Question

Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive.

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.

The equation of tangent at (2, 3) on the curve y² = ax³ + b is y = 4x – 5. Find the values of a and b.

The amount of pollution content added in air in a city due to x-diesel vehicles is given by p(x) = 0·005x³ + 0·02x² + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added.

Consider f :$R_{+}\to[4,\infty)$ given by f(x) = x² + 4 Show that f is invertible with the inverse $f^{-1}$ of f given by $f^{-1}(y)$ = $\sqrt{y-4}$ , where $\textstyle R_{+}$ is the set of all nonnegative real numbers.

Solve the differential equation: $x\,{\frac{d y}{d x}}\ +y=x\cos x+\sin x,$ given $y\left({\frac{\pi}{2}}\right)=1.$

If A = $\left(\begin{array}{l l l }1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1\end{array}\right)$, find $\left(A^{\prime}\right)^{-1}.$