List of top Mathematics Questions asked in CBSE CLASS XII

Evaluate :

\[ I = \int_{-\pi}^{\pi} \frac{2x\,(1+\sin x)}{1+\cos^2 x}\,dx \]

Find the product of the maximum and minimum values of the function \( f(x) = \cos x - \frac{1}{2}\cos 2x,\ x \in \left[0, \frac{\pi}{2}\right] \).

Solve the differential equation \( y + \frac{d}{dx}(xy) = x(\sin x + \log x) \).

Find the value of \(x\) if \(A = B\), where

\[ A = \begin{bmatrix} 6 & 3x \\ 8 & -6 \end{bmatrix} \text{ and } B = \begin{bmatrix} 6 & x^2+2 \\ 8 & x^2-5x \end{bmatrix}. \]

For the following Linear Programming Problem, shade the feasible region on the graph and find the coordinates of its corner points :

Minimize \(Z = 18x + 10y\)

subject to constraints

\[ \begin{aligned} 4x + y &\ge 20 \\ 2x + 3y &\ge 30 \\ x, y &\ge 0. \end{aligned} \]

If \(P(A) = 0.3\), \(P(B) = 0.7\) and \(P(B/A) = 0.5\), then \(P(A \cup B)\) is :

If \(y = \sqrt{\log \left\{ \sin \left( \dfrac{x^2}{3} + 1 \right) \right\}}\), then show that

\[ \frac{dy}{dx} = \frac{x \cot \left( \dfrac{x^2}{3} + 1 \right)}{3y}. \]

Differential of \( \log \sin x \) with respect to \( \sqrt{\cos x} \) is :

\( \int e^{x}\left( \frac{2+\sin 2x}{1+\cos 2x} \right) dx \) is equal to :

The motion of a particle is described as \( S = 2t^{3} - 6t^{2} + 8 \). Its velocity \( \frac{dS}{dt} \), when the acceleration \( \frac{d^{2}S}{dt^{2}} \) vanishes is :

The value of \( \left[ \tan^{-1}(-1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right) \right] \) is :

Solve the differential equation \( (x+y)\,dy + (x-y)\,dx = 0 \), given that \( y(1) = 1 \).

Find the sum of maximum and minimum values of \(f(x) = \sin x\,(1 + \cos x)\) in the interval \([0, \pi]\).

For the given Linear Programming Problem, shade the feasible region on the graph and write the coordinates of the corner points.

Minimize \(Z = 5x + 7y\)

subject to constraints

\[ 2x + y \geq 8 \]

\[ x + 2y \geq 10 \]

\[ x,\ y \geq 0. \]

For events A and B, if \(P(B)=\frac{3}{5}\), \(P(A/B)=\frac{1}{2}\) and \(P(A\cup B)=\frac{4}{5}\), then \(P(A)\) is :

If \(y = \left\{x + \sqrt{x^2 - a^2}\right\}^n\), then prove that \(\dfrac{dy}{dx} = \dfrac{ny}{\sqrt{x^2 - a^2}}\).

Solve the given matrix equation for \(x\), given \(x > 0\):

\[ x\begin{bmatrix} 2x & 2 \\ 3 & x \end{bmatrix} + 2\begin{bmatrix} 8 & 5x \\ 4 & 0 \end{bmatrix} = 2\begin{bmatrix} x^2+8 & 24 \\ 10 & 8 \end{bmatrix} \]

Evaluate :

\[ \int_{0}^{\pi/2} \frac{\sin^{2} x}{1+\sin x \cos x}\, dx \]

If the rate of change of the area of a circle is equal to the rate of change of its diameter, then its radius is :

\(\int \left[\sin(\log x)+\cos(\log x)\right] dx\) is equal to :

If \(y=\sqrt{y+\cos x}\), then \(\frac{dy}{dx}\) is equal to :

The value of \(\cos^{-1}\left[\cos\frac{7\pi}{6}\right]\) is :

(iii) (b) If it is given that the area of the window is \(100\ \text{m}^2\), then find the expression for its perimeter in terms of x.

(ii) From (i), write an expression for the area of the window as a function of x only.

If a part of the newly constructed building falls off, then what is the probability that it was due to the selection of company Y or Z ?