If \( 0 \leq x \leq \frac{\pi}{2} \), then \[ \lim\limits_{x \to a} \frac{2\cos x - 1}{2\cos x - 1} \] Options:
If the function
\[ f(x) = \begin{cases} \frac{(e^x - 1) \sin kx}{4 \tan x}, & x \neq 0 \\ P, & x = 0 \end{cases} \]
is differentiable at \( x = 0 \), then:
When \( |x| < 2 \), the coefficient of \( x^2 \) in the power series expansion of
\[ \frac{x}{(x-2)(x-3)} \]
is:
The mean deviation about the mean for the following data is:
{If \(f(x)\) is a quadratic function such that \(f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{1-x}\right)\), then \(\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} =\)}
A, B, C, D are square matrices such that A + B is symmetric, A - B is skew-symmetric, and D is the transpose of C.
If
\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & 3 & -2 \\ 3 & -4 & 5 \end{bmatrix} \]
and
\[ C = \begin{bmatrix} 0 & 1 & -2 \\ 2 & -1 & 0 \\ 0 & 2 & 1 \end{bmatrix} \]
then the matrix \( B + D \) is:
The number of ways of arranging all the letters of the word "COMBINATIONS" around a circle so that no two vowels come together is
If \( A = \begin{pmatrix} x & y & y \\ y & x & y \\ y & y & x \end{pmatrix} \) and \( 5A^{-1} = \begin{pmatrix} -3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3 \end{pmatrix} \), then \( A^2 - 4A \) is:
A rectangle is formed by the lines \[ x = 4, \quad x = -2, \quad y = 5, \quad y = -2 \] and a circle is drawn through the vertices of this rectangle. The pole of the line \[ y + 2 = 0 \] with respect to this circle is:
If \( a \neq b \neq c \), then
then what is \( 11(a + b + c) \)?
