1.

One possible condition for the three points (a, b), (b, a) and (a², - b²) to be collinear is

• a - b = 2

• a + b = 2

• a = 1 + b

• a = 1 - b

If the mth term and the nth term of an AP are respectively $\frac{1}{\mathrm{n}}$ and $\frac{1}{\mathrm{m}}$, then the mnth term of the AP is

• $\frac{1}{\mathrm{mn}}$

• $\frac{\mathrm{m}}{\mathrm{n}}$

• 1

• $\frac{{\displaystyle \mathrm{n}}}{{\displaystyle \mathrm{m}}}$

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^9\left(\mathrm{x}\right){\cos }^5\left(\mathrm{x}\right)d\mathrm{x}$ equals

• $\frac{1}{20}$

• 20

• 0

• $\frac{1}{330}$

The function f(x) = $\log \left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$ satisfies the equation

• f(x + 2) - 2f(x + 1) + f(x) = 0

• f(x) + f(x + 1) = f{x(x + 1)}

• f(x) + f(y) = $\mathrm{f}\left(\frac{\mathrm{x} + \mathrm{y}}{1 + \mathrm{xy}}\right)$

• f(x + y) = f(x)f(y)

The value of (1 - w + w²)⁵ + (1 + w - w²)⁵, where w and w² are the complex cube roots of unity, is

• 0

• 32w

• - 32

• 32

The value of $\underset{\mathrm{n}\to \infty }{\lim }\left(\frac{1}{\mathrm{n} + 1} + \frac{1}{\mathrm{n} + 2} + ... + \frac{1}{6\mathrm{n}}\right)$ is

• $\log \left(2\right)$

• $\log \left(6\right)$

• 1

• $\log \left(3\right)$

$\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{{\mathrm{a}}^{\cot \left(\mathrm{x}\right)} - {\mathrm{a}}^{\cos \left(\mathrm{x}\right)}}{\cot \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}$

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{\pi }}{2}\right)$

• ${\log }_{\mathrm{e}}\left(2\right)$

• ${\log }_{\mathrm{e}}\left(\mathrm{a}\right)$

• a

The equation of the circle which passes through the points of intersection of the circles x² + y² - 6x = 0 and x² + y² - 6y = 0 and has its centre at $\left(\frac{3}{2}, \frac{{\displaystyle 3}}{{\displaystyle 2}}\right)$, is

• x² + y² + 3x + 3y + 9 = 0

• x² + y² + 3x + 3y = 0

• x² + y² - 3x - 3y = 0

• x² + y² - 3x - 3y + 9 = 0

If 2y = x and 3y + 4x = 0 are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is

• $\sqrt{\frac{2}{3}}$

• $\sqrt{\frac{2}{5}}$

• $\sqrt{\frac{1}{3}}$

• $\sqrt{\frac{1}{2}}$

If t is a parameter, then x = $\mathrm{a}\left(\mathrm{t} + \frac{1}{\mathrm{t}}\right)$, y = $\mathrm{b}\left(\mathrm{t} - \frac{1}{\mathrm{t}}\right)$ represents

• an ellipse

• a circle

• a pair of straight lines

• a hyperbola