UP Board Class 12 Mathematics Question Paper 2025 (Code 324 IZ) Available- Download Here with Solution PDF

At which point is the slope of the curve \( y^2 = 4x \) equal to the slope of the line \( y = x + 3 \)?

• (A) (1, 2)
• (B) (2, 1)
• (C) (-1, 2)
• (D) (1, -2)

If the vector \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) is perpendicular to the vector \( 5\hat{i} - \lambda\hat{j} + 2\hat{k} \), the value of \( \lambda \) is:

• (A) 3
• (B) 0
• (C) 4
• (D) 6

If A is a square matrix and \( A^2 = A \), then \( (A + I)^3 - 7A \) will be:

• (A) A
• (B) 3A
• (C) I - A
• (D) I

The value of \( \int \cos^2 x \, dx \) will be:

• (A) \( -\frac{1}{4}\sin 2x + \frac{x}{4} + C \)
• (B) \( -\frac{1}{2}\sin 2x + \frac{x}{4} + C \)
• (C) \( \cos^2 x - \sin^2 x + C \)
• (D) \( \frac{1}{4}\sin 2x + \frac{x}{2} + C \)

A function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = 3x \) for all \( x \in \mathbb{R} \). Then function \( f \) will be:

• (A) Not one-one
• (B) Not onto
• (C) Onto
• (D) Many-one

If \( y = A + Be^x \), then prove that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \), where A and B are constants.

Solve the differential equation \( \frac{dy}{dx} = \frac{x^2 - 1}{y^2 + 1} \).

Show that \( f(x) = |x| \) is continuous for all values of x.

Given any two events A and B are such that \( P(A) = \frac{1}{2}, P(B) = \frac{1}{4} \) and \( P(A \cap B) = \frac{1}{8} \), then find P(not A and not B).

If \( A = \{1, 2\} \) and \( B = \{3, 4, 5\} \), then find all number of relations from A to B.

If \( y = A \cos \theta + B \sin \theta \), then prove that \( \frac{d^2y}{d\theta^2} = -y \).

Find the Cartesian equation of a line which passes through point (3, -2, -5) and is parallel to the vector \( (3\hat{i} + 2\hat{j} - 2\hat{k}) \).

Solve the differential equation \( \frac{dy}{dx} = e^x \cos x \).

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are vectors and \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), then find the value of \( (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \).

If \( R_1 \) and \( R_2 \) be two equivalence relations on a set A, then prove that \( (R_1 \cap R_2) \) also be an equivalence relation on A.

Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \).

There are two children in a family. It is known that there is at least one child is a boy. Then find the probability that both children are boys.

Find the interval in which the given function \( f(x) = x^2 - 4x + 6 \) is (i) Increasing (ii) Decreasing.

If the matrix A = \(\begin{bmatrix} 0 & 2y & z
x & y & -z
x & -y & z \end{bmatrix}\) satisfies the equation AA' = I, then find the values of x, y and z.

Find the differential coefficient of \(y = x^{\cos x} + (\sin x)^x\) with respect to x.

Find the shortest distance between the lines \(\vec{r} = (2\hat{i} + \hat{j} – \hat{k}) + \mu(3\hat{i} – 5\hat{j} + 2\hat{k})\) and \(\vec{r} = (\hat{i} + \hat{j}) + \lambda(2\hat{i} - \hat{j} + \hat{k})\).

Find the minimum value of Z = 3x + 7y by the graphical method under the following constraints:
\(x + y \le 8, 3x + 5y \ge 0, x \ge 0, y \ge 0\)

If a function f: R \(\to\) \{x \(\in\) R : x \(\in\) (-1, 1)\ is defined as \(f(x) = \frac{x}{1+|x|}\), x \(\in\) R, then prove that f is one-one and onto.

Show that (3\(\hat{i}\) - 4\(\hat{j}\) - 4\(\hat{k}\)), (2\(\hat{i}\) - \(\hat{j}\) + \(\hat{k}\)) and (\(\hat{i}\) - 3\(\hat{j}\) - 5\(\hat{k}\)) are the position vectors of vertices of a right angle triangle.

If \(x\sqrt{1+y} + y\sqrt{1+x} = 0\), \(-1 < x < 1\), then prove that \(\frac{dy}{dx} = -\frac{1}{(1+x)^2}\).

There are 10 white and 5 black balls in a bag. Two balls are drawn one by one. First ball is not placed back before the second is taken out. Assume that the taking out of each ball from the bag is equally likely. What is the probability that both balls taken out are white?

Prove that \(|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|\) is always true for any two vectors \(\vec{a}\) and \(\vec{b}\).

Solve: \(\int \frac{3x - 5}{x^3 - x^2 - x + 1} dx\)

If A = \(\begin{bmatrix} 3 & 3 & 1
3 & 4 & 1
4 & 3 & 1 \end{bmatrix}\), then verify that A(adj A) = \(|\)A\(|\) I and find A\(^{-1}\).

Solve the system of equations
x + y + z = 2,
2x + y - 3z = 0,
x - y + z - 4 = 0 by matrix method.

Prove that: \(\int_0^{\pi/4} \log_e(1 + \tan \theta) d\theta = \frac{\pi}{8} \log_e 2\)

Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.

If \(y = 600 e^{-7x} + 500 e^{7x}\), then show that \(\frac{d^2y}{dx^2} = 49y\).

Find the area bounded by the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).

Solve the integral \(\int \frac{\sec^2 2\theta}{(\cot \theta - \tan \theta)^2} d\theta\)