List of top Questions asked in PSEB XII

Find the area of the region bounded by \( y^2 = 4x \), \( x = 1 \), \( x = 4 \), and the x-axis in the first quadrant.

Evaluate \[ \int_0^{\frac{\pi}{2}} \log \sin x \, dx \]

Solve by Matrix Method
Given: \[ x + 2y - 3z = 6, \quad 3x + 2y - 2z = 3, \quad 2x - y + z = 2 \]

Divide a number 15 into two parts such that the square of one multiplied with the cube of the other is a maximum.

Express the matrix
\[ \begin{bmatrix} 7 & 0 & 3 \\ 2 & 4 & 1 \\ -5 & 6 & 8 \end{bmatrix} \] as the sum of symmetric and skew-symmetric matrices.

If \[ A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}, \] prove that \[ A^2 - 4A - 5I = 0. \] Hence, find \( A^{-1} \).

Discuss the continuity of \( f(x) \) at \( x = 0 \), where

Given: \[ f(x) = \begin{cases} \dfrac{1 - \cos(4x)}{x^2}, & x \neq 0 \\[6pt] 8, & x = 0 \end{cases} \]

Find \( (AB)^{-1} \), if

\[ A = \begin{bmatrix} 3 & 4 \\ 1 & 1 \end{bmatrix}, \quad B^{-1} = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} \]

Verify \( (AB)^T = B^T A^T \).

Given:

\[ A = \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix} \]

If \( A \) is an invertible matrix of order \( n \), then \( \text{adj} (\text{adj} A) = |A|^{n-2} A \).

A 2x2 matrix whose elements are \( a_{ij} = \frac{(t + n)^2}{2} \) is:

The adjoint of the matrix

\[ \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \]

is:

When \( x = at^2 \), \( y = 2at \), then \( \frac{dy}{dx} \) is:

\( \int \frac{(\log x)^2}{x} \, dx \) equals:

Derivative of \( \left( \sin^{-1}x + \cos^{-1}x \right) \) with respect to \( x \) is:

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

If \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \), then \( x \) is:

If

\[ \begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix} \]

then \( x \) is equal to:

If

\[ \begin{bmatrix} 1 & 4 \\ -3 & x \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 8 & -3 \end{bmatrix} \]

then the value of \( x \) is:

The matrix

\[ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

is:

If

\[ A = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix}, \quad f(x) = x^2 + 2x - 3 \]

then find \( f(A) \).

Express

\[ \begin{bmatrix} 5 \\ 3 \end{bmatrix} \]

as the sum of a symmetric matrix and a skew-symmetric matrix.

If

\[ X = \begin{bmatrix} 3 & 4 \\ 2 & -1 \end{bmatrix}, \quad 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]

then find the matrix \( Y \).

Solve: \[ x \log x \frac{dy}{dx} + y = \frac{2}{x} \log x \]