$$\begin{gathered} \mathrm{If} \sin \left(\mathrm{\beta }\right) \mathrm{is} \mathrm{the} \mathrm{harmonic} \mathrm{mean} \mathrm{of} \sin \left(\mathrm{\alpha }\right) \mathrm{and} \cos \left(\mathrm{\alpha }\right), \mathrm{and} \sin \left(\mathrm{\theta }\right) \mathrm{is} \mathrm{the} \mathrm{arithmetic} \mathrm{mean} \\ \mathrm{of} \sin \left(\mathrm{\alpha }\right) \mathrm{and} \cos \left(\mathrm{\alpha }\right), \mathrm{then} \mathrm{which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is}/\mathrm{are} \mathrm{correct} \end{gathered}$$

$$\begin{gathered} 1. \sqrt{2}\sin \left(\mathrm{\alpha } + \frac{\mathrm{\pi }}{4}\right)\sin \left(\mathrm{\beta }\right) = \sin \left(2\mathrm{\alpha }\right) \\ 2. \sqrt{2}\sin \left(\mathrm{\theta }\right) = \cos \left(\mathrm{\alpha } - \frac{\mathrm{\pi }}{4}\right) \\ \mathrm{Select} \mathrm{the} \mathrm{correct} \mathrm{answer} \mathrm{using} \mathrm{the} \mathrm{code} \mathrm{given} \mathrm{below} : \end{gathered}$$

• 1 only

• 2 only

• both 1 and 2

• neither 1 or 2 

$$\begin{gathered} \mathrm{If} {\mathrm{x}}_{\mathrm{i}} > 0, {\mathrm{y}}_{\mathrm{i}} > 0 (\mathrm{i} = 1, 2, 3, ...... \mathrm{n}) \mathrm{are} \mathrm{the} \mathrm{values} \mathrm{oftwo} \mathrm{variables} \mathrm{X} \mathrm{and} \mathrm{Y} \mathrm{with} \\ \mathrm{geometric} \mathrm{means} \mathrm{P} \mathrm{and} \mathrm{Q} \mathrm{respectively}, \mathrm{then} \mathrm{the} \mathrm{geometric} \mathrm{mean} \mathrm{of} \frac{\mathrm{X}}{\mathrm{Y}} \mathrm{is} \end{gathered}$$

• $\frac{\mathrm{P}}{\mathrm{Q}}$

• $\mathrm{antilog}\left(\frac{\mathrm{P}}{\mathrm{Q}}\right)$

• $\mathrm{n}\left(\log \left(\mathrm{P}\right) - \log \left(\mathrm{Q}\right)\right)$

• $\mathrm{n}\left(\log \left(\mathrm{P}\right) +\hspace{0.17em}\log \left(\mathrm{Q}\right)\right)$

33.

Three-digit numbers are formed from the digits 1, 2 and 3 in  such a way that the digits are ot repeated. What is the sum of such three-digit numbers ?

• 1233

• 1322

• 1323

• 1332

What is the sum of the serie 0.3 + 0.33 + 0.333 +... n terms 

• $\frac{1}{3}\left[\mathrm{n} - \frac{1}{9}\left(1 - \frac{1}{{10}^{\mathrm{n}}}\right)\right]$

• $\frac{1}{3}\left[\mathrm{n} - \frac{2}{9}\left(1 - \frac{1}{{10}^{\mathrm{n}}}\right)\right]$

• $\frac{1}{3}\left[\mathrm{n} - \frac{1}{3}\left(1 - \frac{1}{{10}^{\mathrm{n}}}\right)\right]$

• $\frac{1}{3}\left[\mathrm{n} - \frac{1}{9}\left(1 + \frac{1}{{10}^{\mathrm{n}}}\right)\right]$

If the sum of m terms of an AP is n and the sum of n terms is m, then the sum of (m + n) terms is

• mn

• m + n

• 2(m + n)

• - (m + n)

The sum of the roots of the equation x² + bx + c = 0 (where b and c are non-zero) is equal to the sum of the  reciprocals of their squares. Then $\frac{1}{\mathrm{c}}, \mathrm{b}, \frac{\mathrm{c}}{\mathrm{b}}$ are in

• AP

• GP

• HP

• None of the above

The sum of the roots of the equation ax² + x + c = 0 (where a and c are non-zero) is equal to the sum of the reciprocals of their squares. Then a, ca², c² are in

• AP

• GP

• HP

• None of the above

The fifth term of an AP of 11 terms, whose sum is n² - 2n, is

• 5

• 7

• 8

• 15

The sum of all the two-digit odd numbers is

• 2475

• 2530

• 4905

• 5049

The sum of the first n terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + ... = ?$

• $2^n - n - 1$

• $1 - 2^{ - \mathrm{n}}$

• $2^{ - \mathrm{n}} + \mathrm{n} - 1$

• $2^{\mathrm{n}} - 1$