If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in

• G.P.

• A.P.

• Arithmetic − Geometric Progression

• Arithmetic − Geometric Progression

• I₂ > I₁

• I₁ > I₂

• I₃ = I₄

• I₃ = I₄

23.

The locus of a point P (α, β) moving under the condition that the line y = αx + β is a tangent to the hyperbola 

• an ellipse

• a circle

• a parabola

• a parabola

If non-zero numbers a, b, c are in H.P., then the straight line  always passes through a fixed point. That point is

• (-1, 2)

• (-1, -2)

• (1, -2)

• (1, -2)

If the circles x² + y² + 2ax + cy + a = 0 and x² + y² – 3ax + dy – 1 = 0 intersect in two distinct points P and Q then the line 5x + by – a = 0 passes through P and Q for

• exactly one value of a

• no value of a

• infinitely many values of a

• infinitely many values of a

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is 

• an ellipse

• a circle

• a hyperbola

• a hyperbola

If a circle passes through the point (a, b) and cuts the circle x² + y² = p² orthogonally, then the equation of the locus of its centre is 

• x² + y² – 3ax – 4by + (a² + b² – p² ) = 0

• 2ax + 2by – (a² – b² + p² ) = 0

• x² + y² – 2ax – 3by + (a² – b² – p² ) = 0

• x² + y² – 2ax – 3by + (a² – b² – p² ) = 0

An ellipse has OB as semi-minor axis, F and F′ its focii and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

• 

• 1/2

• 1/4

• 1/4

Let A and B be two events such that  where  stands for complement of event A. Then events A and B are

• equally likely and mutually exclusive

• equally likely but not independent

• independent but not equally likely

• independent but not equally likely

A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of 2 cm/s² and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then the lizard will catch the insect after

• 20 s

• 1 s

• 21 s

• 21 s