Mathematics Part 2

Surface Areas and Volumes Question 1.2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid. Solution:Let the length of each edge of the cube be a cm.Then, Volume = 64 cm³ So, a³ = 64 or a = 4When two cubes of equal volumes (i.e […]

Areas Related to Circles Question 1.Find the area of the shaded region in the figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle. Solution:Since ROQ is a diameter, therefore ∠RPQ = 90°.RQ² = RP² + PQ²or RQ² = 7² + 24² or RQ² = 49 +

Question 1.Find the area of a sector of a circle with radius 6 cm, if angle of the sector is 60°. Solution:We know that the area A of a sector of angle θ in a circle of radius r is given by A = θ/360∘ x πr².Here, r = 6 cm, θ = 60°∴ A = [60∘/360∘ ×

Areas Related to Circles Question 1.The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles. Solution:Let r be the radius of the circle whose circumference is equal to the sum of the circumferences

Constructions Question 1.Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Solution:Steps of Construction :1. Take a point 0 as centre and draw a circle of radius 6 cm.2. Mark a point P at a distance

Constructions Question 1.Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts. Solution:Steps of Construction : 1. Draw a line segment AB = 7.6 cm.2. Draw a ray AC making any acute angle with AB, as shown in the figure.3. On ray AC, starting

Circles Question 1.From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is(A) 7 cm(B) 12 cm(C) 15 cm Solution:Since QT is a tangent to the circle at T and OT is the radius.

Circles Question 1.How many tangents can a circle have? Solution:A circle can have an infinite number of tangents. Question 2.Fill in the blanks : Solution: Question 3.A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ =

Some Applications of Trigonometry Question 1.A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see figure). Solution:In rt ∆

Introduction to Trigonometry Question 1.Express the trigonometric ratios sin A, sec A and tan A in terms of cot A. Solution:Consider a ∆ ABC, in which ∠B = 90°. Base = ABPerp. = BCand Hyp. = AC∴ cot A =  Base/Perp  = AB/BCor AB/BC = cot A = cotA/1Let AB = k cot A and BC = k. Question 2.Write the