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A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Given Side of a cube = 12cm Solution We know volume of cube = (side)³ = (12)³ = 1728m³ Surface area of cube = 6a² =... View Article

The following data gives the information on the observed lifetimes (in hours) of 225 electrical components: Determine the modal lifetimes of the components.

Lifetime (in hours) 0-20 20-40 40-60 60-80 80-100 100-120 Frequency 10 35 52 61 38 29 Solution According to the given parameters The... View Article

The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.

Let AB be the height of the tower and C is the point elevation which is 30 m away from the foot of the tower. To Find: AB (height of the tower)... View Article

Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

Given: Two congruent circles which intersect at A and B. PAB is a line segment To prove: BA=BQ Construction: Let us join AB... View Article

Curved surface area of a right circular cylinder is 4.4 m2. If the radius of the base of the base of the cylinder is 0.7 m, find its height. (Assume π = 22/7)

Given Curved surface area of a right circular cylinder is 4.4 m2 Radius of the base of the base of the cylinder is 0.7 m Find out We have... View Article

In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see Fig.) . Show that the line segments AF and EC trisect the diagonal BD.

Given that ABCD is a parallelogram. E and F are the mid-points of sides AB and CD respectively. To Prove AF and EC trisect the diagonal BD.... View Article

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that BAD = ABE and EPA = DPB (see Fig.) . Show that (i) ΔDAP ΔEBP (ii) AD = BE

Solution Given P is the mid-point of line segment AB. BAD = ABE and EPA = DPB To Prove (i) ∆ DAP ≅ ∆ EBP (ii) AD = BE Proof ∠ BAD = ∠ ABE... View Article

A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.

Given Diameter of cylinder = 7 m Radius of cylinder (r)= 7/2m Height of cylinder (h)= 20 m Length of Cuboid (l) = 22 m Breadth of Cuboid (b)... View Article

The area of a trapezium is 34 cm ² and the length of one of the parallel sides is 10 cm and its height is 4 cm Find the length of the other parallel side.

Let the length of the other parallel side be x. Length of one parallel side, a = 10 cm height, (h) = 4 cm and Area of a trapezium is 34... View Article

D, E and F are respectively the mid-points of the sides BC, CA and AB of a ΔABC. Show that (i) BDEF is a parallelogram. (ii) ar(DEF) = 1/4 ar(ABC) (iii) ar (BDEF) = 1/2 ar(ABC)

(i) From ΔABC, EF || BC and EF = ½ BC (by mid point theorem) BD = ½ BC (D is the mid point) So, BD = EF also, BF and DE... View Article

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) 1/4 , -1 (ii) √2, 1/3 (iii) 0, √5 (iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1

From the formulas of sum and product of zeroes, we know, Sum of zeroes = α+β Product of zeroes = α β (i)1/4, -1 We know... View Article

Solve the following pair of linear equations by the substitution method (i) 0.2x + 0.3y = 1.3 0.4x + 0.5y = 2.3 (ii) √2 x + √3 y = 0 √3 x-√8 y = 0 (iii) (3x/2) – (5y/3) = -2 (x/3) + (y/2) = (13/6)

(i) Given equations 0.2x + 0.3y = 1.3 0.4x + 0.5y = 2.3 From 1st equation, we get, x = (1.3- 0.3y)/0.2 _________________(1) Now,... View Article

Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula. (i) 2x² – 7x + 3 = 0 (ii) 2x² + x – 4 = 0 (iii) 4x² + 4√3x + 3 = 0 (iv) 2x² + x + 4 = 0

(i) 2x2 – 7x + 3 = 0 Let us compare the given equation with ax2 + bx + c = 0, we get,... View Article

The given distribution shows the number of runs scored by some top batsmen of the world in one- day international cricket matches. Find the mode of the data.

Run Scored Number of Batsman 3000-4000 4 4000-5000 18 5000-6000 9 6000-7000 7 7000-8000 6 8000-9000 3 9000-10000 1 10000-11000... View Article

Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

Let us assume ABC is a cone. From the cone, the frustum DECB is cut by a plane parallel to its base. Here, r1 and r2 are the radii of... View Article

In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in Fig. Find the area of the design.

Solution Given, Radius of the circular design = r = 32 cm ABC is an equilateral triangle. Area of circle = πr2 = (22/7) × 32 × 32 =... View Article

A hemispherical bowl made of brass has inner diameter 10.5cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm2. (Assume π = 22/7)

Given Inner diameter 10.5cm Find out The cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm2. Solution We know that Formula for... View Article

A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use π = 3.14 and √3 = 1.73)

Radius, r = 12 cm Draw a perpendicular OD on chord AB and it will bisect chord AB. So, AD = DB Radius=r=12cm Area of triangle=1/2*base*height... View Article

Praveen wanted to make a temporary shelter for her car, by making a box – like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up) . Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5m, with base dimensions 4m×3m?

Given l = 4m b = 3m h = 2.5m Find out We have to find the tarpaulin required to make the shelter Area of tarpaulin required = lb + 2lh +... View Article

The radius of a spherical balloon increases from 7cm to 14cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.

Given The radius of a spherical balloon increases from 7cm to 14cm Find out We have to find the ratio of surface areas of the balloon in the two... View Article