List of top Questions asked in UP Board XII- 2026

If \(f(x)=\begin{cases}x+2, & x\ne0\\1, & x=0\end{cases}\), then prove that the function is not continuous at \(x=0\).

Find the angle between vectors \(\hat i-2\hat j+3\hat k\) and \(3\hat i-2\hat j+\hat k\).

If \(P(A)=\dfrac7{13}\), \(P(B)=\dfrac9{13}\) and \(P(A\cap B)=\dfrac4{13}\), then find \(P\left(\dfrac AB\right)\).

Find the value of \(\displaystyle\int_2^3\dfrac1x\,dx\).

If functions \(f:R\to R\) and \(g:R\to R\) are defined respectively as \(f(x)=\cos x\) and \(g(x)=3x^2\), then find \(gof\) and \(fog\).

Find the principal value of \(\sin^{-1}\left(\dfrac{1}{\sqrt2}\right)\).

Direction ratios of a line are 2, \(-1\) and \(-2\). Find the direction cosines of the line.

If vectors \(5\hat i-\lambda\hat j+2\hat k\) and \(2\hat i+3\hat j+4\hat k\) are perpendicular to each other, then the value of \(\lambda\) is:

The relation \(R=\{(a,b): a\le b^2\}\) is defined in the set of real numbers. Then \(R\) is:

If \(x-y=\pi\), then the value of \(\dfrac{dy}{dx}\) is:

Direction cosines of the \(x\)-axis are:

If the function \(f:N\to N\) is defined as \(f(x)=x^2\), then \(f\) is:

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

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

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

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.

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

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.

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 \(\vec a,\vec b,\vec c\).

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.

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