Question

Find the equation of tangents to the curve y = x³ + 2x – 4 which are perpendicular to the line x + 14y – 3 = 0.

Solve the differential equation : $(1+x^{2}){\frac{d y}{d x}}\ +y=e^{\tan^{-1}x}.$

Let f : N → N be defined as $f(n)={\left\{\begin{array}{l l}{\displaystyle{\frac{n+1}{2}},{\mathrm{when~}}n{\mathrm{~is~odd}}}\\ {\displaystyle{\frac{n}{2}},{\mathrm{when~}}n{\mathrm{iseven}}}\end{array}\right.}$ for all n ∈ N. State whether the function f is bijective. Justify your answer.

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

Find the particular solution of the following differential equation. $\cos y\,d x+(1+2e^{-x})\sin y\,d y=0;y(0)={\frac{\pi}{4}}$

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways — telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as

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 ?