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.

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?

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Let f, g and h be functions from R to R. Show that

(f + g)oh = foh + goh

(f . g)oh = (foh) . (goh) 

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north-west (iii) 40°

(iv) 40 watt (v) 10^–19 coulomb (vi) 20 m/s²

Answer the following as true or false.

 \begin{align}(i) \overrightarrow{a}\;  and\; \overrightarrow{-a}\; are\; collinear.\end{align}

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.