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

A relation \(R = \{(a, b) : a = b - 1, b > 4\}\) is defined on set \(\mathbb{N}\), then correct answer will be:

• (A) \((2, 4)\)
• (B) \((4, 5)\)
• (C) \((4, 6)\)
• (D) \((3, 5)\)

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

• (A) \(\pi\)
• (B) \(-\frac{\pi}{2}\)
• (C) \(0\)
• (D) \(2\sqrt{3} \pi\)

Differential coefficient of \(\cos^{-1}(e^x)\) will be:

• (A) \(\sin^{-1}(e^x)\)
• (B) \(\frac{c^x}{\sqrt{1 - e^{2x}}}\)
• (C) \(\frac{-e^x}{\sqrt{1 - e^{-2x}}}\)
• (D) \(\frac{-e^x}{\sqrt{1 - e^{2x}}}\)

The value of \(\int x e^x \, dx\) will be:

• (A) \(e^x\)
• (B) \((1 + x) e^x\)
• (C) \((x - 1) e^x\)
• (D) \((1 - x) e^x\)

The order of the differential equation \(2x^2 \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + y = 0\) will be:

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) None of these

If \( A = \{a, b, c\} \) and \( B = \{1, 2\} \), then find the number of relations from \( A \) to \( B \).

Find the maximum value of \( Z = 3x + 4y \) under the constraints \[ x + y \leq 4, \quad x \geq 0, \quad y \geq 0. \]

If vectors \( 2\hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular to each other, then find the value of \( \lambda \).

Show that \( f(x) = |x| is continuous at x = 0.

If \[ P(A) = \frac{3}{13}, \quad P(B) = \frac{5}{13}, \quad and \quad P(A \cap B) = \frac{2}{13}, \]
\text{then find the value of \( P(B|A) \).

If \[ \begin{bmatrix} x + z
y + z
x + y + z \end{bmatrix} = \begin{bmatrix} 5
7
9 \end{bmatrix}, \]
then find the value of \( x \), \( y \), and \( z \).

Find the general solution of \[ \frac{dy}{dx} = \frac{x - 1}{2 + y}. \]

Prove that the function \( f : \mathbb{R} \to \mathbb{R}^+ \) defined by \( f(x) = e^x \) is one-one.

If \( x = a \cos^2 t, y = b \sin^2 t \), then find \( \frac{dy}{dx} \).

Find the equation of tangent at the point \( (am^2, am^3) \) on the curve \[ ay^2 = x^3. \]

If \[ P(A) = 0.4 \quad and \quad P(B) = 0.5, \quad also, A and B are independent events, then find \]
(i) \( P(A \cup B) \) and (ii) \( P(A \cap B) \).

Show that the points \( A(2, 3, 4) \), \( B(-1, -2, 1) \) and \( C(5, 8, 7) \) are collinear.

Evaluate: \[ \int \sqrt{x^2 + 2x + 5} \, dx \]

Let a relation \( R = \{(a, b) : (a - b) is a multiple of 5 \} \) be defined on the set \( \mathbb{Z} \) (set of integers). Prove that \( R \) is an equivalence relation.

If \[ A = \begin{bmatrix} 8 & 0
4 & -2
3 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & -2
4 & 2
-5 & 1 \end{bmatrix}, \]
and \[ 2A + 3X = 5B, \quad then find the matrix \, X. \]

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

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

Prove that the function \[ f(x) = \tan^{-1} (\sin x + \cos x), \quad x > 0 is always increasing function on (0, \frac{\pi}{4}). \]

Evaluate: \[ \int \sqrt{x^2 - a^2} \, dx. \]

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

Find the particular solution of the differential equation \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, \quad (x \neq 0) \]
given that \( y = 0 \) if \( x = \frac{\pi}{2} \).

Find the shortest distance between two lines: \[ \mathbf{r_1} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda (2\hat{i} + 3\hat{j} + 6\hat{k}) \]
and \[ \mathbf{r_2} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu (2\hat{i} + 3\hat{j} + 6\hat{k}) \]

Find the minimum value of \[ Z = 50x + 70y \]
\text{under the following constraints by graphical method: \[ 2x + y \geq 8, \] \[ x + 2y \geq 10, \quad x \geq 0, \quad y \geq 0. \]

There are three children in a family. If it is known that at least one child is a girl among them, find the probability that all three children are girls.

Evaluate: \[ \int_{-1}^{3/2} |x \sin (\pi x)| \, dx. \]

Find the differential coefficient of \[ y = x^x + (\cos x)^{\tan x}. \]

Prove that every differentiable function is continuous. Examine continuity and differentiability of the function \[ f(x) = |x + 2| \quad at \quad x = -2. \]

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

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