21.

The angle between the curves y = sin(x) and y = cos(x) is

• ${\tan }^{-1}\left(3\sqrt{3}\right)$

• ${\tan }^{-1}\left(5\sqrt{2}\right)$

If f : $\mathrm{R} \to \mathrm{R} \mathrm{and} \mathrm{g} : \mathrm{R} \to \mathrm{R}$ are defined by f(x) = 2x + 3 and g(x) = x² + 7, then the values of x such that g(x) = x² + 7, then values ofx such that g(f(x)) = 8 are

• 1, 2

• - 1, 2

• - 1, - 2

• 1, - 2

Suppose f : [- 2, 2] $\to$ R is defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{- 1, \mathrm{for} - 2 \le \mathrm{x} \le 0 \\ \mathrm{x} - 1, \mathrm{for} 0 \le \mathrm{x} \le 2 \\ \left\{\mathrm{x} \in \left[- 2, 2\right] : \mathrm{x} \le 0\right\} \mathrm{and} \mathrm{f}\left(\left|\mathrm{x}\right|\right) = \mathrm{x}\right\} \end{gathered}$$

is equal to

• {- 1}

• 0

• $\left\{- \frac{1}{2}\right\}$

• $\mathrm{\varphi }$

If f : $\mathrm{R} \to \mathrm{R} \mathrm{and} \mathrm{g} :\hspace{0.17em}\mathrm{R} \to \mathrm{R}$ are given by f(x) = $\left|\mathrm{x}\right|$ and g(x) = [x] for each x $\in$ R, then $\left\{\mathrm{x} \in \mathrm{R} : \mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right) \le \mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)\right\}$ is equal to

• $\mathrm{Z} \cup \left(- \infty , 0\right)$

• $\left(- \infty , 0\right)$

• Z

• R

If t_n = $\frac{1}{4}\left(\mathrm{n} + 2\right)\left(\mathrm{n} + 3\right)$ for n = 1, 2, 3 ..., then $\frac{1}{{\mathrm{t}}_1} + \frac{1}{{\mathrm{t}}_2} + ... + \frac{1}{{\mathrm{t}}_{2003}}$ is equal to

• $\frac{4006}{3006}$

• $\frac{4003}{3007}$

• $\frac{4006}{3008}$

• $\frac{4006}{3009}$

If f(x) = $\frac{1}{\sqrt{\mathrm{x} + 2\sqrt{2\mathrm{x} - 4}}} + \frac{1}{\sqrt{\mathrm{x} - 2\sqrt{2\mathrm{x} - 4}}}$ for x > 2, then f(11) is equal to

• $\frac{7}{6}$

• $\frac{5}{6}$

• $\frac{6}{7}$

• $\frac{5}{7}$

If ${\mathrm{e}}^{\mathrm{f}\left(\mathrm{x}\right)} = \frac{10 + \mathrm{x}}{10 - \mathrm{x}}, \mathrm{x} \in \left(- 10, 10\right)$ and f(x) = $\mathrm{kf}\left(\frac{200\mathrm{x}}{100 + {\mathrm{x}}^2}\right)$, then k is equal to

• 0.5

• 0.6

• 0.7

• 0.8

Let l₁ and l₂ be two lines intersecting at P. If A₁, B₁, C₁ are points on l₁, and A₂, B₂, C₂, D₂, E₂ are points on l₂ and if none of these coincides with P, then the number of triangles formed by these eight points, is :

• 56

• 55

• 46

• 45

Consider the fourteen lines in the plane given by y = x + r, y = - x + r,where r $\in$ {0, 1, 2, 3, 4, 5, 6}. The number of squares formed by these lines, whose sides are of length $\sqrt{2}$, is

• 9

• 16

• 25

• 36

If the coefficient of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)⁴³ are equal, then r is equal to :

• 12

• 14

• 16

• 18