Question

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.

If y(x) is a solution of the differential equation $\left(\frac{\dot{2}+\sin x}{1+y}\right)\frac{d y}{d x}\;=-\cos x\;\mathrm{and}\;y\;(0)=1,$ then find the value of $y\left({\frac{\pi}{2}}\right).$

Solve the differential equation ${\frac{d y}{d x}}\ +\ y\ \cot\ x$= 2 cos x, given that y = 0 when $x={\frac{\pi}{3}}\,.$

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 $({\sqrt{2}}a,b)$ .

The total expenditure (in ₹) required for providing the cheap edition of a book for poor and deserving students is given by R(x) = 3x² + 36x, where x is the number of set of books. If the marginal expenditure is defined as ${\frac{d R}{d x}},$ write the marginal expenditure required for 1200 such sets.

Using properties of determinants, prove that $\begin{vmatrix}x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x\end{vmatrix}$ = x³

Find the point on the curve y = x³ – 11x + 5 at which the equation of tangent is y = x – 11.