Let L denote the line in the xy-plane with x and y intercepts as 3 respectively. Then the image of the point (– 1, – 4) in the line is :

• $\left(\frac{8}{5}, \frac{29}{5}\right)$

• $\left(\frac{11}{5}, \frac{28}{5}\right)$

• $\left(\frac{29}{5}, \frac{8}{5}\right)$

• $\left(\frac{29}{5}, \frac{11}{5}\right)$

$\mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} 2\mathrm{x}(2\mathrm{x} + 1) = 1, \mathrm{then} \mathrm{\beta } \mathrm{is} \mathrm{equal} \mathrm{to} :$

• $2\mathrm{\alpha }\left(\mathrm{\alpha } - 1\right)$

• $2\mathrm{\alpha }\left(\mathrm{\alpha } + 1\right)$

• $2{\mathrm{\alpha }}^2$

• $- 2\mathrm{\alpha }\left(\mathrm{\alpha } + 1\right)$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{tangent} \mathrm{tothe} \mathrm{curve}, \mathrm{y} = \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{x}\right), (\mathrm{x} > 0) \mathrm{at} \mathrm{a} \mathrm{point} (\mathrm{c}, \mathrm{f}(\mathrm{c}\left)\right) \mathrm{is} \mathrm{parallel} \\ \mathrm{to} \mathrm{the} \mathrm{line}-\mathrm{segment} \mathrm{joining} \mathrm{the} \mathrm{points} (1, 0) \mathrm{and} (\mathrm{e}, \mathrm{e}), \mathrm{then} \mathrm{c} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• $\frac{\mathrm{e} - 1}{\mathrm{e}}$

• $\frac{1}{\mathrm{e} - 1}$

• ${\mathrm{e}}^{\left(\frac{1}{\mathrm{e} - 1}\right)}$

• $\mathrm{e}\left({}^{\frac{1}{1 - \mathrm{e}}}\right)$

If the constant term in the binomial expansion of ${\left(\sqrt{\mathrm{x}} - \frac{\mathrm{k}}{{\mathrm{x}}^2}\right)}^{10} \mathrm{is} 405, \mathrm{then} \left|\mathrm{k}\right| = ?$

• 3

• 2

• 1

• 9

5.

If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, the eccentricity e of the ellipse satisfies : 

• ${\mathrm{e}}^4 + 2{\mathrm{e}}^2 - 1 = 0$

• ${\mathrm{e}}^2 + \mathrm{e} - 1$

• ${\mathrm{e}}^2 + 2\mathrm{e} - 1 = 0$

• ${\mathrm{e}}^4 + {\mathrm{e}}^2 - 1 = 0$

The set of all real values for which the function

$\mathrm{f}\left(\mathrm{x}\right) = \left(1 - {\cos }^2\left(\mathrm{x}\right)\right) . \left(\mathrm{\lambda } + \sin \left(\mathrm{x}\right)\right), \mathrm{x} \in \left( - \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right),$

has exactly one maxima and exactly one minima, is

• $\left(- \frac{1}{2}, \frac{1}{2}\right) - \left\{0\right\}$

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

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

• $\left(- \frac{3}{2}, \frac{3}{2}\right) - \left\{0\right\}$

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{statement}: “\mathrm{For} \mathrm{an} \mathrm{integer} \mathrm{n}, \mathrm{if} {\mathrm{n}}^3 – 1 \mathrm{is} \mathrm{even}, \mathrm{then} \mathrm{n} \mathrm{is} \mathrm{odd}”. \mathrm{The} \\ \mathrm{contrapositive} \mathrm{statement} \mathrm{of} \mathrm{this} \mathrm{statement} \mathrm{is}: \end{gathered}$$

• For an integer n, if n is even, then n³ – 1 is odd

• For an integer n, if n is even, then n³ – 1 is even.

• For an integer n, if n³ – 1 is not even, then n is not odd

• For an integer n, if n is odd, then n³ – 1 is even.

The centre of the circle passing through the point (0, 1) and touching the parabola y = x² at the point (2, 4) is:

• $\left(\frac{3}{10}, \frac{{\displaystyle 16}}{5}\right)$

• $\left(\frac{- 53}{10}, \frac{{\displaystyle 16}}{5}\right)$

• $\left(\frac{ - 16}{5}, \frac{{\displaystyle 53}}{10}\right)$

• $\left(\frac{6}{5}, \frac{{\displaystyle 53}}{10}\right)$

The angle of elevation of the summit of a mountain from a point on the ground is 45º. After climbing up one km towards the summitat an inclination of 30º from the ground, the angle of elevation of the summit is found to be 60º. Then the height (in km) of the summit from the ground is

• $\frac{\sqrt{3} + 1}{\sqrt{3} - 1}$

• $\frac{1}{\sqrt{3} + 1}$

• $\frac{1}{\sqrt{3} - 1}$

• $\frac{\sqrt{3} - 1}{\sqrt{3} + 1}$

$\mathrm{Let} \mathrm{\theta } = \frac{\mathrm{\theta }}{5} \mathrm{and} \mathrm{A} = \left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]. \mathrm{If} \mathrm{B} = \mathrm{A} + {\mathrm{A}}^4, \mathrm{Then} \det \left(\mathrm{B}\right) :$

• 0

• 1

• lies in (2, 3)

• lies in (1, 2)