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

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

• $\frac{1}{\sqrt{50}}$

• $\frac{1}{7}$

• $\frac{1}{5}$

• $\frac{1}{\sqrt{10}}$

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

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

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

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

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

If the roots of x³ - kx² + 14x - 8 = 0 are in geometric progression, then k is equal to

• - 3

• 7

• 4

• 0

187.

The number of four-digit numbers formed by using the digits 0, 2, 4, 5 and which are not divisible by 5, is

• 10

• 8

• 6

• 4

$$\begin{gathered} \mathrm{If} \mathrm{x} = \frac{1}{5} + \frac{1 . 3}{5 . 10} + \frac{1 . 3 . 5}{5 . 10 . 5} +\hspace{0.17em}... \infty , \\ 3{\mathrm{x}}^2 + 6\mathrm{x} = ? \end{gathered}$$

• 1

• 2

• 3

• 4

$\mathrm{In} ∆\mathrm{ABC}, \left(\mathrm{a} + \mathrm{b} + \mathrm{c}\right)\left(\mathrm{b} + \mathrm{c} - \mathrm{a}\right) = \mathrm{\lambda bc}, \mathrm{then}$

• $\mathrm{\lambda } < - 6$

• $\mathrm{\lambda } > 6$

• $0 < \mathrm{\lambda } < 4$

• $\mathrm{\lambda } > 4$

The greatest positive integer which divides (n + 16)(n + 17)(n + 18)(n +19), for all positive integers n, is

• 6

• 24

• 28

• 20