In the following figure seg AB ⊥ seg BC, seg DC ⊥ seg BC. If AB = 2 and DC = 3, find `(A(triangleABC))/(A(triangleDCB))`

Concept: Properties of Ratios of Areas of Two Triangles

Find the slope and y-intercept of the line y = -2x + 3.

Concept: Intercepts Made by a Line

In the following figure, in ΔABC, BC = 1, AC = 2, ∠B = 90°. Find the value of sin θ.

Concept: Heights and Distances

Find the diagonal of a square whose side is 10 cm.

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

The volume of a cube is 1000 cm3. Find the side of a cube.

Concept: Surface Area of a Combination of Solids

If two circles with radii 5 cm and 3 cm respectively touch internally, find the distance between their centres.

Concept: Touching Circles

If sin θ =3/5, where θ is an acute angle, find the value of cos θ.

Concept: Trigonometric Ratios of Complementary Angles

Draw ∠ABC of measure 105° and bisect it.

Concept: Basic Geometric Constructions

Find the slope of the line passing through the points A(-2, 1) and B(0, 3).

Concept: Slope of a Line

Find the area of the sector whose arc length and radius are 8 cm and 3 cm respectively.

Concept: Areas of Sector and Segment of a Circle

In the following figure, in Δ PQR, seg RS is the bisector of ∠PRQ.

PS = 3, SQ = 9, PR = 18. Find QR.

Concept: Similarity of Triangles

In the following figure, if m(are DXE) = 90° and m(are AYC) = 30°. Find ∠DBE.

Concept: Areas of Sector and Segment of a Circle

In the following figure, Q is the centre of a circle and PM, PN are tangent segments to the circle. If ∠MPN = 50°, find ∠MQN.

Concept: Number of Tangents from a Point on a Circle

Draw the tangents to the circle from the point L with radius 2.7 cm. Point ‘L’ is at a distance 6.9 cm from the centre ‘M’.

Concept: Construction of Tangents to a Circle

The ratio of the areas of two triangles with the common base is 14 : 9. Height of the larger triangle is 7 cm, then find the corresponding height of the smaller triangle.

Concept: Properties of Ratios of Areas of Two Triangles

Two building are in front of each other on either side of a road of width 10 metres. From the top of the first building which is 40 metres high, the angle of elevation to the top of the second is 45°. What is the height of the second building?

Concept: Heights and Distances

Find the volume and surface area of a sphere of radius 2.1 cm.

`(pi=22/7)`

Concept: Surface Area of a Combination of Solids

Prove that ‘the opposite angles of a cyclic quadrilateral are supplementary’.

Concept: Cyclic Quadrilateral

Prove that sin^{6}θ + cos^{6}θ = 1 – 3 sin^{2}θ. cos^{2}θ.

Concept: Trigonometric Identities

A test tube has diameter 20 mm and height is 15 cm. The lower portion is a hemisphere. Find the capacity of the test tube. (π = 3.14)

Concept: Surface Area of a Combination of Solids

Prove that the angle bisector of a triangle divides the side opposite to the angle in the ratio of the remaining sides.

Concept: Similarity of Triangles

Write down the equation of a line whose slope is 3/2 and which passes through point P, where P divides the line segment AB joining A(-2, 6) and B(3, -4) in the ratio 2 : 3.

Concept: Division of a Line Segment

ΔRST ~ ΔUAY, In ΔRST, RS = 6 cm, ∠S = 50°, ST = 7.5 cm. The corresponding sides of ΔRST and ΔUAY are in the ratio 5 : 4. Construct ΔUAY.

Concept: Division of a Line Segment