If an infinite GP has the first term x and the sum 5, then which of the following is correct ?

• $\mathrm{x} < - 10$

• $- 10 < \mathrm{x} < 0$

• $0 < \mathrm{x} < 10$

• $\mathrm{x} > 10$

Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to

• 2480

• 2481

• 2482

• 2483

The sum of the series $3 - 1 + \frac{1}{3} - \frac{1}{9} + ... =$

• $\frac{20}{9}$

• $\frac{9}{20}$

• $\frac{9}{4}$

• $\frac{4}{9}$

24.

$$\begin{gathered} \mathrm{If} \mathrm{\mu }, \mathrm{\nu } \mathrm{and} \mathrm{\omega } (\mathrm{all} \mathrm{positive}) \mathrm{are} \mathrm{the} \mathrm{pth}, \mathrm{qth} \mathrm{and} \mathrm{rth} \mathrm{terms} \\ \mathrm{of} \mathrm{a} \mathrm{GP}, \mathrm{then} \mathrm{the} \mathrm{determinant} \mathrm{of} \mathrm{the} \mathrm{matrix} \\ \left(\begin{array}{ccc}\ln \left(\mathrm{\mu }\right)& \mathrm{p}& 1\\ \ln \left(\mathrm{\nu }\right)& \mathrm{q}& 1\\ \ln \left(\mathrm{\omega }\right)& \mathrm{r}& 1\end{array}\right) \mathrm{is} \end{gathered}$$

• 0

• 1

• (p - q)(q - r)(r - p)

• $\ln \left(\mathrm{\mu }\right) \times \ln \left(\mathrm{\nu }\right) \times \ln \left(\mathrm{\omega }\right)$

Let T, be the rth term of an AP for r = 1, 2, 3, ... If for some distinct positive integers m and n we have T_m = 1/n and T_n = 1/m, then what is T_mn equal to ?

• ${\left(\mathrm{mn}\right)}^{ - 1}$

• m ^{- 1} + n ^{- 1}

• 1

• 0

The total number of 5-digit numbers that can be composed of distinct digits from 0 to 9 is

• 45360

• 30240

• 27216

• 15120

$$\begin{gathered} \mathrm{If} \sec (\mathrm{\theta } - \mathrm{\alpha }), \sec \left(\mathrm{\theta }\right) \mathrm{and} \sec (\mathrm{\theta } + \mathrm{\alpha }) \mathrm{are} \mathrm{in} \mathrm{AP}, \mathrm{where} \cos \left(\mathrm{\alpha }\right) \ne 1, \\ \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \sin \left(2\mathrm{\theta }\right) + \cos \mathrm{\alpha } ? \end{gathered}$$

• 0

• 1

•  - 1

• $\frac{1}{2}$

If the second term of a GP is 2 and the sum of its infinite terms is 8, then the GP is

• $8, 2, \frac{1}{2}, \frac{1}{8}, ...$

• $10, 2, \frac{2}{5}, \frac{2}{25}, ...$

• $4, 2, \frac{1}{2}, \frac{1}{2^2}, ...$

• $6, 3, \frac{3}{2}, \frac{3}{4}, ...$

If a, b, c are in AP or GP or HP, then $\frac{\mathrm{a} - \mathrm{b}}{\mathrm{b} - \mathrm{c}} = ?$

• $\frac{\mathrm{b}}{\mathrm{a}} \mathrm{or} 1 \mathrm{or} \frac{\mathrm{b}}{\mathrm{c}}$

• $\frac{\mathrm{c}}{\mathrm{a}} \mathrm{or} \frac{\mathrm{c}}{\mathrm{b}} \mathrm{or} 1$

• $1 \mathrm{or} \frac{\mathrm{a}}{\mathrm{b}} \mathrm{or} \frac{\mathrm{a}}{\mathrm{c}}$

• $1 \mathrm{or} \frac{\mathrm{a}}{\mathrm{b}} \mathrm{or} \frac{\mathrm{c}}{\mathrm{a}}$

What is the sum of all three-digit numbers that can be formed using all the digits 3, 4 and 5, when repetition of digits is not allowed ?

• 2664

• 3882

• 4044

• 4444