A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If $\angle$BAC = 90°, and ar($∆$ABC) = 55sq. units, then the abscissa of the vertex C is :

• $1 + \sqrt{5}$

• $2 + \sqrt{5}$

• $1 + 2\sqrt{5}$

• $2\sqrt{5} - 1$

$$\begin{gathered} \mathrm{If} 1 + \left(1 - 2^2 . 1\right) + \left(1 - 4^2 . 3\right) + \left(1 - 6^2 . 5\right) + ... + \left(1 - {20}^2 . 19\right) = \mathrm{\alpha } - 220\mathrm{\beta } \mathrm{then} \\ \mathrm{an} \mathrm{ordered} \mathrm{pair} \left(\mathrm{\alpha }, \mathrm{\beta }\right) = ? \end{gathered}$$

• (11, 97)

• (10, 103)

• (11, 103)

• (10, 97)

Let P(3, 3) be a point on the hyperbola,$\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ . If the normal to it at P intersects the x-axis at (9,0) and e is its eccentricity, then the ordered pair (a², e²) is equal to :

• $\left(\frac{9}{2}, 3\right)$

• $\left(\frac{3}{2}, 2\right)$

• $\left(9, 3\right)$

• $\left(\frac{9}{2}, 2\right)$

The mean and variance of 8 observations are 10 and 13.5 respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :

• 9

• 5

• 3

• 7

Let $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \left(\mathrm{a} > \mathrm{b}\right)$ be agiven ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function $\mathrm{\phi }\left(\mathrm{t}\right) = \frac{5}{12} + \mathrm{t} - {\mathrm{t}}^2$ then a² + b² is equal to

• 145

• 126

• 116

• 135

6.

$$\begin{gathered} \mathrm{Let} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{be} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^2 – 3\mathrm{x} + \mathrm{p} = 0 \mathrm{and} \mathrm{\gamma } \mathrm{and} \mathrm{\delta } \mathrm{be} \mathrm{the} \mathrm{roots} \mathrm{of} \\ {\mathrm{x}}^2 – 6\mathrm{x} + \mathrm{q} = 0. \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }, \mathrm{\delta } \mathrm{from} \mathrm{a} \mathrm{geometric} \mathrm{progression}. \mathrm{Then} \mathrm{ratio} (2\mathrm{q} + \mathrm{p}) : (2\mathrm{q} – \mathrm{p}) \mathrm{is} \end{gathered}$$

• 3 : 1

• 5 : 3

• 9 : 7

• 33 : 31

$$\begin{gathered} \mathrm{If} \mathrm{A} =\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \mathrm{isin}\left(\mathrm{\theta }\right)\\ \mathrm{isin}\left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right], \left(\mathrm{\theta } = \frac{\mathrm{\pi }}{24}\right) \mathrm{and} {\mathrm{A}}^5 = \left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right], \mathrm{where} \mathrm{i} = \sqrt{ - 1}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \\ \mathrm{not} \mathrm{true} ? \end{gathered}$$

• $0 \le {\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{b}}^2 \le 1$

• ${\mathrm{a}}^2 - {\mathrm{d}}^2 = 0$

• ${\mathrm{a}}^2 - {\mathrm{b}}^2 = \frac{1}{2}$

• ${\mathrm{a}}^2 - {\mathrm{c}}^2 = 1$

$$\begin{gathered} \mathrm{Let} \mathrm{y} = \mathrm{y}\left(\mathrm{x}\right) \mathrm{be} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation}, \mathrm{xy}\text{'} – \mathrm{y} = {\mathrm{x}}^2(\mathrm{xcosx} + \mathrm{sinx}), \mathrm{x} > 0. \\ \mathrm{If} \mathrm{y}\left(\mathrm{\pi }\right) = \mathrm{\pi }, \mathrm{then} \mathrm{y}\text{'}\text{'}\left(\frac{\mathrm{\pi }}{2}\right) + \mathrm{y}\left(\frac{\mathrm{\pi }}{2}\right) \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• $1 + \frac{\mathrm{\pi }}{2}$

• $1 + \frac{\mathrm{\pi }}{2} + \frac{{\mathrm{\pi }}^2}{4}$

• $2 + \frac{\mathrm{\pi }}{2} +\hspace{0.17em}\frac{{\mathrm{\pi }}^2}{4}$

• $2 + \frac{\mathrm{\pi }}{2}$

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 2\right| \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = \mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right), \mathrm{x} \in \left[0, 4\right]. \mathrm{Then} \\ {\int }_0^3\left(\mathrm{g}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{x}\right)\right)d\mathrm{x} = ? \end{gathered}$$

• 0

• $\frac{1}{2}$

• $\frac{3}{2}$

• 1

$\mathrm{The} \mathrm{integral} \int {\left(\frac{\mathrm{x}}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\right)}^2\mathrm{dx} = ? \left(\mathrm{Where} \mathrm{Cis} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}\right) :$

• $\tan \left(\mathrm{x}\right) + \frac{\mathrm{xsec}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\tan \left(\mathrm{x}\right) - \frac{\mathrm{xsec}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\sec \left(\mathrm{x}\right) + \frac{\mathrm{xtan}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\sec \left(\mathrm{x}\right) - \frac{\mathrm{xsec}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$