21.

 $\mathrm{If} \mathrm{z} = \sec \left(\mathrm{y} - \mathrm{ax}\right) + \tan \left(\mathrm{y} + \mathrm{ax}\right), \frac{{\partial }^2\mathrm{z}}{\partial {\mathrm{x}}^2} - {\mathrm{a}}^2\frac{{\partial }^2\mathrm{z}}{\partial {\mathrm{y}}^2} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 0

• - z

• z

• 2x

Let A = $\left\{\mathrm{x} \in \mathrm{R}, \mathrm{x} \ne 0, - 4 \le \mathrm{x} \le 4\right\}$ and $\mathrm{f} : \mathrm{A} \to \mathrm{R}$ defined by f(x) = $\frac{\left|\mathrm{x}\right|}{\mathrm{x}}$ for x $\in$ A. Then, the range of f is

• {1, - 1}

• $\left\{\mathrm{x} : 0 \le \mathrm{x} \le 1\right\}$

• (1)

• $\left\{\mathrm{x} : - 4\le \mathrm{x} \le 0\right\}$

$\mathrm{f}\left(\mathrm{x}\right) = \frac{{\cos }^2\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right)}{{\sin }^2\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)}, \mathrm{for} \mathrm{x} \in \mathrm{R}$, then f(2002) is equal to

• 1

• 2

• 3

• 4

The function f : R $\to$ R is defined by f(x) = ${\cos }^2\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right) \mathrm{for} \mathrm{x} \in \mathrm{R}$, then f(R) is equal to

• $(\frac{3}{4}, 1]$

• $[\frac{3}{4}, 1)$

• $\left[\frac{3}{4}, 1\right]$

• $\left(\frac{3}{4}, 1\right)$

If $\log \left(2\right) = \mathrm{a}, \log \left(3\right) = \mathrm{b}, \log \left(7\right) = \mathrm{c}$ and 6^x = 7^{x + 4} then x is equal to

• $\frac{4\mathrm{b}}{\mathrm{c} + \mathrm{a} - \mathrm{b}}$

• $\frac{4\mathrm{c}}{\mathrm{a} + \mathrm{b} - \mathrm{c}}$

• $\frac{4\mathrm{c}}{\mathrm{c} - \mathrm{a} - \mathrm{b}}$

• $\frac{4\mathrm{a}}{\mathrm{a} + \mathrm{b} - \mathrm{c}}$

In the sequence, (1, 2, 3), (4, 5, 6), (7, 8, 9, 10)... of sets, the sum of elements in the 50th set is

• 62525

• 65225

• 56255

• 55765

The least value of the natural number n satisfying C(n, 5) + C(n, 6) > C(n + 1, 5)

• 10

• 11

• 12

• 13

The number of ways that 8 beads of different colours be strung as a necklace is

• 2520

• 2880

• 4320

• 5040

The number of 5-digited number which are not divisible by 5 and which consists of different odd digits is

• 24

• 32

• 96

• 120

The sum of the coefficients in the expansion of (1 + x + x²)ⁿ is

• 2

• 2ⁿ

• 3ⁿ

• 4