JEE Main 2026 April 4 Shift 1 Mathematics Question Paper with Solutions PDF

Question 1:

Let [·] denote the greatest integer function. If the domain of the function \[ f(x) = \cos^{-1}\left(\frac{4x + 2\lfloor x \rfloor}{3}\right) \] is \([\alpha, \beta]\), then \(12(\alpha + \beta)\) is equal to:

• (A) 6
• (B) 8
• (C) 9
• (D) 4

If the set of all solutions of \(|x^2 + x - 9| = |x| + |x^2 - 9|\) is \([\alpha, \beta] \cup [\gamma, \infty)\), then \((\alpha^2 + \beta^2 + \gamma^2)\) is equal to:

• (A) 9
• (B) 18
• (C) 36
• (D) 72

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:

• (A) 9
• (B) 4
• (C) 5
• (D) 1

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

• (A) 48
• (B) 45
• (C) 51
• (D) 35

Let \(S = \left\{ A = \begin{bmatrix} a & b
c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} and A^2 - 4A + 3I = 0 \right\}\) 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:

• (A) 20
• (B) 17
• (C) 21
• (D) 19

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:

• (A) 3
• (B) 4
• (C) -4
• (D) -3

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:

• (A) \(\frac{5}{87}\)
• (B) \(\frac{25}{83}\)
• (C) \(\frac{15}{29}\)
• (D) \(\frac{5}{29}\)

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

• (A) 5040
• (B) 3050
• (C) 3410
• (D) 4320

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:

• (A) 10
• (B) 11
• (C) 12
• (D) 13

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:

• (A) 109
• (B) 117
• (C) 161
• (D) 131

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:

• (A) 5:1
• (B) 1:5
• (C) 2:3
• (D) 3:2

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:

• (A) 309
• (B) 403
• (C) 497
• (D) 524

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) 1
• (B) 3
• (C) 4
• (D) 6

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:

• (A) 1024
• (B) 1014
• (C) 1104
• (D) 1204

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:

• (A) 3
• (B) 6
• (C) 8
• (D) 12

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:

• (A) 3
• (B) -1
• (C) 1
• (D) 2

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:

• (A) 42
• (B) 46
• (C) 56
• (D) 66

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

• (A) \(1 + \frac{\pi^2}{8}\)
• (B) \(2 + \frac{\pi^2}{4}\)
• (C) \(\frac{\pi^2}{8} - 1\)
• (D) \(4 + \frac{\pi^2}{2}\)

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

• (A) 1
• (B) 2
• (C) 4
• (D) 8

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:

• (A) \(\sqrt{3} \tan \left( \frac{11\sqrt{3}}{6} \right)\)
• (B) \(\frac{\sqrt{3}}{2} \tan \left( \frac{11\sqrt{3}}{12} \right)\)
• (C) \(\sqrt{3} \tan \left( \frac{11\sqrt{3}}{12} \right)\)
• (D) \(\frac{\sqrt{3}}{2} \tan \left( \frac{11\sqrt{3}}{6} \right)\)

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 ____.

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 ____.

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 ____.

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 ____.

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 ____.