Each of a and b can take values 1 or 2 with equal probability. The probability that the equation ax² + bx + 1= 0 has real roots, is equal to

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{1}{16}$

Cards are drawn one-by-one without replacement from a well shuffled pack of 52 cards. Then, the probability that a face card (jack, queen or king) will appear for the first time on the third turn is equal to

• $\frac{300}{2197}$

• $\frac{36}{85}$

• $\frac{12}{85}$

• $\frac{4}{51}$

There are two coins, one unbiased with probaility $\frac{1}{2}$ or getting heads and the other one is biased with probability $\frac{3}{4}$ of getting heads. A coin is selected at random and tossed. It shows heads up. Then, the probability that the unbiased coin was selected is

• $\frac{2}{3}$

• $\frac{3}{5}$

• $\frac{1}{2}$

• $\frac{2}{5}$

Lines x + y = 1 and 3y = x + 3 intersect the ellipse x² + 9y² = 9 at the points P,Q and R. The area of the $∆$PQR is

• $\frac{36}{5}$

• $\frac{18}{5}$

• $\frac{9}{5}$

• $\frac{1}{5}$

For the variable , the locus of the point of intersection of the lines 3tx - 2y + 6t = 0 and 3x + 2ty - 6 = 0 is

• $\mathrm{the} \mathrm{ellipse} \frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{9} = 1$

• $\mathrm{the} \mathrm{ellipse} \frac{{\mathrm{x}}^2}{9} + \frac{{\mathrm{y}}^2}{4} = 1$

• $\mathrm{the} \mathrm{hyperbola} \frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{9} = 1$

• $\mathrm{the} \mathrm{hyperbola} \frac{{\mathrm{x}}^2}{9} - \frac{{\mathrm{y}}^2}{4} = 1$

The locus of the mid-points of the chords of an ellipse x² + 4y² = 4 that are drawn from the positive end of the minor axis, is

• a circle with centre $\left(\frac{1}{2}, 0\right)$ and radius 1

• a parabola with focus $\left(\frac{1}{2}, 0\right)$ and directrix x = - 1

• an ellipse with centre $\left(0, \frac{1}{2}\right)$, major axis $\frac{1}{2}$ and minor axis

• a hyperbola with centre $\left(0, \frac{1}{2}\right)$, transverse axis 1 and conjugate axis $\frac{1}{2}$

A point P lies on the circle x² + y² = 169. If Q = (5, 12) and R = (-12, 5) then the $\angle$QPR is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

A point moves, so that the sum of squares of its distance from the points (1, 2) and (- 2, 1) is always 6. Then, its locus is

• the straight line $\mathrm{y} - \frac{3}{2} = - 3\left(\mathrm{x} + \frac{1}{2}\right)$

• a circle with centre $\left(- \frac{1}{2}, \frac{3}{2}\right)$ and radius $\frac{1}{\sqrt{2}}$

• a parabola with focus (1, 2) and directrix passing through (- 2, 1)

• an ellipse with foci (1, 2) and (- 2, 1)

A circle passing through (0, 0), (2, 6), (6, 2) cut the x-axis at the point P $\ne$ (0, 0). Then, the lenght of OP, where O is the origin, is

• $\frac{5}{2}$

• $\frac{5}{\sqrt{2}}$

• 5

• 10

For the variable t, the locus of the points of intersection of lines x - 2y = t and x + 2y = $\frac{1}{\mathrm{t}}$ is

• the straight line x = y

• the circle with centre at the origin and radius 1

• the ellipse with centre at the origin and one focus $\left(\frac{2}{\sqrt{5}}, 0\right)$

• the hyperbola with centre at the origin and one $\left(\frac{\sqrt{5}}{2}, 0\right)$