Question

Evaluate : $\large\textstyle\int{\frac{3x+1}{(x^{3}-x^{2}-x+1)}}d x\,.$

Solve the following differential equation : $(x^{2}-1){\frac{d y}{d x}}~+2x y={\frac{2}{x^{2}-1}},~~x\neq{\bf1}$

P speaks truth in 70% of the cases and Q in 80% of the cases. In what percent of cases are they likely to agree in stating the same fact ?

Prove the following : $\begin{vmatrix}a² & bc & ac+c² \\ a²+ab & b² & ac \\ ab & b²+bc & c²\end{vmatrix}$ = $4a^{2}b^{2}c^{2}.$

Find : $\large\textstyle\int$$\frac{x^{4}+1}{x\left(x^{2}+1\right)^{2}}d x$.

Find the equation of tangent to the curve y = cos (x + y), -2π ≤ x ≤ 0, that is parallel to the line x + 2y = 0.

Find the position vector of a point R, which divides the line joining two points P and Q whose position vectors are $2\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}-3\stackrel{\rightarrow}{b}$ respectively, externally in the ratio 1 : 2 Also, show that P is the mid-point, of line segment RQ.