Question

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.

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x² – 3xy²) dx = (y³ – 3x²y) dy, where C is a parameter.

Find the particular solution of the differential equation $(1+e^{2x})d y+(1+y^{2})e^{x}d x=0,$ given that y = 1, when x = 0.

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 ?

To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap-books and pastel-sheets made by them using recycled paper at the rate of ₹ 20, ₹ 15 and ₹ 10 per unit respectively. School A sold 25 paper bags, 10 scrap-books and 30 pastel-sheets. School B sold 20 paper-bags, 15 scrap-books and 30 pastel-sheets. While school C sold 25 paper-bags, 18 scrap-books and 35 pastel-sheets. Using matrices, find the total amount raised by each school.

Find the particular solution of differential equation : ${\frac{d y}{d x}}=-{\frac{x+y\cos x}{1+\sin x}}$ given that y = 1 when x = 0.

Find : $\large\textstyle\int$$\large\frac{x^{2}\,d x}{x^{4}+x^{2}-12}\;.$