List of top Questions asked in UP Board XII- 2026

The value of the integral \(\displaystyle\int\dfrac{\sin^2x-\cos^2x}{\sin^2x\cos^2x}\,dx\) is

The value of \(\cos(\sec^{-1}x+\text{cosec}^{-1}x)\), \(|x|\ge 1\) is

\(f:X \to Y\) will be an onto function if and only if its range is

Solve: \(\displaystyle\int\left[\sqrt{\cot x}+\sqrt{\tan x}\right]dx\).

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

Find the value of: \(\displaystyle\int_0^{\pi/2}\log\sin x\,dx\).

If \(A=\begin{bmatrix}1&2&3\\2&4&5\\3&5&6\end{bmatrix}\), then find \(A^{-1}\).

Solve the following system of equations by using the Matrix Method: \[ x-y+2z=7 \] \[ 3x+4y-5z=-5 \] \[ 2x-y+3z=12 \]

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

Prove that the height of the cylinder of maximum volume inscribed in a sphere of radius \(R\) is \(\dfrac{2R}{\sqrt3}\). Also find the maximum volume of the cylinder.

Prove that vectors \(2\hat i-\hat j+\hat k\), \(\hat i-3\hat j-5\hat k\) and \(3\hat i-4\hat j-4\hat k\) are vertices of a right-angled triangle.

Evaluate \(\displaystyle\int\dfrac{x^2+1}{x^2-5x+6}\,dx\).

If \(y=Ae^{mx}+Be^{nx}\), then show that \(\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mny=0\).

Maximize \(Z=10x+3y\) by the graphical method under the following constraints: \(x\ge0;\ y\ge0;\ 5x+3y\le15;\ 2x+5y\le10\).

Find the value of the determinant \(\begin{vmatrix}a&b&c\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}\).

Prove that \(\tan^{-1}\dfrac15+\tan^{-1}\dfrac17+\tan^{-1}\dfrac13+\tan^{-1}\dfrac18=\dfrac\pi4\).

Solve the differential equation: \((1+x^2)\,dy+2xy\,dx=\cot x\,dx\).

If \(\cos y=x\cos(a+y)\) and \(\cos a\ne\pm1\), then prove that \(\dfrac{dy}{dx}=\dfrac{\cos^2(a+y)}{\sin a}\).

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

Find the projection of vector \(\hat i+3\hat j+7\hat k\) on vector \(7\hat i-\hat j+8\hat k\).

If \(P(A)=\dfrac13\), \(P(B)=\dfrac12\) and \(P(A\cup B)=\dfrac23\), then show that \(A\) and \(B\) are independent events.

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

If \(y=5\cos x-3\sin x\), then prove that \(\dfrac{d^2y}{dx^2}+y=0\).

Find the vector and Cartesian equations of a line which passes through the point \((1,2,3)\) and is parallel to the vector \(2\hat i+3\hat j+2\hat k\).

Find the value of the determinant \(\begin{vmatrix}x&a&x+a\\y&b&y+b\\z&c&z+c\end{vmatrix}\).