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

Suppose that A = {2, 3, 4, 5} and a relation R on A is defined by R = {(a, b) : a, b \(\in\) A, a - b = 12\. Then the set R is

• (A) \(\emptyset\)
• (B) Not \(\emptyset\)
• (C) 2, 3
• (D) 2, 4, 5

If the function f: N \(\rightarrow\) N, is defined by f(x) = x - 1 for all x \(>\) 2 and f(1) = f(2) = 1, then f is

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

If \(\int x \log x \,dx = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c\), then f(x) is

• (A) \((\log x)^{-1}\)
• (B) \(2\log x\)
• (C) \(\log x\)
• (D) \(3\log x\)

If y = 5x\(^2\) + 4, then at the point with x-coordinate 2, the slope is

• (A) \(3/2\sqrt{14}\)
• (B) \(1/2\sqrt{14}\)
• (C) 20
• (D) 1

The vector function is given by \(\vec{f}(t) = t\hat{i} + t^2\hat{j} + 5\hat{k}\), then at point t = 1 the slope is

• (A) \(\hat{i} + 2\hat{j}\)
• (B) \(\hat{i} + 3\hat{j}\)
• (C) \(2\hat{i} + \hat{j} + \hat{k}\)
• (D) \(\hat{i} + 3\hat{j} + 5\hat{k}\)

Prove that \( \sin^{-1} x = \cos^{-1} \sqrt{1 - x^2} \).

Find the direction cosine of Z-axis.

Obtain the projection of the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + 5\hat{k} \) on the vector \( \vec{b} = \hat{i} + 3\hat{j} + \hat{k} \).

Find the value of \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \).

Prove that \( 0 \leq P(E) \leq 1 \), where P(E) is the probability of the event E.

If the ordered pairs (2x - 3, 5) and (x, y - 1) are equal, then find the numbers x and y.

Obtain the differential equation of the family of curves \(y = \frac{2ce^{2x}}{1+ce^{2x}}\).

The modulus of two vectors \(\vec{a}\) and \(\vec{b}\) are \(\sqrt{3}\) and 4 respectively, and \(\vec{a} \cdot \vec{b} = 6\). Then find the angle between the vectors \(\vec{a}\) and \(\vec{b}\).

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

Find the area of a parallelogram whose adjacent sides are the vectors \( \vec{a} = \hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - 7\hat{j} + \hat{k} \).

A box is formed by a 3 m x 8 m rectangular steel-sheet on cutting the squares of length x m from its each corner to form the box without cover. Then find the maximum volume of the box so formed.

A person has a contract of construction. The probability of being a strike is 0.65. The probabilities of completing the construction work on time in both conditions are 0.80 and 0.32 whether the strike is not happened and it is happened respectively. Then find the probability of completing the construction work in due time.

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

If the function \(f: [0, \frac{\pi}{2}] \rightarrow \mathbb{R}\) is given by \(f(x) = \sin x\) and function \(g: [0, \frac{\pi}{2}] \rightarrow \mathbb{R}\) is given by \(g(x) = \cos x\), then prove that f and g are one-one but \(f + g\) is not one-one.

Minimize Z = 3x + 2y by graphical method under the following constraints: \(x + 2y \leq 10\), \(3x + y \leq 15\), \(x \geq 0\), \(y \geq 0\).

If \(A = \begin{bmatrix} 3 & 1
-1 & 2 \end{bmatrix}\), then show that \(A^2 - 5A + 7I = O\). Using this, obtain \(A^{-1}\).

Differentiate the function \(x^{\cos x}\) with respect to x.

Find the differential equation of the family of curves denoted by \(y = a \sin(x+b)\), where a and b are arbitrary constants.

Find the shortest distance between the lines \( \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \) and \( \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \).

A car is started 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 \( x = t^2(2 - \frac{t}{3}) \). Find the time required by the car to reach at point Q and also find the distance between P and Q.

Find the area enclosed by the curve \( y = x^2 \) and the line \( y = 16 \).

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

If \( E_1 \) and \( E_2 \) are mutually exclusive events, then prove that \( P(E_1) + P(E_2) = P(E_1 \cup E_2) + P(E_1 \cap E_2) \).

If \(A = \begin{bmatrix} 1 & 3 & 3
1 & 4 & 3
1 & 3 & 4 \end{bmatrix}\), then find \(A^{-1}\).

Solve the following system of equations by matrix method:
\(x + y + 2z = 1\)
\(3x + 2y + z = 7\)
\(2x + y + 3z = 2\)

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

Solve : \( (1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0 \).

Prove that \(\int_0^\pi \sqrt{\left(\frac{1+\cos 2x}{2}\right)} dx = 2\).

Prove that \(\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x}} = \frac{\pi}{12}\).