Question

Write the value of $\begin{vmatrix}2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86\end{vmatrix}$

If for any 2 × 2 square matrix A, A(adj A) = $\left(\begin{array}{l l }8 & 0 \\ 0 & 8\end{array}\right)$ , then write the value of |A|.

Write the projection of the vector $\stackrel{\rightarrow}{a}\$ = $2\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ on the vector $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}+2\stackrel{\wedge}{j} +2\stackrel{\wedge}{k}$.

Find the solution of the differential equation ${\frac{d y}{d x}}=x^{3}e^{-2y}$.

Find a unit vector parallel to the sum of vectors $\stackrel{\wedge}{i}+\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ and $2\stackrel{\wedge}{i}-3\stackrel{\wedge}{j} +5\stackrel{\wedge}{k}$

If the following function f(x) is continuous at x = 0, then write the value of k.

Find : $\textstyle\int\!{\frac{1}{x(1+\log x)}}\,d x$