Question

A speaks truth in 60% of the cases, while B in 90% of cases. In what percent of cases are they likely to contradict each other in stating the same fact ? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A ?

Solve the following differential equation : ${\frac{d y}{d x}}\ +\,\sec x{y}=\tan x,\left(0\leq x<{\frac{\pi}{2}}\right)\!.$

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 a vector of magnitude 5 units and parallel to the resultant of $\stackrel{\rightarrow}{a}\$ = $2\stackrel{\wedge}{i}+3\stackrel{\wedge}{j} -\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$.

If f(x) =$\frac{4x+3}{6x-4}\,,\;\;x\;\;\neq\;\;\frac{2}{3}\,,$ then Show that fof(x) = x for all $x\neq{\frac{2}{3}}\,.$ What is the inverse of f?

Find the equations of the tangent and normal to the curve ${\frac{x^{2}}{a^{2}}}-{\frac{y^{2}}{b^{2}}}=1$ at the point $({\sqrt{2}}a,b)$ .

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