$$\begin{gathered} \mathrm{If} {\mathrm{T}}_1, {\mathrm{T}}_2, {\mathrm{T}}_3 .... \mathrm{are} \mathrm{in} \mathrm{A}.\mathrm{P}. \mathrm{such} \mathrm{that} \\ {\mathrm{T}}_1 + {\mathrm{T}}_2 + ... + {\mathrm{T}}_{25} = {\mathrm{T}}_{28} +{\mathrm{T}}_{27} + ... {\mathrm{T}}_{40} \mathrm{and} \mathrm{first} \mathrm{term} \mathrm{is} 3 \\ \mathrm{then} \mathrm{value} \mathrm{of} \mathrm{common} \mathrm{difference} \mathrm{of} \mathrm{A}.\mathrm{P}. \mathrm{is} \end{gathered}$$

• $\frac{1}{2}$

• $\frac{1}{6}$

• 2

• 3

$$\begin{gathered} \mathrm{Let} \frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{3} = 1 \mathrm{is} \mathrm{an} \mathrm{ellipse} \mathrm{and} \mathrm{a} \mathrm{hyperbola} \mathrm{which} \mathrm{is} \mathrm{confocal} \mathrm{with} \mathrm{ellipse} \mathrm{such} \mathrm{that} \\ \mathrm{its} \mathrm{transverse} \mathrm{axis} \mathrm{is} \sqrt{2} \mathrm{then} \mathrm{which} \mathrm{of} \mathrm{following} \mathrm{point} \mathrm{does} \mathrm{not} \mathrm{lie} \mathrm{on} \mathrm{hyperbola} \end{gathered}$$

• $\left(1, - \frac{1}{\sqrt{2}}\right)$

• $\left(-\sqrt{\frac{3}{2}}, 1\right)$

• $\left(\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}\right)$

• None of these

3.

$$\begin{gathered} \mathrm{Let} \mathrm{\alpha }, \mathrm{\beta } \mathrm{are} \mathrm{roots} \mathrm{of} {\mathrm{x}}^2 + \mathrm{px} + 2 = 0 \mathrm{and} \frac{1}{\mathrm{\alpha }}, \frac{1}{\mathrm{\beta }} \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} 2{\mathrm{x}}^2 - 2\mathrm{qx} + 1 = 0. \\ \mathrm{Then} \mathrm{find} \mathrm{the} \mathrm{value} \mathrm{of} \left(\mathrm{\alpha } + \frac{1}{\mathrm{\beta }}\right)\left(\mathrm{\beta } + \frac{1}{\mathrm{\alpha }}\right)\left(\mathrm{\alpha } - \frac{1}{\mathrm{\alpha }}\right)\left(\mathrm{\beta } - \frac{1}{\mathrm{\beta }}\right) \end{gathered}$$

• $\frac{9}{4}\left(9 - {\mathrm{p}}^2\right)$

• $\frac{9}{4}\left(9 + {\mathrm{p}}^2\right)$

• $\frac{4}{9}\left(9 - {\mathrm{q}}^2\right)$

• $\frac{9}{4}\left(9 - {\mathrm{q}}^2\right)$

$\mathrm{S} = \left(2 {}^1\mathrm{P}_0 - 3{}^2\mathrm{P}_1 + 4{}^3\mathrm{P}_2 + ... 51 \mathrm{terms}\right) + \left(1! +\hspace{0.17em}2! + 3! - 4! + ...51\mathrm{terms}\right), \mathrm{find} \mathrm{S}$

• 1 + 51!

• 1 + 52!

• 1 + (50)51!

• 1 + (51)51!

Let the data 4, 10, x, y, 27 be in increasing order. If the median of data is 18 and its mean deviation about mean is 7.6 then the mean of this data is :

• 17

• 16

• 16.5

• 15.5

${\left(\frac{1 + \mathrm{i}}{1 - \mathrm{i}}\right)}^{\frac{\mathrm{m}}{2}} = {\left(\frac{1 + \mathrm{i}}{1 - \mathrm{i}}\right)}^{\frac{\mathrm{n}}{3}} = 1. \mathrm{m}, \mathrm{n} \in \mathrm{N} \mathrm{find} \mathrm{HCF} \mathrm{of} \left(\mathrm{m}, \mathrm{n}\right) \mathrm{for} \mathrm{least} \mathrm{m} \& \mathrm{n}$

• 4

• 3

• 6

• 9

$\mathrm{Let} \left[\begin{array}{cc}\mathrm{x}& 1\\ 1& 0\end{array}\right] \mathrm{be} \mathrm{a} 2 \times 2 \mathrm{matrix} \mathrm{such} \mathrm{that} {\mathrm{A}}^4 = {\left[{\mathrm{a}}_{\mathrm{ij}}\right]}_{2 \times 2}{\mathrm{a}}_{11} = 109, \mathrm{then} \mathrm{find} {\mathrm{a}}_{22}$

• 12

• 4

• - 8

• 10

$2\mathrm{\pi } - \left[{\sin }^{-1}\left(\frac{4}{5}\right) + {\sin }^{-1}\left(\frac{5}{13}\right) + {\sin }^{-1}\left(\frac{16}{65}\right)\right] = ?$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $\frac{3\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

$$\begin{gathered} \underset{\mathrm{x}\to 0}{\lim }\left(\frac{1 - \mathrm{x} + \left|\mathrm{x}\right|}{1 + \left|\mathrm{x}\right| - \mathrm{\lambda }}\right) = \mathrm{L} \left(\mathrm{finite}\right) \mathrm{where} \left[*\right] \mathrm{denotes} \mathrm{the} \\ \mathrm{greatest} \mathrm{integer} \mathrm{function} \mathrm{then} \mathrm{find} \mathrm{L} \end{gathered}$$

• 0

• $\frac{1}{2}$

• 1

• 2

$\mathrm{The} \mathrm{preposition} \mathrm{p}\Rightarrow \left(~\left(\mathrm{p} \wedge ~ \mathrm{q}\right)\right) \mathrm{is} \mathrm{equivalent} \mathrm{to}$

• q

• $~ \mathrm{p} \vee \mathrm{q}$

• $\mathrm{p} \vee ~ \mathrm{q}$

• $~ \mathrm{p} \vee ~ \mathrm{q}$