Let the equation of an ellipse be $\frac{{\mathrm{x}}^2}{144} + \frac{{\mathrm{y}}^2}{25} = 1$. Then, the radius of the circle with centre (0, $\sqrt{2}$) and passing through the foci of the ellipse is

• 9

• 7

• 11

• 5

The straight lines x + y = 0, 5x + y = 4 and x + 5y = 4 form

• an isosceles triangle

• an equilateral triangle

• a scalene triangle

• a right angled triangle

The value of $\mathrm{\lambda }$ for which the curve (7x + 5)² + (7y + 3)² = ${\mathrm{\lambda }}^2$(4x + 3y - 24)² represents a parabola is

• $\pm \frac{6}{5}$

• $\pm \frac{7}{5}$

• $\pm \frac{1}{5}$

• $\pm \frac{2}{5}$

14.

Let f(x) = x + 1/2. Then, the number of real values of x for which the three unequal terms f(x), f(2x), f(4x) are in HP is

• 1

• 0

• 3

• 2

Let f(x) = 2x² + 5x + 1. If we write f(x) as f(x) = a(x + 1)(x - 2) + b(x - 2)(x - 1) + c(x - 1)(x + 1) for real numbers a, b, c then

• there are infinite number of choices for a, b, c

• only one choice for a but infinite number of choices for b and c

• exactly one choice for each of a, b, c

• more than one but finite number of choices for a, b, c

If $\mathrm{\alpha }, \mathrm{\beta }$ are the roots of ax² + bx + c = 0 ($\mathrm{a} \ne 0$) and $\mathrm{\alpha } + \mathrm{h}, \mathrm{\beta } + \mathrm{h}$ are the roots of px² + qx + r = 0 (p $\ne$ 0), then the ratio of the squares of their discriminants is

• a² : p²

• a : p²

• a² : p

• a : 2p

The equation of the common tangent with positive slope to the parabola y² = 8$\sqrt{3}$x and the hyperbola 4x² - y² = 4 is

• $\mathrm{y} = \sqrt{6}\mathrm{x} + \sqrt{2}$

• $\mathrm{y} = \sqrt{6}\mathrm{x} - \sqrt{2}$

• $\mathrm{y} = \sqrt{3}\mathrm{x} + \sqrt{2}$

• $\mathrm{y} = \sqrt{3}\mathrm{x} - \sqrt{2}$

The point on the parabola y² = 64x which is nearest to the line 4x + 3y + 35 = 0 has coordinates

• (9, - 24)

• (1, 81)

• (4, - 16)

• (- 9, - 24)

Let z₁, z₂ be two fixed complex numbers in the argand plane and z be an arbitrary point satisfying $\left|\mathrm{z} - {\mathrm{z}}_1\right| + \left|\mathrm{z} - {\mathrm{z}}_2\right| = 2\left|{\mathrm{z}}_1 - {\mathrm{z}}_2\right|$ Then, the locus of z will be

• an ellipse

• a straight line joining z₁ and z₂

• a parabola

• a bisector of the line segment joining z₁ and z₂

the coefficient of x⁸ in ${\left({\mathrm{ax}}^2 + \frac{1}{\mathrm{bx}}\right)}^{13}$ is equal to the coefficient of x^{- 8} in ${\left(\mathrm{ax} - \frac{1}{{\mathrm{bx}}^2}\right)}^{13}$ then a and b will satisfy the relation

• ab + 1 = 0

• ab = 1

• a = 1 - b

• a + b = - 1