In the following figure RP: PK= 3:2, then find the value of A(ΔTRP):A(ΔTPK).

Concept: Properties of Ratios of Areas of Two Triangles

If two circles with radii 8 cm and 3 cm, respectively, touch internally, then find the distance between their centers.

Concept: Touching Circles

If the angle θ = -60° , find the value of sinθ .

Concept: Trigonometric Ratios of Complementary Angles

Find the slope of the line passing through the points A(2,3) and B(4,7).

Concept: Slope of a Line

The radius of a circle is 7 cm. find the circumference of the circle.

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle

If the sides of a triangle are 6 cm, 8 cm and 10 cm, respectively, then determine whether the triangle is a right angle triangle or not.

Concept: Pythagoras Theorem

In the figure given below, Ray PT is bisector of ∠QPR. If PQ = 5.6 cm, QT = 4 cm and TR = 5 cm, find the value of x .

Concept: Similarity of Triangles

In the following figure, Q is the center of the circle. PM and PN are tangents to the circle. If ∠MPN = 40° , find ∠MQN.

Concept: Number of Tangents from a Point on a Circle

Write the equation 2x - 3y - 4 = 0 in the slope intercept form. Hence, write the slope and y-intercept of the line.

Concept: Intercepts Made by a Line

If `cosθ=1/sqrt(2)`, where θ is an acute angle, then find the value of sinθ.

Concept: Trigonometric Ratios of Complementary Angles

If (4,-3) is a point on the line AB and slope of the line is (-2), write the equation of the line AB.

Concept: General Equation of a Line

Draw a tangent at any point ‘P’ on the circle of radius 3.5 cm and centre O.

Concept: Construction of Tangent to the Circle from the Point on the Circle

In a triangle ABC, line l || Side BC and line l intersects side AB and AC in points P and Q, respectively. Prove that: `"AP"/"BP"="AQ"/"QC"`

Concept: Similarity of Triangles

In figure, ΔABC is an isosceles triangle with perimeter 44 cm. The base BC is of length 12 cm. Side AB and side AC are congruent. A circle touches the three sides as shown in the figure below. Find the length of the tangent segment from A to the circle.

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle

Draw tangents to the circle with center ‘C’ and radius 3.6 cm, from a point B at a distance of 7.2 cm from the center of the circle.

Concept: Construction of Tangents to a Circle

Prove that:

sec^{2}θ + cosec^{2}θ = sec^{2}θ x cosec^{2}θ

Concept: Trigonometric Identities

Write the equation of each of the following lines:

- The x-axis and the y-axis.
- The line passing through the origin and the point (-3, 5).
- The line passing through the point (-3, 4) and parallel to X-axis.

Concept: General Equation of a Line

From the top of a lighthouse, an observer looks at a ship and finds the angle of depression to be 60° . If the height of the lighthouse is 90 meters, then find how far is that ship from the lighthouse? (√3 = 1.73)

Concept: Heights and Distances

Prove that the “the opposite angles of the cyclic quadrilateral are supplementary”.

Concept: Cyclic Properties

The sum of length, breadth and height of a cuboid is 38 cm and the length of its diagonal is 22 cm. Find the total surface area of the cuboid.

Concept: Surface Area of a Combination of Solids

In triangle ABC, ∠C=90°. Let BC= a, CA= b, AB= c and let 'p' be the length of the perpendicular from 'C' on AB, prove that:

1. cp = ab

2. `1/p^2=1/a^2+1/b^2`

Concept: Pythagoras Theorem

Construct the circumcircle and incircle of an equilateral triangle ABC with side 6 cm and centre O. Find the ratio of radii of circumcircle and incircle.

Concept: Division of a Line Segment

There are three stair-steps as shown in the figure below. Each stair step has width 25 cm, height 12 cm and length 50 cm. How many bricks have been used in it, if each brick is 12.5 cm x 6.25 cm x 4 cm?

Concept: Heights and Distances