If A = $\left[ \begin{array}{cc}\mathrm{cos\alpha }& - \mathrm{sin\alpha }\\ \mathrm{sin\alpha }& \mathrm{cos\alpha }\end{array} \right]$, then for what value of α is A an identity matrix?

What is the principal value of ${\cos }^{-1} \left(-\frac{\sqrt{3}}{2}\right)$?

What is the range of the function f(x) = $\frac{\left| \mathrm{x} - 1 \right|}{\left( \mathrm{x} - 1 \right)}$?

Find the minor of the element of second row and third column ( a₂₃ ) in the following determinant:

$\left| \begin{array}{ccc} 2 & -3 & 5\\ 6& 0& 4\\ 1& 5& -7 \end{array}\right|$

If $\left[ \begin{array}{cc}1 & 2\\ 3 & 4\end{array} \right] \left[ \begin{array}{cc}3 & 1\\ 2 & 5\end{array} \right] = \left[ \begin{array}{cc}7& 11\\ \mathrm{k}& 23\end{array} \right]$, then write the value of k.

Evaluate: $\int \frac{\log \mathrm{x}}{\mathrm{x}} \mathrm{dx}$

What is the cosine of the angle which the vector $\sqrt{2} \hat{\mathrm{i}} + \hat{\mathrm{j}} + \mathrm{k}$  makes with y-axis?

8.

Write a vector of magnitude 15 units in the direction of vector  $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\mathrm{k}$

Write the vector equation of the following line:

$\frac{\mathrm{x} - 5}{3} = \frac{\mathrm{y} + 4}{7} = \frac{6 - \mathrm{z}}{2}$

What is the degree of the following differential equation?

5x $5\mathrm{x} {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 - \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 6\mathrm{y} = \log \mathrm{x}$