In the given figure,  is the diameter of a circle with center . Two points  and  are chosen on the circle such that  . When  and  are extended, they meet at . The value of  is

Spheres of unit diameter are centered at (𝑙 𝑚 𝑛) where 𝑙 𝑚 and l take every possible integer values.

The distance between two spheres is computed from the center of one sphere to the center of the other sphere. For a given sphere, x is the distance to its nearest sphere and y is the distance to its next nearest sphere. The value of  is:

In the triangle PQR, the lengths of PT and TR are in the ratio of 3:2.

ST is parallel to QR. Two semicircles are drawn with PS and PQ as diameters, as shown in the figure. Which one of the following statements is true about the shaded area PQS?

Note: The figure shown is representative.

In the following figure,        

Points AFCD create a rhombus.

The length of  (in ) is

A surveyor has to measure the horizontal distance from her position to a distant reference point C. Using her position as the center, a 200 m horizontal line segment is drawn with the two endpoints A and B. Points A, B, and C are not collinear. Each of the angles  and  are measured as . The distance (in m) of the reference point C from her position is nearest to

A square with sides of length 6 cm is given. The boundary of the shaded region is defined by two semi-circles whose diameters are the sides of the square, as shown.

The area of the shaded region is

The letters P, Q, R, S, T and U are to be placed one per vertex on a regular convex hexagon, but not necessarily in the same order.

Consider the following statements:

• The line segment joint R and S is longer than the line segment joining P and Q.
• The line segment joining R and S is perpendicular to the line segment joining P and Q.

The line segment joining R and U is parallel to the line segment joining T and Q.

Based on the above statements, which one of the following options is CORRECT?  

For a regular polygon having 10 sides, the interior angle between the sides of the polygon, in degrees, is:

A transparent square sheet below is folded along the dotted line. The folded sheet will look like __________.

In the figure shown below, each inside square is formed by joining the midpoints of the sides of the next larger square. The area of the smallest square (Shaded) as shown, in  is

Given a semicircle with O as the centre; as shown in the figure, the ratio  is ________. Where  and  are chords.

In a  rectangle grid shown below, each cell is a rectangle. How many rectangles can be observed in the grid?

In a triangle PQR, PS is the angle bisector of  and . What is the length of PS?

Four branches of a company are located at M, N, O, and P. M is north of N at a distance of 4 Km; P is south of O at a distance of 2Km; N is southeast of O by 1Km. What is the distance between M and P in Km?

When a point inside of a tetrahedron (a solid with four triangular surfaces) is connected by straight lines to its corners, how many (new) internal planes are created with these lines?