Question

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing, when the foot of the ladder is 4 m away from the wall ?

Find the particular solution of the following differential equation : $(x\ +1){\frac{d y}{d x}}=2e^{-y}-1;$ y = 0 when x = 0.

Evaluate : $\large\textstyle\int$$\large{\frac{x}{(x-1)^{2}(x+2)}}d x\ .$

Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) =${\frac{x-2}{x-3}},~~\forall~x\in~A.$ Show that f is bijective. Also, find (i) x, if $f^{-1}(x)$ = 4 (ii) $f^{-1}(7)$

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

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.

Solve the following differential equation : (1 + x²) dy + 2xy dx = cot x dx, (x ≠ 0)