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

The modulus function f: R \( \rightarrow \) R\(^+\) given by f(x) = |x| is

• (A) one-one and onto
• (B) many-one and onto
• (C) one-one but not onto
• (D) neither one-one nor onto

A relation R = {(a, b) : a = b - 1, b \( \geq \) 3\ is defined on set N, then

• (A) (2, 4) \( \in \) R
• (B) (4, 5) \( \in \) R
• (C) (4, 6) \( \in \) R
• (D) (1, 3) \( \in \) R

The value of \( \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be

• (A) 0
• (B) \( \frac{\pi}{2} \)
• (C) \( \frac{\pi}{4} \)
• (D) \( \frac{\pi}{8} \)

The degree of differential equation \( 9 \frac{d^2y}{dx^2} = \left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}} \) is

• (A) 1
• (B) 6
• (C) 3
• (D) 2

The value of expression \( \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k} \) is

• (A) 0
• (B) 1
• (C) 2
• (D) 3

Write \( \cot^{-1}\left\{\frac{1}{\sqrt{x^2-1}}\right\}; x > 1 \) in the simplest form.

Prove that the function f(x) = |x| is continuous at x = 0.

Find the degree of the differential equation \( xy \frac{d^2y}{dx^2} + x \left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) = 2 \)

If P(A) = 0.12, P(B) = 0.15 and P(B/A) = 0.18, then find the value of P(A \( \cap \) B).

Find the angle between the vectors \( -2\hat{i} + \hat{j} + 3\hat{k} \) and \( 3\hat{i} - 2\hat{j} + \hat{k} \).

If \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) be functions defined by \( f(x) = \cos x \) and \( g(x) = 3x^2 \) respectively, then prove that \( gof \neq fog \).

Find the general solution of differential equation \( ydx + (x - y^2)dy = 0 \).

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

If \( \begin{bmatrix} x+y & 2
5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2
5 & 8 \end{bmatrix} \), then find the values of x, y, z.

Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x > 0 \).

Find the unit vector perpendicular to each of the vectors \( (\vec{a} + \vec{b}) \) and \( (\vec{a} - \vec{b}) \) where \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \).

If the Cartesian equation of a line is \( \frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2} \), then find its equation in vector form.

There are 4 white and 2 black balls in a bag and in another bag 3 white and 5 black balls. Find the probability of getting both black balls if a ball is drawn from each bag.

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

If \( A = \begin{bmatrix} 0 & -\tan{\frac{\alpha}{2}}
\tan{\frac{\alpha}{2}} & 0 \end{bmatrix} \), then prove that \( (I + A) = (I - A) \begin{bmatrix} \cos{\alpha} & -\sin{\alpha}
\sin{\alpha} & \cos{\alpha} \end{bmatrix} \).

Differentiate \( \tan^{-1}\left(\frac{\sin x}{1 + \cos x}\right) \) with respect to x.

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

If \( y = e^{\tan^{-1}x} \), prove that \( (1 + x^2)\frac{d^2y}{dx^2} + (2x - 1)\frac{dy}{dx} = 0 \).

If \( f(x) = \begin{cases} -2 & if x \le -1
2x & if -1 < x \le 1
2 & if x > 1 \end{cases} \), then test the continuity of the function at \( x = -1 \) and at \( x = 1 \).

If three vectors \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) satisfying the condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \). If \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \) and \( |\vec{c}| = 2 \), then find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).

The radius of an air bubble is increasing at the rate of \( \frac{1}{2} \) cm/s. At what rate is the volume of the bubble increasing while the radius is 1 cm?

Minimize Z = 3x + 2y by graphical method under the following constraints:

x + y \( \ge \) 8,

3x + 5y \( \le \) 15,

x \( \ge \) 0, y \( \ge \) 0

The probability of solving a question by the three students A, B, C are respectively \( \frac{3}{10}, \frac{1}{5} \) and \( \frac{1}{10} \). Find the probability of solving the question.

Find the inverse of the matrix \( A = \begin{bmatrix} 2 & 0 & -1
5 & 1 & 0
0 & 1 & 3 \end{bmatrix} \).

Solve the system of equations by matrix method:
\( 3x - 2y + 3z = 8 \)
\( 2x + y - z = 1 \)
\( 4x - 3y + 2z = 4 \)

Prove that the semi-vertical angle of a cone with given slant height and maximum volume is \( \tan^{-1}(\sqrt{2}) \).

Find a particular solution of the differential equation \( (x - y) (dx + dy) = dx - dy \) when \( y = -1 \) if \( x = 0 \).

Integrate: \( \int \left(\frac{2 + \sin 2x}{1 + \cos 2x}\right) e^x dx \)

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