Let n be a positive even integer. If the ratio of the largest coefficient and the 2nd largest coefficient in the expansion of (1 + x)ⁿ is 11 : 10. Then, the number of terms in the expansion of (1 + x)ⁿ is

• 20

• 21

• 10

• 11

Five numbers are in AP with common difference $\ne$ 0. If the 1st, 3rd and 4th terms are in GP, then

• the 5th term is always 0.

• the 1st term is always 0.

• the middle term is always 0

• the middle term is always - 2.

The sum of series

$\frac{1}{1 \times 2}{}^{25}\mathrm{C}_0 + \frac{1}{2 \times 3}{}^{25}\mathrm{C}_1 + \frac{1}{3 \times 4}{}^{25}\mathrm{C}_2 + ... + \frac{1}{26 \times 27}{}^{25}\mathrm{C}_{25}$

is

• $\frac{2^{27} - 1}{26 \times 27}$

• $\frac{2^{27} - 28}{26 \times 27}$

• $\frac{1}{2}\left(\frac{2^{26} + 1}{26 \times 27}\right)$

• $\left(\frac{2^{26} - 1}{52}\right)$

If P, Q and R are angles of an isosceles triangle and $\angle \mathrm{P} = \frac{\mathrm{\pi }}{2}$,  then the value of

$$\begin{gathered} {\left(\cos \left(\frac{\mathrm{P}}{3}\right) - \mathrm{isin}\left(\frac{\mathrm{P}}{3}\right)\right)}^3 + \left(\cos \left(\mathrm{Q}\right) + \mathrm{isin}\left(\mathrm{Q}\right)\right)\left(\cos \left(\mathrm{R}\right) - \mathrm{isin}\left(\mathrm{R}\right)\right) \\ + \left(\cos \left(\mathrm{P}\right) - \mathrm{isin}\left(\mathrm{P}\right)\right)\left(\cos \left(\mathrm{Q}\right) - \mathrm{isin}\left(\mathrm{Q}\right)\right)\left(\cos \left(\mathrm{R}\right) - \mathrm{isin}\left(\mathrm{R}\right)\right) \end{gathered}$$

• i

• - i

• 1

• - 1

Let f : R $\to$ R  be such that f is injective and f(x) f(y) = f(x + y) for all x, y $\in$if f(x), f(y) and f(z) are in GP, then x, y and z are in

• AP always

• GP always

• AP depending on the values of x, y and z

• GP depending on the values of x, y and z

The number of solutions of the equation

$\frac{1}{2}{\log }_{\sqrt{3}}\left(\frac{\mathrm{x} + 1}{\mathrm{x} +\hspace{0.17em}5}\right) + {\log }_{\mathrm{e}}{\left(\mathrm{x} + 5\right)}^2$ = 1

• 0

• 1

• 2

• infinite

If $\mathrm{P} = 1 + \frac{1}{2 \times 2} + \frac{1}{3 \times 2^2} + ... \mathrm{and} \mathrm{Q} = \frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + ...$,

then

• P = Q

• 2P =Q

• P = 2Q

• P = 4Q

The area of the region bounded by the parabola y = x² - 4x + 5 and the straight line y= x + l is

• $\frac{1}{2}$

• 2

• 3

• $\frac{9}{2}$

If $\mathrm{f}\left(\mathrm{x}\right) = \sin \left(\mathrm{x}\right) + 2{\cos }^2\left(\mathrm{x}\right), \frac{\mathrm{\pi }}{4} \le \mathrm{x} \le \frac{3\mathrm{\pi }}{4}$. Then, f attains its

• $\mathrm{minimum} \mathrm{at} \mathrm{x} = \frac{\mathrm{\pi }}{4}$

• maximum at x = $\frac{\mathrm{\pi }}{2}$

• minimum x = $\frac{\mathrm{\pi }}{2}$

• mamum at x = ${\sin }^{-1}\left(\frac{1}{4}\right)$

50.

A line passing through the point of intersection of x + y = 4 and x - y = 2 makes an angle ${\tan }^{-1}\left(\frac{3}{4}\right)$ with the x-axis. It intersects the parabola y² = 4(x - 3) at points (x₁, y₁) and (x₂, y₂) respectively. Then, $\left|{\mathrm{x}}_1 - {\mathrm{x}}_2\right|$ is equal to

• $\frac{16}{9}$

• $\frac{32}{9}$

• $\frac{40}{9}$

• $\frac{80}{9}$