JEE Main 2026 April 4 Shift 2 Mathematics Question Paper with Solutions PDF

Question 1:

For the function \(f: [1, \infty) \rightarrow [1, \infty)\) defined by \(f(x) = (x - 1)^4 + 1\), among the two statements:

(I) The set \(S = \{x \in [1, \infty) : f(x) = f^{-1}(x)\}\) contains exactly two elements, and

(II) The set \(S = \{x \in [1, \infty) : f(x) = f^{-1}(x + 1)\}\) is an empty set,

Options:

• (A) only (I) is TRUE
• (B) only (II) is TRUE
• (C) both (I) and (II) are TRUE

Let \(S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}\). Then \(\sum_{z \in S} |z + \sqrt{3}i|^2\) is equal to:

• (A) 42
• (B) 23
• (C) 27

If the system of equations:
\(x + y + z = 5\)
\(x + 2y + 3z = 9\)
\(x + 3y + \lambda z = \mu\)

has infinitely many solutions, then the value of \(\lambda + \mu\) is:

• (A) 16
• (B) 18
• (C) 19

If \(\alpha = 1\) and \(\beta = 1 + i\sqrt{2}\), where \(i = \sqrt{-1}\) are two roots of the equation
\(x^3 + ax^2 + bx + c = 0, a, b, c \in \mathbb{R}\), then \(\int_{-1}^{1} (x^3 + ax^2 + bx + c) dx\) is equal to:

• (A) \(-2\)
• (B) \(-4\)
• (C) \(-8\)

If the quadratic equation \((\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \lambda \neq -2\), has two positive roots, then the number of possible integral values of \(\lambda\) is:

• (A) 1
• (B) 2
• (C) 3

Let \(A = \begin{bmatrix} 1 & 2 & 7
4 & -2 & 8
3 & 8 & -7 \end{bmatrix}\) and \(\det(A - \alpha I) = 0\), where \(\alpha\) is a real number. If the largest possible value of \(\alpha\) is \(p\), then the circle \((x - p)^2 + (y - 2p)^2 = 320\), intersects the co-ordinate axes at:

• (A) 1 point
• (B) 2 points
• (C) 3 points

Let \(\alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \infty\) and \(\beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty\). Then the value of \((0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_{5}(\beta)}\) is equal to:

• (A) 4
• (B) 5
• (C) 8

For 10 observations \(x_1, x_2, \dots, x_{10}\), if \(\sum_{i=1}^{10} (x_i + 2)^2 = 180\) and \(\sum_{i=1}^{10} (x_i - 1)^2 = 90\), then their standard deviation is:

• (A) 2
• (B) \(\sqrt{3}\)
• (C) \(2\sqrt{2}\)

In the expansion of \(\left( 9x - \frac{1}{3\sqrt{x}} \right)^{18}, x > 0\), if the term independent of \(x\) is \((221)k\), then \(k\) is equal to:

• (A) 84
• (B) 78
• (C) 168

Let \(P(3\cos\alpha, 2\sin\alpha), \alpha \neq 0\), be a point on the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\). \(Q\) be a point on the circle \(x^2 + y^2 - 14x - 14y + 82 = 0\) and \(R\) be a point on the line \(x + y = 5\) such that the centroid of the triangle \(PQR\) is \(\left( 2 + \cos\alpha, 3 + \frac{2}{3}\sin\alpha \right)\). Then the sum of the ordinates of all possible points \(R\) is:

• (A) 6
• (B) 2
• (C) 4

Let \(H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is \(\frac{8}{3}\). If the line \(x = \alpha\) intersects the hyperbola H at the points A and B such that the area of the triangle AOB is \(4\sqrt{15}\), where O is the origin, then \(a^2\) equals:

• (A) 12
• (B) 16
• (C) 24

\(\max_{0 \leq x \leq \pi} \left( 16 \sin\left(\frac{x}{2}\right) \cos^3\left(\frac{x}{2}\right) \right)\) is equal to:

• (A) 2
• (B) \(3\sqrt{3}\)
• (C) \(4\sqrt{3}\)

The shortest distance between the lines
\(\vec{r} = (\frac{1}{3}\hat{i} + \frac{8}{3}\hat{j} - \frac{1}{3}\hat{k}) + \lambda(2\hat{i} - 5\hat{j} + 6\hat{k})\)

and \(\vec{r} = (-\frac{2}{3}\hat{i} - \frac{1}{3}\hat{k}) + \mu(\hat{j} - \hat{k}), \lambda, \mu \in \mathbb{R}\), is:

• (A) \(\sqrt{5}\)
• (B) 3
• (C) \(2\sqrt{3}\)

If \((2\alpha + 1, \alpha^2 - 3\alpha, \frac{\alpha - 1}{2})\) is the image of \((\alpha, 2\alpha, 1)\) in the line \(\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z}{1}\), then the possible value(s) of \(\alpha\) is (are):

• (A) Only 3
• (B) Only 3 and \(-1\)
• (C) Only \(3, \frac{1}{4}\) and \(-1\)

Let \(\hat{u}\) and \(\hat{v}\) be unit vectors inclined at an acute angle such that \(|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}\). If \(\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} \times \hat{v})\), then \(\lambda\) is equal to:

• (A) \(\frac{4}{3}(\vec{A} \cdot \hat{u}) - \frac{2}{3}(\vec{A} \cdot \hat{v})\)
• (B) \(\frac{2}{3}(\vec{A} \cdot \hat{u}) - \frac{1}{3}(\vec{A} \cdot \hat{v})\)
• (C) \(\frac{4}{3}(\vec{A} \cdot \hat{u}) + \frac{2}{3}(\vec{A} \cdot \hat{v})\)

Let for some \(\alpha \in \mathbb{R}\), \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function satisfying \(f(x + y) = f(x) + 2y^2 + y + \alpha xy\) for all \(x, y \in \mathbb{R}\). If \(f(0) = -1\) and \(f(1) = 2\), then the value of \(\sum_{n=1}^{5} (\alpha + f(n))\) is:

• (A) 110
• (B) 140
• (C) 150

Let \(A = \{ (a, b, c) : a, b, c are non-negative integers and a + b + 2c = 22 \}\). Then \(n(A)\) is equal to:

• (A) 121
• (B) 124
• (C) 144

The area of the region bounded by the curves \(x + 3y^2 = 0\) and \(x + 4y^2 = 1\) is equal to:

• (A) \(\frac{1}{3}\)
• (B) \(\frac{2}{3}\)
• (C) \(\frac{4}{3}\)

Let \(y = y(x)\) be the solution of the differential equation:
\(\frac{dy}{dx} + \left( \frac{6x^2 + (3x^2 + 2x^3 + 4)e^{-2x}}{(x^3 + 2)(2 + e^{-2x})} \right)y = 2 + e^{-2x}\), \(x \in (-1, 2)\), satisfying \(y(0) = \frac{3}{2}\). If \(y(1) = \alpha(2 + e^{-2})\), then \(\alpha\) is equal to:

• (A) \(\frac{13}{8}\)
• (B) \(\frac{6}{13}\)
• (C) \(\frac{12}{13}\)

The integral \(\int_{0}^{1} \cot^{-1}(1 + x + x^2) dx\) is equal to:

• (A) \(2 \tan^{-1} 2 + \frac{1}{2} \log_e (\frac{5}{4}) + \frac{\pi}{2}\)
• (B) \(2 \tan^{-1} 2 + \frac{1}{2} \log_e (\frac{5}{4}) - \frac{\pi}{2}\)
• (C) \(2 \tan^{-1} 2 - \frac{1}{2} \log_e (\frac{5}{4}) + \frac{\pi}{2}\)

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to \(a/b\), where \(a, b \in \mathbb{N}\) and \(\gcd(a, b) = 1\), then \(a + b\) is equal to _______

Let \(f(x) = \begin{cases} e^{x-1}, & x < 0
x^2 - 5x + 6, & x \ge 0 \end{cases}\) and \(g(x) = f(|x|) + |f(x)|\). If the number of points where \(g\) is not continuous and is not differentiable are \(\alpha\) and \(\beta\) respectively, then \(\alpha + \beta\) is equal to _______.

Let A, B be points on the two half-lines \(x - \sqrt{3}|y| = \alpha, \alpha > 0\) at a distance of \(\alpha\) from their point of intersection P. The line segment AB meets the angle bisector of the given half-lines at the point Q. If \(PQ = \frac{9}{2}\) and R is the radius of the circumcircle of \(\Delta PAB\), then \(\frac{\alpha^2}{R}\) is equal to ________

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola \(y^2 = 16x\). Let the vertex B containing the right angle be \((4, 8)\) and the locus of the centroid of \(\Delta ABC\) be a conic \(C_0\). Then three times the length of latus rectum of \(C_0\) is _______.

Let \(f\) be a twice differentiable function such that \(f(x) = \int_0^x \tan(t-x) dt - \int_0^x f(t) \tan t dt, x \in (-\frac{\pi}{2}, \frac{\pi}{2})\). Then \(f''(\frac{\pi}{6}) + 12 f'(-\frac{\pi}{6}) + f(\frac{\pi}{6})\) is equal to ________.