the differential equation $(1 + y^2) + \left(x - 2e^{\tan^{-1} y}\right) \frac{dy}{dx} = 0, \quad y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$(1+y2)+(x−2etan−1y)dxdy​=0,y∈(−2π​,2π​) with $f(0) = 1$f(0)=1, then $f\left( \frac{1}{\sqrt{3}} \right)$f(3​1​) is equal to: