egin{align} intleft(1-x ight).sqrt | Class 12 Mathematics Chapter Integrals, Integrals NCERT Solutions

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f : R → R defined by f(x) = 3 – 4x

(ii) f : R → R defined by f(x) = 1 + x² 

 Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

 Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

 Find gof and fog, if

(i) f(x) = | x | and g(x) = | 5x – 2 |
(ii) f(x) = 8x3 and g(x) = x^1/3 .

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Determine order and degree(if defined) of differential equation \begin{align}\frac{d^2y}{dx^2}=\cos3x  + sin3x\end{align}

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?