11.

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is :

• 3000

• 2250

• 2255

• 1500

If the sum of the first 20 terms of the series log$\left({\log }_{\left(7^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}\left(\mathrm{x}\right)\right) + \left({\log }_{\left(7^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}\left(\mathrm{x}\right)\right) + \left({\log }_{7^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\left(\mathrm{x}\right)\right) + ... \mathrm{is} 460$,

then x = ?

• e²

• $7^{\raisebox{1ex}{$46$}\!\left/ \!\raisebox{-1ex}{$21$}\right.}$

• 7²

• $7^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Which of the following points lies on the tangent to the curve${\mathrm{x}}^4{\mathrm{e}}^{\mathrm{y}} + 2\sqrt{\mathrm{y} + 1} = 3$ at the point (1, 0)

• (- 2, 4)

• (2, 2)

• ( - 2, 6)

• (2, 6)

If x = 1 is a critical point of the function f(x) = (3x² + ax –2 – a) e^x, then

• x = 1 and x = $- \frac{2}{3}$ are local minima of f

• x = 1 is a local maxima and x =  $- \frac{2}{3}$is a local minima of f.

• $\mathrm{x} = 1 \mathrm{is} \mathrm{a} \mathrm{local} \mathrm{minima} \mathrm{and} \mathrm{x} = - \frac{2}{3} \mathrm{is} \mathrm{a} \mathrm{local} \mathrm{maxima} \mathrm{of} \mathrm{f}.$

• $\mathrm{x} = 1 \mathrm{and} \mathrm{x} = - \frac{2}{3}\mathrm{are} \mathrm{local} \mathrm{maxima} \mathrm{of} \mathrm{f}.$

$\mathrm{The} \mathrm{derivative} \mathrm{of} {\tan }^{-1}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - 1}{\mathrm{x}}\right) \mathrm{with} \mathrm{respect} \mathrm{to} {\tan }^{-1}\left(\frac{2\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}}{1 - 2{\mathrm{x}}^2}\right) \mathrm{at} \mathrm{x} = \frac{1}{2} \mathrm{is} :$

• $\frac{\sqrt{3}}{10}$

• $\frac{\sqrt{3}}{12}$

• $\frac{2\sqrt{3}}{5}$

• $\frac{2\sqrt{3}}{3}$

If the system of linear equations

x + y + 3z = 0

x + 3y + k²z = 0

3x + y + 3z = 0

has a non–zero solution (x, y, z) for some k $\in$ R, then x $\mathrm{x} + \left(\frac{\mathrm{y}}{\mathrm{z}}\right)$ is equal to :

•  - 9

•  - 3

• 9

• 3

If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non–zero distinct real numbers then $\left|\begin{array}{ccc}\mathrm{x}& \mathrm{a} + \mathrm{y}& \mathrm{x} + \mathrm{a}\\ \mathrm{y}& \mathrm{b} + \mathrm{y}& \mathrm{y} + \mathrm{b}\\ \mathrm{z}& \mathrm{c} + \mathrm{y}& \mathrm{z} + \mathrm{c}\end{array}\right| = ?$

• y(a - b)

• 0

• y(b - a)

• y(a - c)

$\mathrm{If} \int \frac{\cos \left(\mathrm{\theta }\right)}{5 + 7\sin \left(\mathrm{\theta }\right) - 2{\cos }^2\left(\mathrm{\theta }\right)}\mathrm{d\theta } = {\mathrm{Alog}}_{\mathrm{e}}\left|\mathrm{B}\left(\mathrm{\theta }\right)\right| + \mathrm{C}, \mathrm{where} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}, \mathrm{then} \frac{\mathrm{B}\left(\mathrm{\theta }\right)}{\mathrm{A}}\mathrm{can} \mathrm{be}:$

• $\frac{2\sin \left(\mathrm{\theta }\right) + 1}{5\left(\sin \left(\mathrm{\theta }\right) + 3\right)}$

• $\frac{5\left(2\sin \left(\mathrm{\theta }\right) + 1\right)}{\sin \left(\mathrm{\theta }\right) + 3}$

• $\frac{2\sin \left(\mathrm{\theta }\right) + 1}{\sin \left(\mathrm{\theta }\right) + 3}$

• $\frac{5\left(\sin \left(\mathrm{\theta }\right) + 3\right)}{2\sin \left(\mathrm{\theta }\right) + 1}$

The area (in sq. units) of the region A = {x, y) : (x –1) [x] $\le \mathrm{y} \le 2\sqrt{\mathrm{x}}, 0 \le \mathrm{x} \le 2$,where [t] denotes the greatest integer function, is :

• $\frac{4}{3}\sqrt{2} - \frac{1}{2}$

• $\frac{8}{3}\sqrt{2} - 1$

• $\frac{4}{3}\sqrt{2} + 1$

• $\frac{8}{3}\sqrt{2} - \frac{1}{2}$

$$\begin{gathered} \mathrm{Let} \mathrm{y} = \mathrm{y}\left(\mathrm{x}\right) \mathrm{be} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \cos \left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{ysin}\left(\mathrm{x}\right) = \sin \left(2\mathrm{x}\right), \mathrm{x} \in \left(0, \frac{\mathrm{\pi }}{2}\right) \\ \mathrm{If} \mathrm{y}\left(\mathrm{\pi }/3\right) = 0, \mathrm{then} \mathrm{y}\left(\mathrm{\pi }/4\right) = ? \end{gathered}$$

• $\frac{1}{\sqrt{2}} - 1$

• 2 - $\sqrt{2}$

• $\sqrt{2} - 2$

• 2 + $\sqrt{2}$