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

In the set \(\{1, 2, 3, 4, 5, 6\}\), the relation \(R\) defined by \(R = \{(a, b) : b = a+1\}\) will be:

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

The function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^{2} + 5\) will be:

• (A) one-one and onto
• (B) many-one and onto
• (C) one-one, but not onto
• (D) neither one-one nor onto

If \(A\) is a square matrix such that \(A^{2} = A\), then \((I + A)^{3} - 7A\) will be:

• (A) \(A\)
• (B) \(I - A\)
• (C) \(I\)
• (D) \(3A\)

On the vector \(\hat{i} + \hat{j}\) the projection of vector \(\hat{i} - \hat{j}\) will be:

• (A) \(\dfrac{1}{\sqrt{2}}\)
• (B) \(\sqrt{2}\)
• (C) \(1\)
• (D) \(0\)

The order of the differential equation \(\left(\dfrac{d^{3}y}{dx^{3}}\right)^{2} + x^{2}\left(\dfrac{d^{2}y}{dx^{2}}\right)^{3} + 7y = \sin x\) will be:

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

Find the value of \(\cos(\sec^{-1}x)\).

If \[ A = \begin{bmatrix} 1 & -2 & 3
-4 & 2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3
4 & 5
2 & 1 \end{bmatrix} \]
find the matrix \(AB\).

Test whether the function \(f : \mathbb{R} \to \mathbb{R}\) defined by \[ f(x) = \begin{cases} x^3 + 3, & x \neq 0
1, & x = 0 \end{cases} \]
is continuous at \(x=0\).

Evaluate: \[ \int \sec x \, (\sec x + \tan x) \, dx \]

If \(A\) and \(B\) are two events such that \(P(A) = 0.6\), \(P(B) = 0.3\) and \(P(A \cap B) = 0.15\), find \(P(A|B)\).

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

Evaluate: \[ \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \]

Find the value of \(\vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times(\vec{c}+\vec{a})+\vec{c}\times(\vec{a}+\vec{b})\).

Find the distance of the point \((2,5,-3)\) from the plane \[ \vec{r}\cdot(6\hat{i}-3\hat{j}+2\hat{k})=4. \]

Find the vector and Cartesian equation of the plane which passes through the point \((1,0,-2)\) and on which the vector \(\hat{i} + \hat{j} - \hat{k}\) is perpendicular.

Eliminating arbitrary constants \(a\) and \(b\), find the differential equation represented by the family of curves: \[ y^2 = a(b^2 - x^2) \]

If \[ x = a\left(\cos t + \log \tan \frac{t}{2}\right), \quad y = a \sin t, \]
find the value of \(\dfrac{dy}{dx}\).

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

Prove that \[ \begin{vmatrix} x & x^{2} & yz
y & y^{2} & zx
z & z^{2} & xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx). \]

Prove that in the set \(\mathbb{Z}\) of integers the relation \(R\) given by \(R = \{(a,b): 5 divides (a-b)\}\) is an equivalence relation.

Find the point on the curve \[ y = x^3 - 11x + 5 \]
at which the tangent is \[ y = x - 11. \]

Evaluate: \[ \int \sqrt{x^2 - 8x + 7}\, dx \]

From a well-shuffled pack of 52 cards, three cards were drawn one after another without any replacement. What is the probability that the first two cards be King and the third be Ace?

Prove that \[ \int \sin^{-1}\!\left(\frac{2x}{1+x^2}\right) dx = 2x \tan^{-1}x - \log(1+x^2) + C. \]

Find the equation of the curve passing through \((-2,3)\) on which the gradient of the tangent at any point \((x,y)\) is \[ \frac{dy}{dx} = \frac{2x}{y^2}. \]

The probabilities of solving a particular problem by \(A\) and \(B\) independently are respectively \(\tfrac{1}{2}\) and \(\tfrac{1}{3}\). If both try to solve the problem independently, find the probability that (i) the problem is solved, (ii) only one of them solves the problem.

Solve by matrix method the system of equations: \[ \begin{aligned} 2x - 3y + 5z &= 11,
3x + 2y - 4z &= -5,
x + y - 2z &= -3. \end{aligned} \]

Prove that \[ \int_{0}^{\pi} \frac{x\sin x}{1+\cos^2 x}\, dx = \frac{\pi^2}{4}. \]

Minimize \(Z = 6x + 3y\) under the constraints: \[ 4x + y \geq 80, \quad x + 5y \geq 115, \quad 3x + 2y \leq 150, \quad x \geq 0, \; y \geq 0. \]

If the area bounded by the curve \(x=y^{2}\) and the line \(x=4\) is divided into two equal parts by the line \(x=a\), find the value of \(a\).