There are two circles C₁ , and C₂ of radii 3 and 8 units respectively. The common internal tangent T, touches the circles at points P and Q respectively. The line joining the centers of the circles intersects T at X. The distance of X from the center of the smaller circle is 5 units. What is the length of the line segment PQ?

Triangle ABC is a right-angled triangle. D and E are midpoints of AB and BC respectively. Read the following statements. (i) AE = 19 (ii) CD = 22 (iii) Angle B is right angle. Which of the following statements would be sufficient to determine the length of AC?

There are two squares S₁ and S₂ with areas 8 and 9 units, respectively. S1 is inscribed within S₂ , with one corner of S₁ on each side S₂ . The corners of the smaller square divides the sides of the bigger square into two segments, one of length ‘a’ and the other of length ‘b’, where, b > a. A possible value of ‘b/a’, is:

A circular road is constructed outside a square field. The perimeter of the square field is 200 ft. If the width of the road is 7 x $$\sqrt[]{2}$$   ft. and cost of construction is Rs.100 per sq. ft. Find the lowest possible cost to construct 50% of the total road.

In the diagram below, CD = BF = 10 units and $$\angle CED$$   = $$\angle BAF$$   = 30°. What would be the area of triangle AED? (Note: Diagram below may not be proportional to scale.)

The parallel sides of a trapezoid ABCD are in the ratio of 4: 5. ABCD is divided into an isosceles triangle ABP and a parallelogram PBCD (as shown below). ABCD has a perimeter equal to 1120 meters and PBCD has a perimeter equal to 1000 meters. Find Sin $$\angle ABC$$  , given 2 $$\angle DAB=\angle BCD.$$  

A person is standing at a distance of 1800 meters facing a giant clock at the top of a tower. At 5.00 p.m., he can see the tip of the minute hand of the clock at 30 degree elevation from his eye-level. Immediately, the person starts walking towards the tower. At 5.10 pm., the person noticed that the tip of the minute hand made an angle of 60 degrees with respect to his eye-level. Using three-dimensional vision, find the speed at which the person is walking. The length of the minutes hand is $$200\ \sqrt[]{3}$$   meters $$\left(\sqrt[]{3}=1.732\right)$$  

∆ABC and ∆XYZ  are equilateral triangles of 54 cm sides. All smaller triangles like ∆ANM, ∆OCP, ∆QPX etc. are also equilateral triangles. Find the area of the shape MNOPQRM.

In the figure below, two circular curves create 60° and 90° angles with their respective centers. If the length of the bottom curve Y is $$10\pi$$  , find the length of the other curve.

AB, CD and EF are three parallel lines, in that order. Let d₁ and d₂ be the distances from CD to AB and EF respectively. d₁ and d₂ are integers, where d₁ :d₂ = 2:1, P is a point on AB, Q and S are points on CD and R is a point on F-F, If the area of the quadrilateral PQRS is 30 square units, what is the value of QR when value of SR is the least?

The volume of a pyramid with a square base is 200 cm³ . The height of the pyramid is 13 cm. What will be the length of the slant edges (i.e. the distance between the apex and any other vertex), rounded to the nearest integer?