List of practice Questions

To whom is communalism harmful?
Where is NOT federal government in the following countries?
System of democratic government is better than other systems because this ............
In which state of India did printing press start first of all in 16\textsuperscript{th century?}
How many languages are mentioned in the Constitution besides Hindi?
Where was Bretton Woods Institution organized?
What was the first and main cause of the suspension of the Non Co-operation Movement?
The first main cause of economic depression was –
What is the number of Earth satellites?
If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), then show that \(A^2 - 5A + 7I = O\). Using this, obtain \(A^{-1}\).
If \( y = \sin^{-1} x \), then prove that \( (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 0 \).
If \(A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}\), then find \(A^{-1}\).
Solve: \( (1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0 \).
Prove that \(\int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx = 2\).
At \( t = 2 \), the slope of the vector function \( \vec{f}(t) = 2\hat{i} + 3\hat{j} + 5t^2\hat{k} \) is
If the relation \( R \) is given by \[ R = \{(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)\}, \] then find \( R^{-1} \circ R^{-1} \).
If \(A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1/4 \\ 0 & 0 \\ 1/2 & 1/8 \end{bmatrix}\), then prove that \(|C| = 1\), where \(C = (A')' B\).
If \( A \) and \( B \) are two matrices of order \( n \) which are invertible, then prove that \( (AB)^{-1} = B^{-1}A^{-1} \).
If function \( f \) is defined as \[ f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \] then prove that \( f \) is continuous.
If \(y = \cos^{-1} x\), then show that \((1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = 0\).
For matrix \( A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix} \) show that \( A^3 - 6A^2 + 5A + 11I = 0 \) and with the help of this find \( A^{-1} \).
If the matrices \(A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & 1/4 \\ 0 & 0 \\ 1/2 & 1/8 \end{bmatrix}\), then prove that \((A')' B = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\).
Find the value of \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \).
The vector function is given by \(\vec{f}(t) = t\hat{i} + t^2\hat{j} + 5\hat{k}\), then at point \(t = 1\) the slope is
If \(\int x \log x \, dx = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c\), then \( f(x) \) is