UP Board Class 12 Mathematics Question Paper 2023 with Answer Key and Solutions PDF (February 27, Code 324 BB)

If \(f: X \rightarrow Y\) is an onto function, if and only if the range of \(f\) will be:

• (A) \(X\)
• (B) \(X \cap Y\)
• (C) \(Y\)
• (D) \(X \cup Y\)

The value of \(\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)\) will be:

• (A) \(\pi\)
• (B) \(-\dfrac{\pi}{3}\)
• (C) \(\dfrac{\pi}{3}\)
• (D) \(\dfrac{2\pi}{3}\)

If \[ \left[ \begin{array}{cc} 2x - y & x + 2y
2 & 3 \end{array} \right] = \left[ \begin{array}{cc} 1 & 3
2 & 3 \end{array} \right], \]
then the value of \(x\) and \(y\) will be:

• (A) \(x = 1, y = 1\)
• (B) \(x = 2, y = 1\)
• (C) \(x = \dfrac{1}{2}, y = \dfrac{1}{2}\)
• (D) \(x = 1, y = \dfrac{1}{2}\)

The value of \[ \int \frac{dx}{x^2 - a^2} \]
will be:

• (A) \(\frac{1}{2a^2} \log \left| \frac{x - a}{x + a} \right| + C\)
• (B) \(\frac{1}{2a} \log \left| \frac{x - a}{x + a} \right| + C\)
• (C) \(\frac{1}{4a} \log \left| \frac{x + a}{x - a} \right| + C\)
• (D) \(\frac{1}{4a^2} \log \left| \frac{x + a}{x - a} \right| + C\)

The nature of the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y \]
will be:

• (A) Multipower
• (B) Power one and linear
• (C) Homogeneous and power zero
• (D) Homogeneous and power one

Prove that a one-one function \( f : \{2, 3, 4\} \to \{2, 3, 4\} \) is onto.

If \( A = \begin{bmatrix} \cos a & -\sin a
\sin a & \cos a \end{bmatrix} \) and \( A + A' = I \), then find the value of \( a \).

If \[ \Delta = \begin{vmatrix} 3 & 2 & 3
2 & 2 & 3
3 & 2 & 3 \end{vmatrix} \]
then find the value of \(|\Delta|\).

Find the value of the expression \( i \cdot i + j \cdot j + 2k \cdot k \).

Find the degree of the differential equation \[ x y \left( \frac{d^2 y}{dx^2} \right)^2 + x \frac{dy}{dx} - y = 2. \]

Prove that \( f(x) = \tan x \) for all \( x \in \mathbb{R} \) is a continuous function.

If \( y = A \sin x + B \cos x \), then find the differential equation of it.

Find those points on the curve \[ \frac{x^2}{4} + \frac{y^2}{25} = 1, \]
on which the normal is parallel to the x-axis.

If the length of a rectangle is decreasing at the rate of 3 cm/min and width is increasing at the rate of 2 cm/min, then find the rate of change in perimeter of the rectangle when \( x = 10 \) cm and \( y = 6 \) cm, where \( x \) = length and \( y \) = width.

Prove that \( f(x) = |x - 2| \) is not differentiable at \( x = 2 \).

Evaluate: \[ \int \tan^4 x \sec^2 x \, dx. \]

If the position vectors of the points A, B, C and D are \( \mathbf{A} = 3\hat{i} + 2\hat{j} - 3\hat{k}, \) \( \mathbf{B} = \hat{i} + \hat{j} + \hat{k}, \) \( \mathbf{C} = 2\hat{i} + 5\hat{j}, \)
and \( \mathbf{D} = \hat{i} - 6\hat{j} - \hat{k}, \)
respectively. Prove that the points are collinear.

By graphical method solve the LPP under the following constraints:
\[ x + 2y \geq 10, \] \[ 3x + 4y \leq 24, \] \[ x \geq 0, \, y \geq 0, \]
then find the minimum value of \( z = 200x + 500y \).

Find the shortest distance between the lines \[ \vec{r_1} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2t) \hat{k} \]
and \[ \vec{r_2 = (s + 1) \hat{i} + (2s - 1) \hat{j} - (2s + 1) \hat{k}. \]

There are 500 students in a school, of which 230 are girls. Also, 10% of 230 girls are studying in class XII. Find the probability that a randomly chosen student is of XII class and is a girl.

Find the equation of the curve which is passing through the point \( (1, 1) \), and whose differential equation is \[ x \, dy = (2x^2 + 1) \, dx, \quad (x \neq 0). \]

For any two vectors \( \vec{a} \) and \( \vec{b} \), prove that always \[ |\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}|. \]

Evaluate: \[ \int \left( \sqrt{\cot x} + \sqrt{\tan x} \right) dx \]

A relation \( R = \{(x, y) : Number of pages in x and y are equal \} \) is defined on the set \( A \) of all books in a college library. Prove that \( R \) is an equivalence relation.

Differentiate: \( y = x^{x} \).

Let \[ A = \begin{pmatrix} 2 & -1
3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 2
7 & 4 \end{pmatrix}, \quad C = \begin{pmatrix} 2 & 5
3 & 8 \end{pmatrix}. \]
Then find a matrix D such that \( CD - AB = 0 \).

If \[ A = \begin{bmatrix} 1 & 3 & 3
1 & 4 & 3
1 & 3 & 4 \end{bmatrix}, \]
then prove that \[ A \cdot \text{adj \, A = |A| \cdot I. \]

For any positive constant \( a \), evaluate \[ \frac{dy}{dx}, where y = a \frac{t+1}{t} and x = (t + 1)^{\alpha}. \]

Solve the system of linear equations by matrix method: \[ 3x + 2y + 3z = 5, \] \[ -2x + y + z = -4, \] \[ -x + 3y - 2z = 3. \]

Find the equation of the plane which passes through the intersecting point of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \quad and \quad \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5, \]
and the point \( (1, 1, 1) \).

An Apache helicopter of the enemy is flying along the curve \[ y = x^2 + 7. \]
A soldier placed at the point \( (3, 7) \), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

Find the particular solution of the differential equation: \[ \frac{dy}{dx} + y \cot x = 4x \csc x \quad ( x \neq 0 ). \]
Given that \( y = 0 \) when \( x = \frac{\pi}{2} \).

Coloured balls are distributed in three containers according to the following table:
Container & Black & White & Red
I & 3 & 4 & 5
II & 2 & 2 & 2
III & 1 & 2 & 3
A ball is drawn out from a container randomly chosen. If the ball is black, then find the probability that the ball is drawn from Container-III.

Find the maximization of \( z = x + y \), under the following constraints: \[ x - y \leq -1, \quad -x + y \leq 0, \quad x \geq 0, \quad y \geq 0. \]