UP Board Class 12 Mathematics Question Paper 2025 PDF (Code 324 IY) is available for download here. The Mathematics exam was conducted on March 3, 2025 in the Morning Shift from 8:30 AM to 11:45 AM. The total marks for the theory paper are 100. Students reported the paper to be easy to moderate.
UP Board Class 12 Mathematics Question Paper 2025 (Code 324 IY) with Solutions
| UP Board Class Mathematics Question Paper with Answer Key |  Download |
Check Solutions |
If \( f: \mathbb{R} \to \mathbb{R} \), is given by \( f(x) = (3 - x^3)^{1/3} \), then \( fof(x) \) is equal to :
A relation R is defined in the set N as follows :
R = {(x, y) : x = y – 3, y > 3}
Then which of the following is correct?
The value of \( \int_1^{\sqrt{3}} \frac{dx}{1 + x^2} \) will be :
The order of the differential equation \( \frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y \cos x = 0 \) is :
The value of \( \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \) is :
Find the principal value of \( \sec^{-1}(-\sqrt{2}) \).
Does the function \( f(x) = \begin{cases} x+5 & if x \le 1
x-5 & if x > 1 \end{cases} \) continuous at \( x = 1 \)?
Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \cdot \sin x = 0 \).
Find the direction cosines of a line joining two points \( (-2, 4, -5) \) and \( (1, 2, 3) \).
If P(A) = 0.3, P(B) = 0.4, then find P(A/B) if A and B are independent events.
If \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \) and \( A + A' = I \), then find the value of \( \alpha \).
If \( y = x^{x^{x^{\dots ad inf}}} \), then prove that \( x\frac{dy}{dx} = \frac{y^2}{1 - y \log x} \).
Find the angle between the pair of lines \( \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \).
For two invertible matrices A and B of order n, prove that \( (AB)^{-1} = B^{-1}A^{-1} \).
Show that the given function \( f(x) = \cos x \) is increasing in \( (\pi, 2\pi) \).
If \( [a_{ij}] = 2i - j \), then determine a matrix A of order \( 2 \times 3 \).
Find the area of a parallelogram whose adjacent sides are given by vectors \( \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} \).
The volume of a cube is increasing at the rate of 9 cm\(^3\)/s. If the length of its edge is 10 cm, then its surface area is increasing with which rate?
Does the relation defined by R = {(x, y) : y is divisible by x} on the set A = {1, 2, 3, 4, 5, 6}, an equivalence relation?
Express the matrix \( A = \begin{bmatrix} 3 & 3 & -1
-2 & -2 & 1
-4 & -5 & 2 \end{bmatrix} \) as the sum of a symmetric and a skew-symmetric matrix.
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \) for \( -1 < x < 1 \), then prove that \( \frac{dy}{dx} = -\frac{1}{(1+x)^2} \).
Find the angle between the pair of lines:
\( \vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k}) \) and
\( \vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k}) \).
Find the shortest distance between the lines:
\( \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1} \) and
\( \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1} \).
Find the equations of the tangent and normal to the curve \( x^{2/3} + y^{2/3} = 2 \) at the point (1, 1).
Find the area of the region enclosed by the parabola \( y^2 = 4ax \) and its latus rectum.
Solve the differential equation \( ydx - (x + 2y^2) dy = 0 \).
Find the maximum value of Z = 8x + 5y under the constraints \( 5x + 3y \le 15 \), \( 2x + 5y \le 10 \) and \( x \ge 0, y \ge 0 \) by graphical method.
There are three groups of children having 3 girls and one boy, 2 girls and 2 boys, one girl and 3 boys respectively. One child is selected at random from each group. Find the probability that the three selected children have one girl and 2 boys.
If \( A = \begin{bmatrix} 1 & 3 & 3
1 & 4 & 3
1 & 3 & 4 \end{bmatrix} \), then verify that \( A \cdot adj(A) = |A| I \) and find \( A^{-1} \).
Solve the following system of linear equations by matrix method:
\( x + y + z = 6 \)
\( y + 3z = 11 \)
\( x + z = 2y \)
Evaluate: \( \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \)
Show that \( y = c_1 e^{ax} \cos(bx) + c_2 e^{ax} \sin(bx) \), where \( c_1, c_2 \) are constants, is a solution of the differential equation \( \frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0 \).
Prove that: \( \int_0^{\pi} \log(1 + \cos x) dx = -\pi \log_e 2 \)
Find the particular solution of the differential equation \( \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, (x \neq 0) \). It is given that \( y = 0 \) if \( x = \frac{\pi}{2} \).
Comments