Let f(n) = $\left[\frac{1}{4} + \frac{\mathrm{n}}{1000}\right], \mathrm{where} \left[\mathrm{x}\right] \mathrm{denotes} \mathrm{the} \mathrm{integral} \mathrm{part} \mathrm{of} \mathrm{x}. \mathrm{Then} \mathrm{the} \mathrm{value} \mathrm{of} \sum _{\mathrm{n} = 1}^{1000}\mathrm{f}\left(\mathrm{n}\right) \mathrm{is} :$

• 251

• 250

• 1

• 0

Given that the arithmetic mean and standard deviation of a sample of 15 observations are 24 and 0 respectively. Then which one of the following is the arithmetic mean of the smallest five observations in the data ?

• 0

• 8

• 16

• 24

If a variable takes values 0, 1, 2, 3, ... , n with frequencies

$$\begin{gathered} 1. \mathrm{C}(\mathrm{n}, 1), \mathrm{C}(\mathrm{n}, 2), \mathrm{C}(\mathrm{n}, 3), ... , \mathrm{C}(\mathrm{n}, \mathrm{n}) \\ \mathrm{respectively}, \mathrm{then} \mathrm{the} \mathrm{arithmetic} \mathrm{mean} \mathrm{is} : \end{gathered}$$

• 2n

• n + 1

• n

• $\frac{\mathrm{n}}{2}$

If x₁, and x₂, are positive quantities, then the condition for the difference between the arithmetic mean and the geometric mean to be greater than 1 is :

• ${\mathrm{x}}_1 + {\mathrm{x}}_2 > 2\sqrt{{\mathrm{x}}_1{\mathrm{x}}_2}$

• $\sqrt{{\mathrm{x}}_1} + \sqrt{{\mathrm{x}}_2} > \sqrt{2}$

• $\left|\sqrt{{\mathrm{x}}_1} - \sqrt{{\mathrm{x}}_2}\right| > \sqrt{2}$

• ${\mathrm{x}}_1 + {\mathrm{x}}_2 < 2\left(\sqrt{{\mathrm{x}}_1{\mathrm{x}}_2} +\hspace{0.17em}1\right)$

Given that ${\log }_{\mathrm{x}}\left(\mathrm{y}\right), {\log }_{\mathrm{z}}\left(\mathrm{x}\right), {\log }_{\mathrm{y}}\left(\mathrm{z}\right)$ are in GP, xyz = 64 and x³ , y³, z³ are in AP. Which one of the following is correct ? x, y and z are

• in AP only

• in GP only

• in both AP and GP

• neither in AP nor in GP

Given that ${\log }_{\mathrm{x}}\left(\mathrm{y}\right), {\log }_{\mathrm{z}}\left(\mathrm{x}\right), {\log }_{\mathrm{y}}\left(\mathrm{z}\right)$ are in GP, xyz = 64 and x³ , y³, z³ are in AP. Which one of the following is correct ? xy, yz and zx are

• in AP only

• in GP only

• in both AP and GP

• neither in AP nor in GP

What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25

• 860

• 854

• 840

• 824

The mean of the series x₁, x₂ ... x_n, is $\overline{\mathrm{X}}$. If x, is replaced by $\mathrm{\lambda }$, then what is the new mean ?

• $\overline{\mathrm{X}} - {\mathrm{x}}_2 + \mathrm{\lambda }$

• $\frac{\overline{\mathrm{X}} - {\mathrm{x}}_2 - \mathrm{\lambda }}{\mathrm{n}}$

• $\frac{\overline{\mathrm{X}} - {\mathrm{x}}_2 + \mathrm{\lambda }}{\mathrm{n}}$

• $\frac{\mathrm{n} . \overline{\mathrm{X}} - {\mathrm{X}}_2 + \mathrm{\lambda }}{\mathrm{n}}$

If m is the eometric mean of

$$\begin{gathered} {\left(\frac{\mathrm{y}}{\mathrm{z}}\right)}^{\log \left(\mathrm{yz}\right)}, {\left(\frac{\mathrm{z}}{\mathrm{x}}\right)}^{\log \left(\mathrm{zx}\right)} \mathrm{and} {\left(\frac{\mathrm{x}}{\mathrm{y}}\right)}^{{}^{\log \left(\mathrm{xy}\right)}} \\ \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{m} ? \end{gathered}$$

• 1

• 3

• 6

• 9

What is the number of four-digit decimal numbers ( < 1) in which no digit is repeated ?

• 3024

• 4536

• 5040

• None of the above