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

• (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

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

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

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

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

If a, b, c are distinct and the roots of (b - c)x² + (c - a)x + (a - b) = 0 are equal, then a, b and c are in

• anthmet1c progression

• geometnc progression

• harmonic progression

• arithmetico-geometnc progression

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

• - 3

• 7

• 4

• 0

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

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

• $\mathrm{\lambda } < - 6$

• $\mathrm{\lambda } > 6$

• $0 < \mathrm{\lambda } < 4$

• $\mathrm{\lambda } > 4$

348.

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

• 6

• 24

• 28

• 20

If x_1, x_2, x₃ as well as y_1, y_2, y₃ are in geometric progression with the same common ratio,then the points, $\left({\mathrm{x}}_1, {\mathrm{y}}_1\right), \left({\mathrm{x}}_2, {\mathrm{y}}_2\right), \left({\mathrm{x}}_3, {\mathrm{y}}_3\right)$ are

• vertices of an equilateral triangle

• vertices of a right angled triangle

• vertices of a right angled isosceles triangle

• collinear

$\mathrm{If} {12}^{4 + 2{\mathrm{x}}^2} = {\left(24\sqrt{3}\right)}^{3{\mathrm{x}}^2 - 2}, \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\pm \sqrt{\frac{13}{12}}$

• $\pm \sqrt{\frac{14}{5}}$

• $\pm \sqrt{\frac{12}{13}}$

• $\pm \sqrt{\frac{5}{14}}$