List of top Mathematics Questions asked in CBSE CLASS XII

Find : \( \int \frac{dx}{x^{1/2} + x^{1/3}} \)

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

Evaluate : \( \int_{-\pi/2}^{\pi/2} \frac{\cos^2 x}{2^x + 1} \, dx \)

Find : \( \int \tan^{-1}\left(\frac{1 - x}{1 + x}\right) \, dx \)

Find the vector equation of a line passing through the origin and perpendicular to both the lines \( \vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k}) \) and \( \vec{r} = \mu(\hat{i} - \hat{j} + \hat{k}) \).

Find the general solution of the differential equation \( (x^2 + y^2) \, dy = xy \, dx \).

Find the particular solution of the differential equation \( \frac{dy}{dx} - 3y \cot x = \sin 2x \), given that \( y = 2 \) when \( x = \frac{\pi}{2} \).

Find : \( \int \frac{x + 3}{x^2 + 4x + 5} \, dx \)

If \( x = e^{t + \frac{1}{t}} \) and \( y = e^{t - \frac{1}{t}} \), then find \( \frac{dy}{dx} \) at \( t = -2 \).

Assertion (A) : The vectors \( \vec{a} \) and \( (-2\vec{a}) \), where \( \vec{a} \neq \vec{0} \), are collinear vectors.
Reason (R) : \( \vec{a} \cdot (-2\vec{a}) = 0 \).

If for two unit vectors \( \vec{a} \) and \( \vec{b} \), \( |\vec{a} + 2\vec{b}| = |2\vec{a} - \vec{b}| \), then find the angle between \( \vec{a} \) and \( \vec{b} \).

Assertion (A) : One of the particular solutions of the differential equation \( \frac{dy}{dx} = e^{x+y} \) can be \( e^x + e^{-y} = -2 \).
Reason (R) : \( e^x + e^{-y} = C \) is the general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \).

If the lines \( \frac{x - 3}{1} = \frac{1 - y}{1} = \frac{z + 2}{p} \) and \( \frac{2 - x}{3} = \frac{y + 1}{5} = \frac{z + 56}{2p} \) are perpendicular to each other, then find the value(s) of p.

Find the value of \( \sin [\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))] \).

If a square matrix A is such that \( A^2 = A \) and \( (I - A)^3 = xA + I \), then value of x must be :

If a matrix B is such that \( B \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 0 & 1 & 1 2 & 0 & 1 \end{bmatrix} \), then the order of matrix B is :

If \( A^2 = 4A + 3I \) and \( A^{-1} = xA + yI \), then the value of \( (x + y) \) is :

If A and B are skew-symmetric matrices of same order, then \( AB' + BA' \) is a/an :

If \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198 \) and \( |\vec{a}| = 10 |\vec{b}| \), then :

The domain of \( \sin^{-1}(1 - 2x) \) is :

For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true ?

If \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are direction cosines of lines \( L_1 \) and \( L_2 \) respectively and \( \theta \) is the acute angle between them, then :

Direction ratios of lines \( l_1 \) and \( l_2 \) respectively are \( \langle 1, -2, 3 \rangle \) and \( \langle -2, p, -6 \rangle \). The value of p for which \( l_1 \parallel l_2 \), is :

The area of the shaded region of the circle given below is equal to :

The integral \( \int_{-1}^{1} (1 - |x|) \, dx \) is equal to :