UP Board Class 12 Mathematics Question Paper 2026 (Set 324 DB) with Solutions

If \(A=\{1,2,3\}\), then the number of equivalence relations on \(A\) containing \((1,2)\) is:

• (A) 1
• (B) 2
• (C) 3
• (D) 4

The function \(f:Z\to Z\) defined by \(f(x)=x^3\ \forall x\in Z\) is:

• (A) Onto
• (B) Neither one-one nor onto
• (C) One-one but not onto
• (D) Many-one

\(\sin(\tan^{-1}x),\ |x|<1\) is equal to:

• (A) \(\dfrac{x}{\sqrt{1-x^2}}\)
• (B) \(\dfrac{1}{\sqrt{1-x^2}}\)
• (C) \(\dfrac{1}{\sqrt{1+x^2}}\)
• (D) \(\dfrac{x}{\sqrt{1+x^2}}\)

\(A=[a_{ij}]_{m\times n}\) is a square matrix if:

• (A) m < n
• (B) m > n
• (C) m = n
• (D) None of these

The value of \(\displaystyle\int x\cos x\,dx\) is:

• (A) \(\cos x+x\sin x+c\)
• (B) \(x\sin x+c\)
• (C) \(x\cos x+c\)
• (D) \(\sin x+x\cos x+c\)

Find the value of \(\displaystyle\int_{-1}^{+1}(x+1)\,dx\).

Find the integrating factor of the differential equation \(x\dfrac{dy}{dx}+2y=x^2\ (x\neq0)\).

Find the projection of vector \((\hat i-\hat j)\) on vector \((\hat i+\hat j)\).

If \(P(A)=\dfrac12\), \(P(B)=\dfrac13\) and \(A,B\) are independent events, find the value of \(P(A/B)\).

If \(X=\{1,2,3\}\) and \(Y=\{a,b\}\), then find the number of functions from \(X\) to \(Y\).

Write \(\tan^{-1}\left(\dfrac{x}{\sqrt{a^2-x^2}}\right),\ |x|<a\) in simplest form.

If \(A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}\) and \(B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}\), then find the value of \((2A-B)\).

Prove that \(3\cos^{-1}x=\cos^{-1}(4x^3-3x),\ x\in\left[\dfrac12,1\right]\).

Differentiate the function \(\sin^2x\) with respect to the function \(e^{\cos x}\).

Prove that the given function \(f(x)=3x+17\) is increasing on \(\mathbb{R}\).

If \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be functions such that \(f(x)=\cos x\) and \(g(x)=3x^3\), then find \(f\circ g\).

Find the general solution of the differential equation \(\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}\).

Find the magnitude of the vectors \(\vec a\) and \(\vec b\), if having equal magnitudes and angle \(60^\circ\) between them and their scalar product is \(\dfrac12\).

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

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

Show that the line passing through the points \((1,-1,2)\) and \((3,4,-2)\) is perpendicular to the line passing through the points \((0,3,2)\) and \((3,5,6)\).

If \(A=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}\) and \(I=\begin{bmatrix}1&0\\0&1\end{bmatrix}\) and \(A^2=KA-2I\), then find the value of \(K\).

Prove that the function defined by \(f(x)=\begin{cases}x^2\sin\dfrac1x, & x\neq0\\0, & x=0\end{cases}\) is continuous.

A circular disc having radius 3 cm is warmed and due to expansion its radius is increasing at the rate 0.05 cm/s. Find the increasing rate of its area when its radius is 3.2 cm.

Find the minimum value of the linear programming problem \(Z=200x+500y\) under the following constraints: \(x+2y\ge10\), \(3x+4y\le24\), \(x\ge0,y\ge0\) by graphical method.

A die is thrown twice and found that the sum of the numbers is 6. Find the conditional probability of getting number 4 at least once.

If vectors \(\vec a,\vec b,\vec c\) are mutually perpendicular and having equal magnitude, then show that the vector \(\vec a+\vec b+\vec c\) is equally inclined to the vectors \(\vec a,\vec b,\vec c\).

Find the area of the circle \(x^2+y^2=r^2\).

If \(A=\begin{bmatrix}2&-3&5\\3&2&-4\\1&1&-2\end{bmatrix}\), then find \(A^{-1}\).

Solve the system of linear equations by Matrix Method: \(2x+3y+3z=5\), \(x-2y+z=-4\), \(3x-y-2z=3\).

Prove that the radius of the largest (maximum surface area) right circular cylinder that can be inscribed in a cone is half of the radius of the cone.

Evaluate: \(\displaystyle\int_0^{\pi/2}\log\sin x\,dx\).

Find the value of \(\displaystyle\int_0^{\pi}\dfrac{x\,dx}{a^2\cos^2x+b^2\sin^2x}\).

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