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

197.

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)

$\mathrm{If} {\left(\mathrm{a} + \mathrm{bx}\right)}^{ - 3} = \frac{1}{27} + \frac{1}{3}\mathrm{x} + ... , \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} \left(\mathrm{a}, \mathrm{b}\right) = ?$

• (3, - 27)

• $\left(1, \frac{1}{3}\right)$

• (3, 9)

• (3, - 9)

The value of the sum 1 · 2 . 3 + 2 . 3 . 4 + 3 . 4 . 5 +  ... upto n terms is equal to

• $\frac{1}{6}{\mathrm{n}}^2\left(2{\mathrm{n}}^2 + 1\right)$

• $\frac{1}{6}\left({\mathrm{n}}^2 - 1\right)\left(2\mathrm{n} - 1\right)\left(2\mathrm{n} + 3\right)$

• $\frac{1}{8}\left({\mathrm{n}}^2 + 1\right)\left({\mathrm{n}}^2 +\hspace{0.17em}5\right)$

• $\frac{1}{4}\mathrm{n}\left(\mathrm{n} +\hspace{0.17em}1\right)\left(\mathrm{n} + 2\right)\left(\mathrm{n} + 3\right)$