A circular sector of perimeter 60 m with maximum area is to be constructed. The radius of the circular arc in metre must be :

• 20

• 5

• 15

• 10

The tangent and the normal drawn to the curve y = x² - x + 4at P(1, 4) cut the x-axis at A and B respectively. If the length of the subtangent drawn to the curve at P is equal to the length of the subnormal, then the area of the triangle PAB in sq unit is :

• 4

• 32

• 8

• 16

13.

Which one of the following is not true always?

• If f(x) is not continuous at x = a, then it is not differentiable at x = a

• If f(x) is continuous at x = a, then it is differentiable at x = a

• If f(x) and g(x) are differentiable at x = a, then f(x) + g(x) is also differentiable at x = a

• If a function f(x) is continuous at x = a, then $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ f(x) exists

If y = $1 + \frac{1}{\mathrm{x}} + \frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{x}}^3} + ... \infty \mathrm{with} \left|\mathrm{x}\right| > 1, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ :

• $\frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}$

• x²y²

• $\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}$

• $\frac{- {\mathrm{y}}^2}{{\mathrm{x}}^2}$

If f(x) and g(x) are two functions with g(x) = $\mathrm{x} - \frac{1}{\mathrm{x}}$and fog(x) = ${\mathrm{x}}^3 - \frac{1}{{\mathrm{x}}^3}$, then f' (x) is :

• 3x² + 3

• ${\mathrm{x}}^2 - \frac{1}{{\mathrm{x}}^2}$

• $1 +\hspace{0.17em}\frac{1}{{\mathrm{x}}^2}$

• $3{\mathrm{x}}^2 + \frac{3}{{\mathrm{x}}^4}$

If sin(x + y) + cos(x + y) = log(x + y), then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is :

• $\frac{- \mathrm{y}}{\mathrm{x}}$

• 0

• - 1

• 1

If f(x) is a function such that f''(x) + f(x) = 0 and g(x) = [f(x)]² + [f'(x)]² and (3) = 3 then g(8) is equal to :

• 5

• 0

• 3

• 8

If$\overrightarrow{\mathrm{p}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}, \overrightarrow{\mathrm{q}} = 4\hat{\mathrm{k}} - \hat{\mathrm{j}} \mathrm{and} \overrightarrow{\mathrm{r}} = \hat{\mathrm{i}} + \hat{\mathrm{k}}$ then the unit vector in the direction of $3\overrightarrow{\mathrm{p}} + \overrightarrow{\mathrm{q}} - 2\overrightarrow{\mathrm{r}}$ is :

• $\frac{1}{3}\left(\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(\hat{\mathrm{i}} -\hspace{0.17em}2\hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(\hat{\mathrm{i}} -\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$

• $\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are the two vectors such that $\left|\overrightarrow{\mathrm{a}}\right| = 3\sqrt{3}, \left|\overrightarrow{\mathrm{b}}\right| = 4$ and $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| = \sqrt{7}$, then the angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is :

• $120°$

• $60°$

• $30°$

• $150°$

If $\overrightarrow{\mathrm{a}}$ is vector perpendicular to both $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$, then :

• $\overrightarrow{\mathrm{a}} + \left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) = \overrightarrow{0}$