List of top Mathematics Questions asked in CBSE CLASS XII

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

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

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

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

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

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

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 :

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

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

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

The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :

\( \frac{dy}{dx} = F(x, y) \) will be a homogeneous differential equation for which of the following functions ?
(i) \( F(x, y) = 3x + 2y \)
(ii) \( F(x, y) = \sin \frac{y}{x} + \log y - \log x \)
(iii) \( F(x, y) = e^{y/x} + 1 \)
(iv) \( F(x, y) = \sqrt{x^2 + y^2} - y \)

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

If \( \int \frac{dx}{\sqrt{e^{-2x} - 1}} \) is equal to :

If \( \tan^{-1} x = y \), then \( \frac{dy}{dx} \) is equal to :

A rectangle of perimeter \(24\text{ cm}\) is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.

A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line \(\vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k})\). Find the co-ordinates of the point of intersection. Hence, write the vector equation of a line passing through the point of intersection and perpendicular to both the lines.

Evaluate the definite integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin |x| + \cos |x|)_~dx \]