Question

If matrix A = $\left(\begin{array}{l l }3 & -3 \\ -3 & 3\end{array}\right)$ and A² = λA, then write the value of λ.

Write the differential equation obtained by eliminating the arbitrary constant C in the equation representing the family of curves xy = C cos x.

If $\stackrel{\rightarrow}{a}\$ and $\stackrel{\rightarrow}{b}\$ are perpendicular vectors, $|\stackrel{\rightarrow}{a}\$ + $\stackrel{\rightarrow}{b}|\$ = 13 and $|\stackrel{\rightarrow}{a}|\$ =5,find the value of $|\stackrel{\rightarrow}{b}|\$.

If A = $\left(\begin{array}{l l}{{0}}&{{3}}\\ {{2}}&{{-5}}\end{array}\right)$ and $K A\ =\ {\left(\begin{array}{l l}{0}&{4a}\\ {-8}&{5b}\end{array}\right)}$ find the values of k and a.

Evaluate $\textstyle\int\!{\frac{(1+\log x)^{2}}{x}}\,d x\,.$

Write the integrating factor of the following differential equation. $(1+y^{2})+(2x y-\mathrm{cot}\;y){\frac{d y}{d x}}\;=0$

Find the integrating factor of the differential equation