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Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Solution Given AB = CD i.e. two equal chords. To Prove AB = CD Proof From ΔAOB and ΔCOD, OA = OC and OB =... View Article

The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

Let us say that the diameter of the earth is “d”. Hence, the radius of the earth will be d/2 The diameter of the moon... View Article

Suppose you drop a die at random on the rectangular region shown in Fig. What is the probability that it will land inside the circle with diameter 1m?

At first, calculate the area of the rectangle and the area of the circle. Here, the area of the rectangle is the possible outcome and the area... View Article

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17 ? Perform the division to check your answer.

Solution: 1/17 Dividing 1 by 17: There are 16 digits in the repeating block of the decimal expansion of 1/17. The remainder never becomes... View Article

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see Fig 6.41) . Show that ΔABC ~ ΔPQR.

Given that ΔABC and ΔPQR, AB, BC and median AD of ΔABC are proportional to sides PQ, QR and median PM of ΔPQR To... View Article

You know that 1/7 = 0.142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?

Solution Given that... View Article

A car takes 2 hours to reach a destination by travelling at the speed of 60 km/hr. How long will it take when the car travels at the speed of 80 km/hr?

Solution Let the number of machines required be x. The number of machines and the number of days are inversely proportional. Therefore, from... View Article

Find the area of a rhombus whose side is 5 cm and whose altitude is 4.8 cm. If one of the diagonals is 8 cm long, find the length of the other diagonal.

We know that a rhombus is also a kind of a parallelogram. Formula Area of rhombus = Base×Altitude On substituting the known values we... View Article

The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency f.

Daily Pocket Allowance(in c) 11-13 13-15 15-17 17-19 19-21 21-23 23-35 Number of children 7 6 9 13 f 5 4 Solution To determinet... View Article

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig.) . Show that these altitudes are equal.

Solution Given AB=AC, BE and CF are altitudes To Prove BE=CF Proof From ∆AEB and ∆AFC, ∠E=∠F ∠A=∠A AB=AC... View Article

What will be the unit digit of the squares of the following numbers? i) 81 ii) 272 iii) 799 iv) 3853 v) 1234 vi) 26387 vii) 52698 viii) 99880 ix) 12796 x) 55555

Solution To find the unit digit of the square of any number, we have to find the square of the unit digit of that number and its unit digit will... View Article

In Fig., A,B and C are three points on a circle with centre O such that BOC = 30° and AOB = 60°. If D is a point on the circle other than the arc ABC, find ADC.

Given: ∠BOC=30° and ∠AOB=60°To find: ∠ADCSolution: ∠AOC=2∠ADC [Angle subtended by an arc at the centre is double... View Article

The number of hours for which students of a particular class watched television during holidays is shown through the given graph. Answer the following: (i) For how many hours did the maximum number of students watch T.V.? (ii) How many students watched TV for less than 4 hours? (iii) How many students spent more than 5 hours in watching TV?

Solution (i) 32 students watched T.V for 4-5 hours. ∴, The maximum number of students who watched T.V. for 4 – 5 hours. From the... View Article

Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000?

Given data The incomes of Subba Rao increases every year by Rs.200 and hence, forms an AP. Find out We need to find out the year his income... View Article

In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 12 m high. If the length if the ship is 28 m, how long is the model ship?

Let the length of the model ship be x . The length of the mast and the actual length of the ship are in direct proportion. So, 12/9... View Article

The capacity of a closed cylindrical vessel of height 1m is15.4 liters. How many square meters of metal sheet would be needed to make it? (Assume π = 22/7)

Given Height of cylinder (h) = 1 m = 100 cm Volume of cylinder (V) = 15.4 liters Find out We have to find out how many square meters of the... View Article

In a hot water heating system, there is cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system. (Assume π = 22/7)

Given Length of cylindrical pipe (h)= 28m = 2800 cm Diameter of cylindrical pipe=5 cm Find out We have to find thetotal radiating surface in the... View Article

Curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find (i) radius of the base and (ii) total surface area of the cone. (Assume π = 22/7)

Given Curved surface area of a cone is 308 cm2 Slant height is 14 cm. Find out We have to find (i) radius of the base (ii) total... View Article

In Fig, O is a point in the interior of a triangle. ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that: (i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2 , (ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2.

Given In ΔABC, O is a point in the interior of a triangle. And OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. To Prove (i)... View Article

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Solution Given: Equal chord AB & CD intersect at E. To find: AE=ED and EB=EC Construction: Draw OM perpendicular AB, ON... View Article