21.

A point on curve xy² = 1 which is at minimum distance from the origin is

• (1, 1)

• (1/4, 2)

• (2^1/6, 2^{- 1/3})

• (2^{- 1/3}, 2^1/6)

A spherical iron ball ofradius 10 cm, coated with a layer of ice of uniform thickness, melts at a rate of 100 $\mathrm{\pi }$ cm/min. The rate at which the thickness of decreases when the thickness of ice is 5 cm, is

• 1 cm/min

• 2 cm/min

• $\frac{1}{376} \mathrm{cm}/\min$

• 5 cm/min

If ax² + bx + 4 attains its minimum value - 1 at x = 1, then the values of a and bare respectively

• 5, - 10

• 5, - 5

• 5, 5

• 10, - 5

The function f(x) = (9 - x²)² increases in

• $\left(- 3, 0\right) \cup \left(3, \infty \right)$

• $\left(- \infty , - 3\right) \cup \left(3, \infty \right)$

• $\left(- \infty , - 3\right) \cup \left(0, 3\right)$

• (- 3, 3)

Let $$\begin{gathered} \mathrm{g}\left(\mathrm{x}\right) = \left\{2\mathrm{e}, \mathrm{if} \mathrm{x} \le 1 \\ \log \left(\mathrm{x} - 1\right), \mathrm{if} \mathrm{x} > 1\right\} \end{gathered}$$. The equation  of the normal to y = g(x) at the point (3, log(2)), is

• y - 2x = 6 + log(2)

• y + 2x = 6 + log(2)

• y + 2x = 6 - log(2)

• y + 2x = - 6 + log(2)

A flagpole stands on a building of height 450 ft and an observer on a level ground is 300 ft from the base of the building. The angle of elevation of the bottom of the flagpole is 30° and the height of the flagpole is SO ft. If 8 is the angle of elevation of the top of the flagpole, then tan$\left(\mathrm{\theta }\right)$ is equal to

• $\frac{4}{3\sqrt{3}}$

• $\frac{\sqrt{3}}{2}$

• $\frac{9}{2}$

• $\frac{\sqrt{3}}{5}$

If A (0, 0), B (12, 0), C (12, 2), D (6, 7) and E (0, 5) are the vertices of the pentagon ABCDE, then its area in square units, is

• 58

• 60

• 61

• 63

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are position vectors of the vertices of the triangle ABC , then $\frac{\left|\left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}}\right) \times \left(\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{a}}\right)\right|}{\left(\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}\right) . \left(\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{a}}\right)}$ is equal to

• cot(A)

• cot(C)

• - tan(C)

• tan(A)

If $\overrightarrow{\mathrm{a}}$ is a vector of magnitude 50, collinear with the vector $\overrightarrow{\mathrm{b}} = 6\hat{\mathrm{i}} - 8\hat{\mathrm{j}} - \frac{15}{2}\hat{\mathrm{k}}$ and makes an acute angle with the positive direction of z-axis, then $\overrightarrow{\mathrm{a}}$ is equal to

• $- 24\hat{\mathrm{i}} + 32\hat{\mathrm{j}} + 30\hat{\mathrm{k}}$

• $24\hat{\mathrm{i}} - 32\hat{\mathrm{j}} - 30\hat{\mathrm{k}}$

• $12\hat{\mathrm{i}} - 16\hat{\mathrm{j}} - 15\hat{\mathrm{k}}$

• $- 12\hat{\mathrm{i}} + 16\hat{\mathrm{j}} - 15\hat{\mathrm{k}}$

If the volume of a parallelopiped with $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$ as coterminus edges is 9 cu units, then the volume of the parallelopiped with $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right), \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) \times \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right), \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right) \times \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$ as coterminus edges is

• 9 cu unit

• 729 cu unit

• 81 cu unit

• 27 cu unit