Question

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

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.

Using properties of determinants, prove that : $\begin{vmatrix}x²+1 & xy & xz \\ xy & y²+1 & yz \\ xz & yz & z²+1\end{vmatrix}$

Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1, when x =0.

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 the particular solution of the following differential equation given that at x = 2, y = 1 $x\,{\frac{d y}{d x}}\,+\,2y=x^{2},\,(x\,\neq\,0).$

Find the equations of tangents to the curve y = (x²– 1) (x – 2) at the points, where the curve cuts the X-axis.