UP Board Class 12 Mathematics Question Paper 2023 with Solutions PDF (February 27, Code 324 AX)

The degree of differential equation \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \]
will be

• (A) 2
• (B) 5
• (C) 1
• (D) \( \frac{1}{5} \)

The value of \[ \int \cos^2 x \, dx \]
will be

• (A) \( \frac{x}{2} + \frac{1}{4} \sin(2x) + c \)
• (B) \( \frac{x}{4} - \frac{1}{2} \sin(2x) + c \)
• (C) \( \cos^2 x - \sin^2 x + c \)
• (D) \( 2 \cos x \sin x + \frac{x}{2} + c \)

The angle between the vectors \[ \mathbf{A} = 2\hat{i} + \hat{j} + 3\hat{k} \quad and \quad \mathbf{B} = 3\hat{i} - 2\hat{j} + \hat{k} \]
will be

• (A) \( 90^\circ \)
• (B) \( 60^\circ \)
• (C) \( 30^\circ \)
• (D) \( \cos^{-1} \left( \frac{1}{14} \right) \)

If the numbers of elements of two finite sets \( A \) and \( B \) are \( m \) and \( n \) respectively, then the total number of relations from \( A \) to \( B \) will be

• (A) \( 2^{m+n} \)
• (B) \( 2^{mn} \)
• (C) \( m \times n \)
• (D) \( m + n \)

If \( A = \{1, 2, 3\} \), \( B = \{2, 3, 4\} \), then the function from A to B will be

• (A) \( \{ (1, 2), (1, 3), (2, 3), (3, 3) \} \)
• (B) \( \{ (1, 3), (2, 4) \} \)
• (C) \( \{ (1, 3), (2, 2), (3, 3) \} \)
• (D) \( \{ (1, 2), (2, 3), (3, 2), (3, 4) \} \)

Prove that the function \[ f(x) = \begin{cases} x^3 - 3 & if x \leq 2
x^2 + 1 & if x > 2 \end{cases} \]
is continuous at \( x = 2 \).

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

Prove that the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x - 1 \), when \( x > 2 \), and \( f(1) = f(2) = 1 \), is onto but not one-to-one.

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

Solve the inequality \( 8x + 4 < 7x + 8 \).

Find the coordinates of the point which divides the line joining the points \( (2, -5, 1) \) and \( (1, 4, -6) \) internally in the ratio 2 : 3.

Find the area of the triangle whose two sides are represented by the vectors \[ \vec{a} = 3\hat{i} - \hat{j} + 5\hat{k}, \quad \vec{b} = \hat{i} + 2\hat{j} - \hat{k}. \]

If \[ A = \begin{bmatrix} \cos \theta & \sin \theta
-\sin \theta & \cos \theta \end{bmatrix}, \]
prove that \[ A^3 = \begin{bmatrix} \cos 3\theta & \sin 3\theta
-\sin 3\theta & \cos 3\theta \end{bmatrix}. \]

Two integers among 1 to 11 are selected at random. If their sum is even, then find the probability that both integers are odd.

If \( f : \mathbb{R} \to \mathbb{R} \), where \( f(x) = \sin x \) and \( g : \mathbb{R} \to \mathbb{R} \), where \( g(x) = x^2 \), then find the range of \( f(x) \) and \( g(x) \).

If \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A \cup B) = \frac{2}{3} \), prove that the events \( A \) and \( B \) are independent.

If \( \vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k} \), \( \vec{b} = -\hat{i} + 2\hat{j} + \hat{k} \), and \( \vec{c} = 3\hat{i} + \hat{j} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then find the value of \( \lambda \).

Solve the differential equation \( (x - y) \, dy - (x + y) \, dx = 0 \).

Prove: \[ \left| \begin{matrix} 1+\alpha & 1 & 1
1+\beta & 1 & 1
1 & 1 & 1+\gamma
\end{matrix} \right| = abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 \right) \]

If \( f(x) = x + \frac{1}{x} \), prove that \( \left[ f(x) \right]^3 = f(x^3) + 3f\left( \frac{1}{x} \right) \).

Find the differential coefficient of \( \tan^{-1}\left( \frac{2x}{1-x^2} \right) \) \text{with respect to \( x \).

Find the equation of the plane passing through the points \((-2i + 6j - 6k), (-3i + 10j - 9k), (-5i - 6j - 6k)\).

Find the equation of normal at the point \( (1, 1) \) of the curve \( x^{2/3} + y^{2/3} = 2 \).

If two dice are thrown together, then find the probability of getting at least one 6.

Evaluate: \[ \int \frac{\sec^2(2x)}{(\cot x - \tan x)^2} \, dx. \]

If the coordinates of mid-points of the sides of a triangle are \( (1, 5, -1), (0, 4, -2) \) and \( (2, 3, 4) \), then find the coordinates of its vertices.

Find the value of \( \int_a^b x^2 \, dx \) with the help of definite integral as the limit of a sum.

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

Solve the following system of equations by matrix method: \[ 3x - 2y + 3z = 8, \quad 2x + y - z = 1, \quad 4x - 3y + 2z = 4. \]

Solve the differential equation: \[ \tan^{-1}(y - x) \, dy = (1 + y^2) \, dx. \]

Evaluate: \[ \int_0^\pi \frac{x \sin x}{1 + \cos^2 x} \, dx. \]

Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x + \cos x}} \, dx. \]

If the normal of the curve \[ \frac{x^{2/3}}{a^{2/3}} + \frac{y^{2/3}}{a^{2/3}} = 1 \]
makes an angle \( \theta \) with the x-axis, prove that the equation of the normal is \[ y \cos \theta - x \sin \theta = a \cos 2\theta. \]

Solve the following linear programming problem by the graphical method, under the following constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]
Find the minimum and maximum values of \[ Z = 3x + 9y. \]

Find the inverse of the matrix \[ A = \begin{pmatrix} 2 & 0 & -1
5 & 1 & 0
0 & 1 & 3 \end{pmatrix} \]
by elementary transformations.