Mathematics Tes...

Two parabolas with a common vertex and with axes along x-axis and y-axis intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is

The letters of the word OUGHT are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word TOUGH in this dictionary.

Let P be a point on the parabola x² = 4y. If the distance of P from the centre of the circle x² + y² + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P is

Let the population of rabbits surviving at a time t be governed by the differential equation  If p(0) = 100, then p(t) equals

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag. Its colour is seen and this ball, along with two additional balls of the same colour, is returned to the bag. If now, a ball is drawn at random from the bag, then the probability that this drawn ball is red is:

The length of projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane x + y + z = 7 is

O is the origin and A is the point (a, b, c). Deduce the equation of the plane through A at right angles to OA.

Tangents drawn from the point (-8, 0) to the parabola y² = 8x touch the parabola at P and Q. If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to

The sides of a rhombus ABCD are parallel to the lines x - y + 2 = 0 and 7x - y + 3 = 0. If the diagonals of the rhombus intersect at P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the ordinate of A is:

If f(x) is a quadratic expression such that f(1) + f(2) = 0, and -1 is a root of f(x) = 0, then the other root of f(x) = 0 is: