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

If the function \( f : \mathbb{R} \to \mathbb{R} \) is defined as \( f(x) = 3x \) then f is

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

If \( X = \{ a, b, c \} \) and \( Y = \{ 1, 2, 3 \} \) and the mapping f is given by \( f(a)=2, f(b)=3, f(c)=1 \), then

• (A) \( f(X) \subset Y \)
• (B) \( f(X) = Y \)
• (C) \( f(X) \supset Y \)
• (D) All of these

If \( \int \log x \, dx = x \log x + k(x) + c \) then

• (A) \( k(x) = \log x \)
• (B) \( k(x) = -\log x \)
• (C) \( k(x) = -x \)
• (D) \( k(x) = -x^2 \)

The number of arbitrary constants in a general solution of a differential equation of fifth order is

• (A) 0
• (B) 4
• (C) 3
• (D) 5

At \( t = 2 \), the slope of the vector function \( \vec{f}(t) = 2\hat{i} + 3\hat{j} + 5t^2\hat{k} \) is

• (A) \( 20\hat{k} \)
• (B) \( 10\hat{k} \)
• (C) \( 5\hat{k} \)
• (D) \( 12\hat{k} \)

Prove that \( \sin^{-1} x = \tan^{-1} [x / \sqrt{1-x^2}] \).

Find the unit vector along the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \).

Find the direction-cosines of the sum of the vectors \( \vec{a} = 3\hat{i} + 4\hat{j} - 3\hat{k} \) and \( \vec{b} = -2\hat{i} - 3\hat{j} + \hat{k} \).

If the functions \( f:\mathbb{R} \to \mathbb{R} \) and \( g:\mathbb{R} \to \mathbb{R} \) are defined as \( f(x)=\cos x \) and \( g(x)=3x^2 \) respectively then find gof.

If two dice are thrown together then find the probability of getting the sum eight.

If the position vectors of the points A and B are \(\hat{i}+\hat{j}+\hat{k}\) and \(2\hat{i}+5\hat{j}\) respectively, then find the unit vector along the straight line AB.

If the relation R is given by \(R = \{(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)\}\), then find \(R^{-1} \circ R^{-1}\).

Find the differential equation representing the family of curves \(y=2mx\).

If \(A = \begin{bmatrix} 3 & \sqrt{3} & 2
4 & 2 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1/4
0 & 0
1/2 & 1/8 \end{bmatrix}\), then prove that \(|C| = 1\), where \(C = (A')' B\).

Find the vector equation of a straight line passing through the point (5, 2, -4) and parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).

Without cover a box is formed by 6 m x 16 m rectangular steel sheet on cutting the squares of length x m from its each corner. Then find the maximum volume of the box.

Let A and B are independent events and P(A)=0.3 and P(B)=0.4 ; then find P(B/A).

Find the area of a triangle whose vertices are A(2, 2, 2), B(2, 1, 3) and C(3, 2, 1).

Let function \( f:N \to Y \) is defined as \( f(x)=4x+3 \) where \( Y=\{y \in N : y=4x+3 for x \in N\} \). Prove that f is invertible, also find the inverse of the function f.

Find the minimum value of the objective function \( Z = -50x + 20y \) by graphical method under the following constraints :
\( 2x - y \geq -5 \)
\( 3x + y \geq 3 \)
\( 2x - 3y \leq 12 \)
\( x \geq 0, y \geq 0 \)

If A and B are two matrices of order n which are invertible, then prove that \( (AB)^{-1} = B^{-1}A^{-1} \).

For the two vectors \( \vec{a} \) and \( \vec{b} \), prove that \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \) when \( \vec{a} \neq \vec{0} \) and \( \vec{b} \neq \vec{0} \).

If \( I = \int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}} \), then prove that \( I = \frac{\pi}{12} \).

If function f is defined as \[ f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & if x \neq 0
0, & if x = 0 \end{cases} \]
then prove that f is continuous.

A car is started to move from a point P at time t = 0 and is stopped at the point Q. The distance x metre covered by the car in t second is given by \(x = t^2\left(3-\frac{t}{2}\right)\). Find the time required by the car to reach at the point Q and also find the distance between P and Q.

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

If \(y=\cos^{-1} x\), then show that \((1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 0\).

If x, y, z are three independent events, then prove that \(P(X \cap Y \cap Z) = P(X) \cdot P\left(\frac{Y}{X}\right) \cdot P\left(\frac{Z}{X \cap Y}\right)\).

For matrix \( A = \begin{bmatrix} 1 & 1 & 1
1 & 2 & -3
2 & -1 & 3 \end{bmatrix} \) show that \( A^3 - 6A^2 + 5A + 11I = 0 \) and with the help of this find \( A^{-1} \).

Solve the following system of equations by matrix method :

2x + 3y + 3z = 5

x - 2y + z = -4

3x - y - 2z = 3

Solve: \(\frac{dy}{dx} = \frac{x+y+1}{2x+2y+3}\).

Solve: \((1+y^2) dx = (\tan^{-1} y - x) dy\).

Prove that \( \int_{0}^{\pi/2} \sqrt{\frac{1+\cos 4x}{2}} \, dx = 1 \).

If \( I = \int_{0}^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \), find the value of I.