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

The relation R, defined by R = { (\(T_1\), \(T_2\)) : \(T_1\) is similar to \(T_2\) \, in the set A of all triangles, is

• (A) reflexive and symmetric, but not transitive
• (B) reflexive and transitive, but not symmetric
• (C) symmetric and transitive, but not reflexive
• (D) reflexive, symmetric and also transitive

Let [x] represents the greatest integer which is less than or equal to x. Then the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = [x]\) will be

• (A) One-one and onto
• (B) One-one, but not onto
• (C) Onto, but not one-one
• (D) Neither one-one nor onto

The order of the differential equation \( \left( \frac{d^3y}{dx^3} \right)^2 + \log x \left( \frac{d^2y}{dx^2} \right)^3 + 5y = \cos x \) will be

• (A) 2
• (B) 3
• (C) 5
• (D) 6

If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) will be

• (A) \( -3\hat{i} - 5\hat{j} + 4\hat{k} \)
• (B) \( 3\hat{i} + 5\hat{j} + 4\hat{k} \)
• (C) \( -3\hat{i} - 5\hat{j} - 4\hat{k} \)
• (D) \( 3\hat{i} + 5\hat{j} - 4\hat{k} \)

If \( 2X + Y = \begin{bmatrix} 1 & 0
-3 & 2 \end{bmatrix} \) and \( Y = \begin{bmatrix} 3 & 2
1 & 4 \end{bmatrix} \), then X will be

• (A) \( \begin{bmatrix} -1 & -1
  -2 & -1 \end{bmatrix} \)
• (B) \( \begin{bmatrix} -1 & -1
  -1 & -2 \end{bmatrix} \)
• (C) \( \begin{bmatrix} -2 & -1
  -1 & -1 \end{bmatrix} \)
• (D) \( \begin{bmatrix} 1 & -1
  -1 & -1 \end{bmatrix} \)

Find the principal value of \( cosec^{-1}(-\sqrt{2}) \).

Test whether the function defined by \( f(x) = x^2 - \sin(x) + 5 \) is continuous at \( x = \pi \).

Evaluate: \( \int cosec\,x(cosec\,x + \cot x) \,dx \).

If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A/B) = \frac{2}{5} \), then find \( P(A \cup B) \).

Find the direction-cosines of the y-axis.

Differentiate \(x^{\sin x}\) with respect to \(x\), while \(x > 0\).

Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).

Let a relation R be defined in the set \(\mathbb{N} \times \mathbb{N}\) as follows: \((a, b) R (c, d)\) if and only if \(a + d = b + c\). Prove that R is an equivalence relation.

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

Evaluate : \( \int \frac{dx}{(x+1)(x+2)} \).

In a leap year, selected at random, find the probability that there are 53 Tuesdays.

If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha
-\sin \alpha & \cos \alpha \end{bmatrix} \), verify that \( A'A=I \).

Show that the given function \( f, f(x) = x^3 - 3x^2 + 4x, x \in \mathbb{R} \) is an increasing function in \( \mathbb{R} \).

Find the value of the determinant \( \begin{vmatrix} a^2+1 & ab & ac
ab & b^2+1 & bc
ca & cb & c^2+1 \end{vmatrix} \).

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

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

Prove that \(\tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x\), where \(-\frac{1}{\sqrt{2}} \leq x \leq 1\).

Find the general solution of the differential equation \(x\frac{dy}{dx} + 2y = x^2\) (\(x \neq 0\)).

A die was thrown twice and the sum of the numbers which appeared was found to be 6. Find the conditional probability that the number 4 appears at least once.

Maximize \(Z = x + 2y\) by graphical method under the constraints \(x+y \le 1, -x+y \le 0, x \ge 0, y \ge 0\).

Find the area of the parallelogram whose diagonals are \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} - 3\hat{j} + 4\hat{k} \).

If \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), prove that \( \sin\frac{\theta}{2} = \frac{1}{2} |\hat{a} - \hat{b}| \).

Sand is falling from a pipe at the rate of 12 cm\(^3\)/second. The falling sand forms such a cone on the ground that its height is always one-sixth of the radius of its base. At which rate is the height of the cone formed by sand increasing while its height is 4 cm?

Solve by matrix method the following system of equations:
\(2x + y + z = 1\)
\(x - 2y - z = \frac{3}{2}\)
\(3y - 5z = 9\)

If \( F(x) = \begin{bmatrix} \cos x & -\sin x & 0
\sin x & \cos x & 0
0 & 0 & 1 \end{bmatrix} \), prove that \(F(x)F(y) = F(x+y)\).

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

Prove that \(\int_{0}^{\pi/4} \log(1 + \tan x) dx = \frac{\pi}{8} \log 2\).

Find the area of the region bounded by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Find the general solution of the differential equation \(y - x \frac{dy}{dx} = x + y \frac{dy}{dx}\).