List of top Questions asked in JEE Main- 2026

The increase in the pressure required to decrease the volume (\(\Delta V\)) of water is \(6.3 \times 10^7\) N/m². The percentage decrease in the volume is ______. (Bulk modulus of water = \(2.1 \times 10^9\) N/m².)

If \(A = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}\) and \(B = \tan 81^\circ - \tan 3^\circ\), then \(\frac{B}{A}\) is equal to ____.

The number of points, at which the function \(f(x) = \max\{6x, 2 + 3x^2\} + |x - 1| \cos|x^2 - \frac{1}{4}|\), \(x \in (-\pi, \pi)\), is not differentiable, is ____.

In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm. If the circular scale has 100 divisions, the least count of screw gauge is ______ mm.

Consider the parabola \(P : y^2 = 4kx\) and the ellipse \(E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Let the line segment joining the points of intersection of \(P\) and \(E\), be their latus rectum. If the eccentricity of \(E\) is \(e\), then \(e^2 + 2\sqrt{2}\) is equal to ____.

Let \(\vec{a}_k = (\tan \theta_k) \hat{i} + \hat{j}\) and \(\vec{b}_k = \hat{i} - (\cot \theta_k) \hat{j}\), where \(\theta_k = \frac{2^{k-1}\pi}{2^n+1}\), for some \(n \in \mathbb{N}\), \(n>5\). Then the value of \(\frac{\sum_{k=1}^{n} |\vec{a}_k|^2}{\sum_{k=1}^{n} |\vec{b}_k|^2}\) is ____.

Let \(\int_{-2}^{2} (|\sin x| + |\cos x|) \, dx = 2(3 - \cos 2) + \beta\). Then \(\beta \sin \left( \frac{\beta}{2} \right)\) equals:

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is \(p\), then \(96p\) is equal to ____.

Let \(y = y(x)\) be the solution of the differential equation \(\frac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)\), \(y(0) = \frac{1}{2}\). Then \((2y(1) - 1)\) is equal to:

Let \(f\) be a real polynomial of degree \(n\) such that \(f(x) = f'(x)f''(x)\), for all \(x \in \mathbb{R}\). If \(f(0) = 0\), then \(36(f''(2) + f''(2) + \int_0^2 f(x)\,dx)\) is equal to:

The area of the region \(\{(x, y): y \le \pi - |x|, y \le |x \sin x|, y \ge 0\}\) is:

If \( y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1 + \sqrt{1 - x^2}}\right) \), then \(\frac{dy}{dx}\) at \(x = \frac{\sqrt{5}}{2}\) is equal to:

Suppose that two chords, drawn from the point (1, 2) on the circle \(x^2 + y^2 + x - 3y = 0\) are bisected by the y-axis. If the other ends of these chords are R and S, and the midpoint of the line segment RS is \((\alpha, \beta)\), then \(6(\alpha + \beta)\) is equal to:

A line with direction ratios 1, -1, 2 intersects the lines \(\frac{x}{2} = \frac{y}{3} = \frac{z+1}{3}\) and \(\frac{x+1}{-1} = \frac{y-2}{1} = \frac{z}{4}\) at the points P and Q, respectively. If the length of the line segment PQ is \(\alpha\), then \(225\alpha^2\) is equal to:

Let the vertex A of a triangle ABC be (1, 2), and the mid-point of the side AB be (5, -1). If the centroid of this triangle is (3, 4) and its circumcenter is \((\alpha, \beta)\), then \(2(10\alpha + \beta)\) is equal to:

Suppose that the mean and median of the non-negative numbers 21, 8, 17, \(a\), 51, 103, \(b\), 13, 67, \((a>b)\), are 40 and 21, respectively. If the mean deviation about the median is 26, then \(2a\) is equal to:

Let the line \(L_1 : x + 3 = 0\) intersect the lines \(L_2 : x - y = 0\) and \(L_3 : 3x + y = 0\) at the points A and B, respectively. Let the bisector of the obtuse angle between the lines \(L_2\) and \(L_3\) intersect the line \(L_1\) at the point C. Then \(BC^2 : AC^2\) is equal to:

Let the smallest value of \(k \in \mathbb{N}\), for which the coefficient of \(x^3\) in \((1+x)^3 + (1+x)^4 + (1+x)^5 + \dots + (1+x)^{99} + (1 + kx)^{100}\), \(x \neq 0\), is \((43n + \frac{101}{4}) \binom{100}{3}\) for some \(n \in \mathbb{N}\), be \(p\). Then the value of \(p + n\) is:

The square of the distance of the point (-2, -8, 6) from the line \(\frac{x-1}{1} = \frac{y-1}{2} = \frac{z}{1}\) along the line \(\frac{x+5}{1} = \frac{y+5}{1} = \frac{z}{2}\) is equal to:

The first term of an A.P. of 30 non-negative terms is \(\frac{10}{3}\). If the sum of this A.P. is the cube of its last term, then its common difference is:

Let \(S = \left\{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0 \right\}\) be a set of \(2 \times 2\) matrices. Then the number of matrices in \(S\), for which the sum of the diagonal elements is equal to 4, is:

The number of functions \(f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}\), which are not onto, is:

Let \(A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1\\ 1 & 3 & 5 \end{bmatrix}\). Then the sum of all elements of the matrix \(\operatorname{adj}(\operatorname{adj}(2(\operatorname{adj}A)^{-1}))\) is equal to:

Let \(z\) be a complex number such that \(|z + 2| = |z - 2|\) and \(\arg\left(\frac{z+3}{z-i}\right) = \frac{\pi}{4}\). Then \(|z|^2\) is equal to:

The number of ways of forming a queue of 4 boys and 3 girls such that all the girls are not together, is: