If the normal at one end of the latusrectum of an ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ passes through the one end of the minor axis, then

• e⁴ - e² + 1 = 0

• e² - e + 1 = 0

• e² + e + 1 = 0

• e⁴ + e² - 1 = 0

The equation of the curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact, is

• a parabola

• an ellipse

• a circle

• a hyperbola

The solution of cos(x + y)dy = dx, is

• y = $\tan \left(\frac{\mathrm{x} + \mathrm{y}}{2}\right) + \mathrm{C}$

• $\mathrm{y} = {\cos }^{-1}\left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right)$

• $\mathrm{y} = \sec \left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right) + \mathrm{C}$

• None of the above

Let E' denote the complement of an event E. Let E, F, G be pairwise independent events such that P(G) > 0 and P(E ∩ F ∩ G) = 0. Then, P(E' $\cap$ F' /G) equals

• P(E') + P(F')

• P(E') - P(F')

• P(E') - P(F)

• P(E) - P(F')

If the relation between direction ratios of two lines are given by a + b + c = 0 and 2ab + 2ac - bc = 0, then the angle between the lines is

• $\mathrm{\pi }$

• $\frac{2\mathrm{\pi }}{3}$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

$\underset{\mathrm{x}\to \mathrm{\alpha }}{\lim }{\left(\tan \left(\mathrm{x}\right)\cot \left(\mathrm{x}\right)\right)}^{\frac{1}{\mathrm{x} - \mathrm{\alpha }}}$ is equal to

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

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

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

• None of these

If the slope of the curve $\mathrm{y} = \frac{\mathrm{ax}}{\mathrm{b} - \mathrm{x}}$ at the point (1, 1) is 2, then

• a = 1, b = - 2

• a = - 1, b = 2

• a = 1, b = 2

• None of these

Let w be the complex number $\cos \left(\frac{2\mathrm{\pi }}{3}\right) + \mathrm{isin}\left(\frac{2\mathrm{\pi }}{3}\right)$ Then, the number of distinct complex number z satisfying

$\left|\begin{array}{ccc}\mathrm{z} + 1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& \mathrm{z} + {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{z} + \mathrm{w}\end{array}\right|$ = o is equal to

• 1

• 0

• 2

• 3

If A is a square matrix such that A² = A and B = I - A, then AB + BA + I - (I - A)² is equal to

• A

• 2A

• - A

• I - A

20.

If A = $\frac{1}{3}\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& - 2\\ \mathrm{a}& 2& \mathrm{b}\end{array}\right]$ is an orthagonal matrix, then

• a = 2, b = 1

• a = - 2, b = - 1

• a = 2, b = - 1

• a = - 2, b = 1