List of top Questions asked in UP Board XII- 2026

Evaluate: \(\displaystyle\int_0^1 \frac{\tan^{-1}x}{1+x^2}\, dx\)

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

Minimize \(z=3x+2y\) under the following constraints by graphical method: \(x+y\ge8,\ x\ge0,\ 3x+5y\le15,\ y\ge0\).

Prove that \((\vec a+\vec b)\cdot(\vec a+\vec b) = |\vec a|^2+|\vec b|^2\) if and only if \(\vec a\) and \(\vec b\) are perpendicular. It is given that \(\vec a\ne0,\ \vec b\ne0\).

If \(P(A)=\dfrac{7}{13}\), \(P(B)=\dfrac{9}{13}\) and \(P(A\cap B)=\dfrac{4}{13}\), then find \(P(A/B)\).

Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3\).

Differentiate \(\log(\log x)\), \(x>1\) with respect to \(x\).

Prove that on \(R\), the function \(f(x)=e^{2x}\) is increasing.

Find the direction cosines of the \(x\), \(y\) and \(z\) axes.

Find the area of the parallelogram whose adjacent sides are given by \(\vec a = 3\hat i+\hat j+4\hat k\) and \(\vec b = \hat i-\hat j+\hat k\).

If the ordered pairs \((2x-3,-5)\) and \((x,y-1)\) are equal, then find the value of \(x\) and \(y\).

If \(A^2=A\), then simplify \((I+A)^2 - 7A\), where \(A\) is a square matrix.

Find the order and degree of the differential equation \(\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \sin\left(\dfrac{dy}{dx}\right) + 1 = 0\).

Find the area of the region bounded by the curve \(y=\sin x\), \(x=0\) and \(x=2\pi\).

The value of \(\displaystyle\int_0^{2/3} \frac{dx}{4+9x^2}\) is:

Evaluate: \(\displaystyle\int_{-\pi/2}^{\pi/2} (x^3 + x\cos x)\, dx\)

The value of \(\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})\) is:

If \(A\) is any matrix and \(K\) is any constant, then \((KA)'\) is:

If the radius of a circle is \(r = 6\) cm, then the rate of change of Area with respect to radius is:

The function \(f: R \to R\) defined by \(f(x) = 3x + 5,\ \forall x \in R\) is:

Prove that \(\displaystyle\int_0^{\pi/2}\log\sin x\,dx=-\dfrac{\pi}{2}\log2\).

Find the value of \(\displaystyle\int\left(\sqrt{\cot x}+\sqrt{\tan x}\right)dx\).

Solve by matrix method the system of equations: \(x-y+2z=7,\ 3x+4y-5z=-5,\ 2x-y+3z=12\).

Prove that the height of the cylinder of maximum volume inscribed in a right circular cone of semi-vertical angle \(\alpha\) and height \(h\) is one-third of the height of the cone, and the maximum volume of the cylinder is \(\dfrac{4}{27}\pi h^3\tan^2\alpha\).

Find the area of the circle \((x-2)^2+y^2=4\).