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

If \(y = 4t\) and \(x = \dfrac{4}{t}\), then the value of \(\dfrac{dy}{dx}\) will be:

• (A) \(-t^{2}\)
• (B) \(-\dfrac{1}{t^{2}}\)
• (C) \(-\dfrac{1}{t}\)
• (D) \(t^{3}\)

If \(A = \begin{bmatrix} 0 & 1
1 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0
0 & -1 \end{bmatrix}\), then \(BA\) will be:

• (A) \(\begin{bmatrix} -1 & 0
  0 & 1 \end{bmatrix}\)
• (B) \(\begin{bmatrix} 0 & -1
  1 & 0 \end{bmatrix}\)
• (C) \(\begin{bmatrix} 0 & -1
  -1 & 0 \end{bmatrix}\)
• (D) \(\begin{bmatrix} 0 & 1
  -1 & 0 \end{bmatrix}\)

The slope of the normal to the curve \(y = 2x^{2} + 3\sin x\) at \(x = 0\) will be:

• (A) \(\dfrac{1}{3}\)
• (B) \(3\)
• (C) \(-\dfrac{1}{3}\)
• (D) \(-3\)

The value of \(\int_{1}^{\sqrt{3}} \dfrac{dx}{1+x^{2}}\) will be:

• (A) \(\dfrac{\pi}{6}\)
• (B) \(\dfrac{\pi}{12}\)
• (C) \(\dfrac{\pi}{3}\)
• (D) \(\dfrac{2\pi}{3}\)

The modulus function \(f: \mathbb{R} \to \mathbb{R}^{+}\) given by \(f(x) = |x|\) will be:

• (A) one-one and onto
• (B) many-one and onto
• (C) one-one and into
• (D) many-one and into

Evaluate: \(\int x^{2}\sin(x^{3})\,dx\)

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

Solve \(\dfrac{dy}{dx} = \dfrac{x + e^x}{y}\).

If \(\sin^{-1}\left(\dfrac{1}{2}\right) = \tan^{-1}x\), find the value of \(x\).

If \(A = \begin{bmatrix} 2 & 4
3 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 3
-2 & 5 \end{bmatrix}\), then find the value of \((A + B)\) and \((A - B)\).

If \(y = A\cos\theta + B\sin\theta\), then prove that \(\dfrac{d^{2}y}{d\theta^{2}} = -y\).

Solve the inequality \(3x + 4y \leq 12\), \(4x + 3y \leq 12\), \(x \geq 0\), \(y \geq 0\) by graphical method.

Prove that \[ \begin{vmatrix} a+b+2c & a & b
c & b+c+2a & b
c & a & c+a+2b \end{vmatrix} = 2(a+b+c)^3 \]

If \(A\) and \(B\) are two non-singular square matrices of order \(n\), then prove that \[ (AB)^{-1} = B^{-1}A^{-1} \]

Find the equation of tangent at \(t = \dfrac{\pi}{2}\) on the curve \(x = a \sin^{3}t\), \(y = b \cos^{3}t\).

One die is thrown two times. If the sum of the appeared numbers on their faces is 6, find the conditional probability of appearing number 4 at least one time in that.

Find the vector equation of the line \(\dfrac{x+3}{2} = \dfrac{y-5}{4} = \dfrac{z+6}{2}\).

Solve the differential equation \(\log\left(\dfrac{dy}{dx}\right) = 3x + 4y\).

Prove that the relation \(R = \{(a,b) \in \mathbb{Z} \times \mathbb{Z} \;|\; (a-b)\) is divisible by \(2\}\) is an equivalence relation.

If \(A = \begin{bmatrix} 2 & 3
1 & -4 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & -2
-1 & 3 \end{bmatrix}\), then prove that \((AB)^{-1} = B^{-1}A^{-1}\).

Differentiate \(y = (\cos x)^{\tan x} + x^{x}\).

If the function \(f(x) = \begin{cases} ax+1 & x \leq 3
bx+3 & x > 3 \end{cases}\) is continuous at \(x = 3\), then find the values of \(a\) and \(b\).

Find the interval in which the function \[ f(x) = 4x^3 - 6x^2 - 72x + 30 \]
is (i) increasing and (ii) decreasing.

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

Find the shortest distance between the lines \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \] \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \]

If a die is thrown three times, then find the probability of getting at least one appearing number odd.

Find the area included between the circle \(x^{2} + y^{2} = 8x\), parabola \(y^{2} = 4x\) and upper part of \(x\)-axis.

Evaluate \(\displaystyle \int \left[\log(\log x) + \frac{1}{(\log x)^{2}}\right] dx\).

Prove that the matrix \[ A = \begin{bmatrix} 2 & 3
1 & 2 \end{bmatrix} \]
satisfies the equation \[ A^{2} - 4A + I_{2} = 0, \]
where \(I_{2}\) is the \(2 \times 2\) identity matrix and \(0\) is the \(2 \times 2\) zero matrix. Also, find \(A^{-1}\) with the help of this.

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

Maximize \(Z = 8000x + 12000y\) under the constraints: \[ 9x + 12y \leq 180, \quad 3x + 4y \leq 60, \quad x + 3y \leq 30, \quad x \geq 0, \; y \geq 0 \]

Evaluate \[ \int_0^{\pi/2} \frac{\sin^4x}{\sin^4x + \cos^4x} \, dx \]

Evaluate \[ \int_0^{\pi/2} \ln(\sin x) \, dx \]

If \(y = \sin^{-1}x\), then prove that \((1 - x^{2}) \dfrac{d^{2}y}{dx^{2}} - x \dfrac{dy}{dx} = 0\).

Find the equation of tangent of the curve \(y = \cos(x+y)\), \(-2\pi \leq x \leq 2\pi\), which is parallel to the line \(x + 2y = 0\).

Find the vector equation of a plane which passes through the point of intersection 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 \((1,1,1)\).

Solve the differential equation: \[ \frac{dy}{dx} - y = \cos x \]