List of top Questions asked in UP Board XII

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

Suppose that \( A = \{2, 3, 4, 5\} \) and a relation \( R \) on \( A \) is defined by \( R = \{(a, b) : a, b \in A, a - b = 12\} \). Then the set \( R \) is

Prove that \(\int_{0}^{\pi/4} \log(1 + \tan x) \, dx = \frac{\pi}{8} \log 2\).

If \( F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \), prove that \(F(x)F(y) = F(x+y)\).

Prove that \(\cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right) = \frac{x}{2}\), \(x \in (0, \pi/4)\).

Find the shortest distance between the lines whose vector equations are:
\(\vec{r} = (1-t)\hat{i} + (t-2)\hat{j} + (3-2t)\hat{k}\) and \(\vec{r} = (s+1)\hat{i} + (2s-1)\hat{j} - (2s+1)\hat{k}\).

If \(A = \begin{bmatrix} 0 & -\tan(\alpha/2) \\ \tan(\alpha/2) & 0 \end{bmatrix}\) and \(I\) is the identity matrix of order 2, prove that \(I+A = (I-A)\begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}\).

The relation \( R \), defined by \( R = \{ (T_1, T_2) : T_1 \text{ is similar to } T_2 \} \), in the set \( A \) of all triangles, is

If \( 2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} \) and \( Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \), then \( X \) will be

Show that the given function \( f, f(x) = x^3 - 3x^2 + 4x, x \in \mathbb{R} \) is an increasing function in \( \mathbb{R} \).

Find the value of the determinant \( \begin{vmatrix} x+y+2z & x & y \\ z & y+z+2x & y \\ z & x & z+x+2y \end{vmatrix} \).

Evaluate: \( \int \frac{dx}{(x+1)(x+2)} \).

If \(A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}\), find \((A+2B)'\).

If \(P(B) = \frac{9}{13}\) and \(P(A \cap B) = \frac{4}{13}\), find \(P(A|B)\).

If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \), verify that \( A'A=I \).

Find the value of the determinant \( \begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{vmatrix} \).

Prove that \(\tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x\), where \(-\frac{1}{\sqrt{2}} \leq x \leq 1\).

Test whether the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \[ f(x) = \begin{cases} x+5, & \text{if } x \le 1 \\ x-5, & \text{if } x > 1 \end{cases} \] is continuous at \(x=1\).