For the two circles x² + y² = 16 and x² + y² - 2y = 0 there is/are

• one pair of common tangents

• only one common tangent

• three common tangents

• no common tangent

If C is a point on the line segment joining A(- 3, 4) and B(2, 1) such that AC = 2BC, then the coordinate of C is

• $\left(\frac{1}{3}, 2\right)$

• $\left(2, \frac{1}{3}\right)$

• (2, 7)

• (7, 2)

13.

If a, b, c are real, then both the roots of the equation (x - b)(x - c) + (x - c)(x - a) + (x - a)(x - b) = 0

• positive

• negative

• real

• imaginary

The point (- 4, 5) is the vertex of a square and one of its diagonals is 7x - y + 8 = 0. The equation of the other diagonal is

• 7x - y + 23 = 0

• 7y + x = 30

• 7y + x = 31

• x - 7y = 30

The domain of definition of the function

$\mathrm{f}\left(\mathrm{x}\right) = \sqrt{1 + {\log }_{\mathrm{e}}\left(1 - \mathrm{x}\right)}$

• $- \infty < \mathrm{x} \le 0$

• $- \infty < \mathrm{x} \le \frac{\mathrm{e} - 1}{\mathrm{e}}$

• $- \infty < \mathrm{x} \le 1$

• $\mathrm{x} \ge 1 - \mathrm{e}$

For what value of m, $\frac{{\mathrm{a}}^{\mathrm{m} + 1} + {\mathrm{b}}^{\mathrm{m} + 1}}{{\mathrm{a}}^{\mathrm{m}} + {\mathrm{b}}^{\mathrm{m}}}$  is the arithmetic mean of 'a' and 'b' ?

• 1

• 0

• 2

• None of these

The value of the limit $\underset{\mathrm{x}\to 1}{\lim }\frac{\sin \left({\mathrm{e}}^{\mathrm{x} - 1} - 1\right)}{\log \left(\mathrm{x}\right)}$

• 0

• e

• $\frac{1}{\mathrm{e}}$

• 1

Let f(x) = $\frac{\sqrt{\mathrm{x} + 3}}{\mathrm{x} + 1}$, then the value of $\underset{\mathrm{x}\to - 3 - 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ is

• 0

• does not exist

• $\frac{1}{2}$

• $- \frac{1}{2}$

$\tan \left[\frac{\mathrm{\pi }}{4} + \frac{1}{2}{\cos }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right] + \tan \left[\frac{\mathrm{\pi }}{4} - \frac{1}{2}{\cos }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right]$ is equal to

• $\frac{2\mathrm{a}}{\mathrm{b}}$

• $\frac{2\mathrm{b}}{\mathrm{a}}$

• $\frac{\mathrm{a}}{\mathrm{b}}$

• $\frac{\mathrm{b}}{\mathrm{a}}$

If i = $\sqrt{- 1}$ and n is a positive integer, then iⁿ + i^{n + 1} + i^{n + 3} is equal to

• 1

• i

• iⁿ

• 0