JEE Main 2026 April 8 Shift 2 Mathematics Question Paper with Solutions PDF

If \(\alpha = 3\sin^{-1}\left(\frac{6}{11}\right)\) and \(\beta = 3\cos^{-1}\left(\frac{4}{9}\right)\), consider statements:
Statement 1: \(\cos(\alpha + \beta) > 0\)
Statement 2: \(\cos\alpha < 0\)
Then which of the following is true?

• (1) Statement 1 and 2 are correct
• (2) Only statement 1 is correct
• (3) Only statement 2 is correct
• (4) None of these statements are correct

If \((x\sqrt{1-x^2}) \, dy - (y\sqrt{1-x^2} - x^2 \cos^{-1} x) \, dx = 0\) and \(\lim_{x \to 1^-} y(x) = 1\), then \(y\left(\frac{1}{2}\right)\) is

• (1) \(\frac{\pi^2}{36}\)
• (2) \(\frac{\pi^2 + 18}{36}\)
• (3) \(\frac{\pi^2 - 18}{36}\)
• (4) \(\frac{\pi^2}{18}\)

Statement 1: \(f(x) = e^{|\sin x|} - |x|\) is differentiable for all \(x \in \mathbb{R}\).
Statement 2: \(f(x)\) is increasing in \(x \in (-\pi, -\frac{\pi}{2})\)

• (1) Statement 1 and 2 are false
• (2) Statement 1 is true and statement 2 is false
• (3) Statement 1 is false and statement 2 is true
• (4) Statement 1 and 2 are true

A person goes to college either by bus, scooter or car. The probability that he goes by bus is \(\frac{2}{5}\), by scooter is \(\frac{1}{5}\) and by car is \(\frac{3}{5}\). The probability that he entered late in college if he goes by bus is \(\frac{1}{7}\), by scooter is \(\frac{3}{7}\) and by car is \(\frac{1}{7}\). If it is given that he entered late in college, then the probability that he goes to college by car is

• (1) \(\frac{3}{7}\)
• (2) \(\frac{3}{8}\)
• (3) \(\frac{4}{7}\)
• (4) \(\frac{5}{8}\)

Let \(\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}\), \(\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}\) and a vector \(\vec{c}\) be such that \(2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{c}) = 0\). If \(\vec{a} \cdot \vec{c} = 15\), then the value of \(\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})\) is

• (1) 5
• (2) -5
• (3) 3
• (4) -3

The line passing through point of intersection of \(3x + 4y = 1\) and \(4x + 3y = 1\) intersects axes at P and Q, then locus of midpoint of PQ is

• (1) \(\frac{1}{x} + \frac{1}{y} = 14\)
• (2) \(\frac{3}{x} + \frac{4}{y} = 14\)
• (3) \(\frac{4}{x} + \frac{3}{y} = 14\)
• (4) \(x + y = 14\)

If \(\alpha = 3 + 4 + 8 + 9 + 13 + \dots\) upto 40 terms and \((\tan \beta)^{\frac{\alpha}{1020}}\) is the root of the equation \(x^2 - x - 2 = 0\), then the value of \(\sin^2 \beta + 3\cos^2 \beta\) is

• (1) 1
• (2) \(\frac{2}{3}\)
• (3) \(\frac{5}{3}\)
• (4) \(\frac{1}{3}\)

A person has 3 different bags \& 4 different books. The number of ways in which he can put these books in the bags so that no bag is empty, is

• (1) 36
• (2) 24
• (3) 32
• (4) 30

Let \(\frac{x^2}{f(a^2 + 2a + 7)} + \frac{y^2}{f(3a + 14)} = 1\) represents an ellipse. The major axis of given ellipse is y-axis and \(f\) is a decreasing function. If the range of \(a\) is \(R - [\alpha, \beta]\) then \(\alpha + \beta\) is

• (1) 1
• (2) 2
• (3) 3
• (4) 4

Locus of the mid-point of chord of circle \(x^2 + y^2 - 6x - 8y - 11 = 0\), subtending a right angle at the center is

• (1) \(x^2 + y^2 - 6x - 8y + 7 = 0\)
• (2) \(x^2 + y^2 - 6x - 8y - 7 = 0\)
• (3) \(x^2 + y^2 + 6x + 8y - 7 = 0\)
• (4) \(x^2 + y^2 - 6x + 8y + 7 = 0\)

The number of values of \(z \in \mathbb{C}\) satisfying \(|z-4-8i| = \sqrt{10}\) and \(|z-3-5i| + |z-5-11i| = 4\sqrt{5}\) is

• (1) 0
• (2) 1
• (3) 2
• (4) 3

Let \(A = \{-2, -1, 0, 1, 2\}\). A relation \(R\) is defined on set \(A\) such that \(aRb \Rightarrow 1 + ab > 0\).
Statement-1: It is an equivalence relation.
Statement-2: Number of elements in \(R\) is 17.

• (1) Statement 1 and 2 are false
• (2) Statement 1 is true and statement 2 is false
• (3) Statement 1 is false and statement 2 is true
• (4) Statement 1 and 2 are true

Let \(O\) be the vertex of the parabola \(y^2 = 4x\). Let \(P\) and \(Q\) be two points on parabola such that chords \(OP\) and \(OQ\) are perpendicular to each other. If the locus of mid-point of segment \(PQ\) is a conic \(C\), then latus rectum of \(C\) is

• (1) 1
• (2) 2
• (3) 3
• (4) 4

The mean \& variance of \(x_1, x_2, x_3, x_4\) is 1 and 13 respectively and the mean and variance of \(y_1, y_2, \dots, y_6\) be 2 and 1 respectively, the variance of \(x_1, x_2, \dots, x_4, y_1, y_2, \dots, y_6\) will be

• (1) 6.04
• (2) 6.00
• (3) 5.85
• (4) 5.99

The value of \(\int_0^2 \sqrt{\dfrac{x(x^2 + x + 1)}{(x+1)(x^4 + x^2 + 1)}} \, dx\) is

• (1) \(\dfrac{1}{3} \ln(2^{1/2} + 3)\)
• (2) \(\ln(2^{1/2} + 3)\)
• (3) \(\dfrac{2}{3} \ln(2^{3/2} + 3)\)
• (4) \(\dfrac{2}{3} \ln(2^{1/2} + 3)\)

If \(\int_0^{\pi/4} \left[ \cot(x - \frac{\pi}{3}) - \cot(x + \frac{\pi}{3}) + 1 \right] dx = \alpha \log(\sqrt{3} - 1)\) then \(9\alpha\) is

• (1) 12
• (2) 14
• (3) 9
• (4) 17

If \(\lim_{x \to \frac{\pi}{2}} \dfrac{b(1 - \sin x)}{(\pi - 2x)^2} = \frac{1}{3}\), then \(\int_0^{3b-6} |x^2 + 2x - 3| \, dx\) is equal to

• (1) 3
• (2) 4
• (3) \(\frac{7}{4}\)
• (4) 2

Let \(f(x) = \dfrac{x-1}{x+1}\), \(f^{(1)}(x) = f(x)\), \(f^{(2)}(x) = f(f(x))\) and \(g(x) + f^{(2)}(x) = 0\). The area of the region enclosed by the curves \(y = g(x)\), \(y = 0\), \(x = 4\) and \(2y = 2x - 3\) is

• (1) \(\frac{1}{4} + \ln 2\)
• (2) \(\frac{1}{8} + 2\ln 2\)
• (3) \(\frac{1}{4} + 2\ln 2\)
• (4) \(\frac{1}{8} + \ln 2\)

Consider the system of equations
\(x + y + z = 6\)
\(x + 2y + 5z = 18\)
\(2x + 2y + \lambda z = \mu\)
If the system of equations has infinitely many solutions, then the value of \((\lambda + \mu)\) is equal to

If \(26 \left( \dfrac{2^3 \cdot {}^{12}C_2}{3} + \dfrac{2^5 \cdot {}^{12}C_4}{5} + \dots + \dfrac{2^{13} \cdot {}^{12}C_{12}}{13} \right) = 3^{13} - \alpha\), then find the value of \(\alpha\).