If a, b and c form a geometric progression with common ratio r, then the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve x + 2y² = 0 is

• $- \frac{{\mathrm{r}}^2}{2}$

• $- \frac{\mathrm{r}}{2}$

• $\frac{\mathrm{r}}{2}$

• r

The point (3, 2) undergoes the following three transformations in the order given

(i) Reflection about the line y = x.

(ii) Translation by the distance 1 unit in the positive direction of x-axis.

(iii) Rotation by an angle $\frac{\mathrm{\pi }}{4}$ about the origin in  anti-clockwise direction.

Then, the final position of the point is

• $\left(- \sqrt{18}, \sqrt{18}\right)$

• (- 2, 3)

• $\left(0, \sqrt{18}\right)$

• (0, 3)

If X is a poisson variate such that a = P(X = 1) = P(X = 2), then P(X = 4) is equal to

• $2\mathrm{\alpha }$

• $\frac{\mathrm{\alpha }}{3}$

• ${\mathrm{\alpha e}}^{ - 2}$

• ${\mathrm{\alpha e}}^2$

If pth, qth, rth terms of a geometric progression are the positive numbers a, b and c respectively,then the angle· between the vectors (log(a))²i + (log(b))²j + (log(c))²k and (q - r)i + (r - p)j + (p - q)k is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• ${\sin }^{-1}\left(\frac{1}{\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2}}\right)$

• $\frac{\mathrm{\pi }}{4}$

If x is small, so that x² and higher powers can be neglected, then the approximate value for $\frac{{\left(1 - 2\mathrm{x}\right)}^{ - 1}{\left(1 - 3\mathrm{x}\right)}^{ - 2}}{{\left(1 - 4\mathrm{x}\right)}^{ - 3}} \mathrm{is}$

• 1 - 2x

• 1 - 3x

• 1 - 4x

• 1 - 5x

$\mathrm{If} \frac{1}{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1} = \frac{\mathrm{Ax} +\hspace{0.17em}\mathrm{B}}{{\mathrm{x}}^2 + \mathrm{x} +\hspace{0.17em}1 } + \frac{\mathrm{Cx} + \mathrm{D}}{{\mathrm{x}}^2 - \mathrm{x} +\hspace{0.17em}1}, \mathrm{then} \mathrm{C} + \mathrm{D} = ?$

• - 1

• 1

• 2

• 0

If the roots of ${\mathrm{x}}^3 - 42{\mathrm{x}}^2 + 336\mathrm{x} - 512 = 0$, are in increasing geometric progression, then its common ratio is

• 2 : 1

• 3 : 1

• 4 : 1 

• 6 : 1

The equation of the pair of lines passing through the orign whose sum and product of slopes are respectively the arithmetic mean and geometric mean of 4 and 9 is

• 12x² - 13xy + 2y² = 0

• 12x² + 13xy + 2y² = 0

• 12x² - 15xy + 2y² = 0

• 12x² + 15xy - 2y² = 0

The equation ${\mathrm{x}}^2 - 5\mathrm{xy} + {\mathrm{py}}^2 + 3\mathrm{x} - 8\mathrm{y} + 2 = 0$ represents a pair of straight lines. If $\mathrm{\theta }$ is the angle between them, then $\sin \left(\mathrm{\theta }\right)$ is equal to

• $\frac{1}{\sqrt{50}}$

• $\frac{1}{7}$

• $\frac{1}{5}$

• $\frac{1}{\sqrt{10}}$

$\sum _{\mathrm{k} = 1}^{2\mathrm{n} + 1}{\left(- 1\right)}^{\mathrm{k} - 1} . {\mathrm{k}}^2 = ?$

• (n - 1)(2n - 1)

• (n + 1)(2n + 1)

• (n + 1)(2n - 1)

• (n - 1)(2n + 1)