If 2x + 3y + 12 = 0 and x - y + 4$\mathrm{\lambda }$ = 0 are conjugate with respect to the parabola y = 8x, then $\mathrm{\lambda }$ is equal to

• 2

• - 2

• 3

• - 3

For an ellipse with eccentricity $\frac{1}{2}$ the centre is at the origin. If one directrix is x = 4, then the equation of the ellipse is

• $3{\mathrm{x}}^2 + 4{\mathrm{y}}^2 = 1$

• $3{\mathrm{x}}^2 + 4{\mathrm{y}}^2 = 12$

• $4{\mathrm{x}}^2 + 3{\mathrm{y}}^2 = 1$

• $4{\mathrm{x}}^2 + 3{\mathrm{y}}^2 = 12$

The distance between the foci of the hyperbola x² - 3y² - 4x - 6y - 11 = 0 is

• 4

• 6

• 8

• 10

The radius of the circle with the polar equation

r² - 8r($\sqrt{3}$cos$\mathrm{\theta }$ + sin$\mathrm{\theta }$) + 15 = 0 is

• 8

• 7

• 6

• 5

$\underset{\mathrm{x}\to 0}{\lim } \frac{\left(1 - {\mathrm{e}}^{\mathrm{x}}\right)\sin \left(\mathrm{x}\right)}{{\mathrm{x}}^2 + {\mathrm{x}}^3} = ?$

• - 1

• 0

• 1

• 2

$$\begin{gathered} \mathrm{If} \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \left[\mathrm{x} - 3\right] + \left|\mathrm{x} - 4\right| \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \\ \underset{\mathrm{x}\to 3^{ -}}{\lim }\mathrm{f}\left(\mathrm{x}\right) = ? \end{gathered}$$

• - 2

• - 1

• 0

• 1

$\mathrm{If} \mathrm{x} = \mathrm{a}\left\{\cos \left(\mathrm{\theta }\right) + \log \left(\tan \left(\frac{\mathrm{\theta }}{2}\right)\right)\right\} \mathrm{and} \mathrm{y} = \mathrm{asin}\left(\mathrm{\theta }\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\cot \left(\mathrm{\theta }\right)$

• $\tan \left(\mathrm{\theta }\right)$

• $\sin \left(\mathrm{\theta }\right)$

• $\cos \left(\mathrm{\theta }\right)$

$\left|\begin{array}{ccc}\mathrm{a} - \mathrm{b} - \mathrm{c}& 2\mathrm{a}& 2\mathrm{a}\\ 2\mathrm{b}& \mathrm{b} - \mathrm{c} - \mathrm{a}& 2\mathrm{b}\\ 2\mathrm{c}& 2\mathrm{c}& \mathrm{c} - \mathrm{a} - \mathrm{b}\end{array}\right|$ is equal to

• 0

• a + b + c

• (a + b + c)²

• (a + b + c)³

49.

If : R$\to$C is defined by f(x) =e^ix for x ∈ R, then f is (whereC denotes the set of all complex numbers)

• one-one

• onto

• one-one and onto

• neither one-one nor onto

$$\begin{gathered} \mathrm{If} \mathrm{A} =\left[\begin{array}{cc}1& - 2\\ 4& 5\end{array}\right] \mathrm{and} \mathrm{f}\left(\mathrm{t}\right) = {\mathrm{t}}^2 - 3\mathrm{t} +\hspace{0.17em}7, \mathrm{then} \\ \mathrm{f}\left(\mathrm{A}\right) + \left[\begin{array}{cc}3& 6\\ - 12& - 9\end{array}\right] = ? \end{gathered}$$

• $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$