In the trapezium ABCD the sides AB and CD are parallel. The value of sin∠BAC / sin∠BAD is

log(97 − 56√3) / log √(7 + 4√3) equals which of the following?

In the picture below, EFGH, ABCD are squares, and ABE, BCF, CDG, DAH are equilateral triangles (each built outward on a side of ABCD, with apex at a corner of EFGH). What is the ratio of the area of the square EFGH to that of ABCD? [Figure: see page 16 of source PDF]

Let C be a circle of radius √20 cm. Let L1, L2 be the lines given by 2x − y − 1 = 0 and x + 2y − 18 = 0, respectively. Suppose that L1 passes through the center of C and that L2 is tangent to C at the point of intersection of L1 and L2. If (a, b) is the center of C, which of the…

What is the maximum number of points that can be placed on a circular disk of radius 1 metre (some of the points could be placed on the bounding circle of the disk) such that no two points are at a distance of less than 1 metre from each other?

Let P be the point of intersection of the lines 3x + 4y = 2a and 7x + 2y = 2018, and Q the point of intersection of the lines 3x + 4y = 2018 and 5x + 3y = 1. If the line through P and Q has slope 2, the value of a is:

The figure below shows two right angled triangles ΔOAB and ΔOQP with right angles at vertex A and P, respectively, having the common vertex O. The lengths of some of the sides are indicated in the figure: OA = 3, AB = 1, OP = 2, PQ = 1. (Note that the figure is not drawn to scale…

When opening his fruit shop for the day a shopkeeper found that his stock of apples could be perfectly arranged in a complete triangular array: that is, every row with one apple more than the row immediately above, going all the way up ending with a single apple at the top. Durin…

Consider the function f(x) = (x + 4)(x + 6)(x + 8) ⋯ (x + 98). The number of integers x for which f(x) < 0 is: