Question

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

The total cost associated with provision of free mid-day meals to x students of a school in primary classes is given by C(x) = 0·005x³ – 0·02x² + 30x + 50. If the marginal cost is given by rate of change ${\frac{d c}{d x}}$ of total cost, write the marginal cost of food for 300 students.

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

Find the particular solution of the following differential equation : ${\frac{d y}{d x}}-y=\cos x\ {\mathrm{~for~}}x=0,y=1.$

Solve the differential equation : $3e^{x}\tan y\,d x\ +(2-e^{x})\,\mathrm{sec}^{2}\,y\,d y=0,$ given that when x = 0, y = ${\frac{\pi}{4}}.$ 

Show that the relation R on the set Z of all integers defined by (x, y) ∈ $R\Leftrightarrow(x-y)$ is divisible by 3 is an equivalence relation.

Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by $f(x)\,=\,\left({\frac{x-1}{x-2}}\right)\!,$ show that f is one-one and onto. Hence find $f^{-1}.$