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QUESTION: 1

A deer and a rabbit can complete a full round on a circular track in 9 minutes and 5 minutes respectively. P, Q, R and S are the four consecutive points on the circular track which are equidistant from each other. P is opposite to R and Q is opposite to S. After how many minutes will they meet together for the first time at the starting point, when both have started simultaneously from the same point in same direction?

Solution:

Time taken by a deer to complete one round = 9 minutes

Time taken by a rabbit to complete one round = 5 minutes

They meet together for the first time at the starting point = LCM of 9 and 5 = 45 minutes.

QUESTION: 2

A deer and a rabbit can complete a full round on a circular track in 9 minutes and 5 minutes respectively. P, Q, R and S are the four consecutive points on the circular track which are equidistant from each other. P is opposite to R and Q is opposite to S. After how many minutes will they meet together for the first time, when both have started simultaneously from the same point in same direction(in min)?

Solution:

Circumference of the track = LCM of 9 and 5 = 45 m.

Ratio of time of deer and rabbit = 9 : 5

Ratio of speed of deer and rabbit = 5 : 9

Relative Speed = 4 m/min

They meet together for the first time at the starting point = 45/4 min

QUESTION: 3

A cylindrical cistern whose diameter is 14 cm is partly filled with water. If a rectangular block of iron 22 cm in length, 14 cm in breadth and 7 cm in thickness is wholly immersed in water, by how many centi metre will the water level rise?

Solution:

Volume of the block = 22 * 14 * 7

Radius of the cistern = 14/2 = 7

Volume of the Cylinder = 22/7 * R^{2} * h

22/7 * R^{2} * h = 22/7 * 7 * 7 * h

22/7 * 7 * 7 * h = 22 * 14 * 7 ⇒ h = 14

QUESTION: 4

A well with 28 m inside diameter is dug out 18 m deep. The earth taken out of it has been evenly spread all around it to a width of 21 m to form an embarkment. Find the height of the embarkment.

Solution:

22/7[(R^{2}) – (r^{2})] * h = 22/7(7*7*18)

[(35^{2}) – (7^{2})]h = 14 * 14 * 18

(42*28)h = 14*14*18

h = 3 m

QUESTION: 5

The radii of two cylinders are in the ratio 4:5 and their heights are in the ratio 5:7, What is the ratio of their curved surface areas?

Solution:

2πr1h1/2πr2h2= [4/5 * 5/7] = 4:7

QUESTION: 6

The ratio between the sides of a room is 3:2. The cost of white washing the ceiling of the room at 5 Rs per square metre is Rs. 2500 and the cost of papering the walls at Rs. 2 per square metre is Rs. 960. The height of the room is?

Solution:

Area of Ceiling = Total Cost / Cost of 1 sq. Unit

= 2500/5 = 500

l:b = 2x:3x

l*b = 5x^{2} = 500

l = 30m & b = 20 m

Area of the 4 wall = 960/2 = 480

Height = 480 / 2(30 + 20) = 4.8m

QUESTION: 7

A park is in the form of a square one of whose sides is 50 m. The area of the park excluding the circular lawn in the centre of the park is 1884 m². The radius of the circular lawn is ?

Solution:

Area of park = 50 x 50 = 2500 m²

Area of circular lawn = Area of park – area of park excluding circular lawn

= 2500 – 1884

= 616

Area of circular lawn = (22/7) x r² = 616 m²

⇒ r² = (616 x 7) / 22

= 28 x 7

= 2 x 2 x 7 x 7

∴ r = 14 m

QUESTION: 8

The perimeter of a rectangle and a square is 160 cm each. If the difference between their areas is 600 cm. Find the area of the rectangle.

Solution:

Perimeter of rectangle = Perimeter of Square = 160

4a = 160 ⇒ a = 40

Area of square = 1600

1600 – lb = 600

lb = 1000 cm²

QUESTION: 9

The length of a plot is four times its breath. A playground measuring 400 square meters occupies one fourth of the total area of a plot. What is the length of the plot in meter.?

Solution:

Area of the plot = (4 x 400) m²

= 1600 m²

Breadth = y meter

Length = 4y meter

Now area = 4y x y = 1600 m²

⇒ y² = 400 m²

⇒ y = 20 m

∴ Length of plot = 4y =80 m

QUESTION: 10

If the radius of the cone is doubled, keeping the height constant, what is the ratio of the volume of the smaller cone to larger cone?

Solution:

v1/v2 = (r1²) * h1 / (r2²) * h2 (h1=h2)

(r)²/(2r)²

v1/v2 = 1:4

QUESTION: 11

A room has floor size of 15*6sq cm. What is the height of the room , if the sum of the areas of the base and roof is equal to the sum of the areas of the four walls ?

Solution:

lb+lb = lh+hb+lh+hb

2lb = 2h(l+b)

h = lb/l+b

h=15*6/15+6 = 4.29 cm

QUESTION: 12

What is the volume of a right cone whose cross section is isosceles triangle with a base 10cm and slant height 13cm ?

Solution:

Base of the cone(r) =10/2 = 5 cm

Height of the cone (h)= √169-25 =√144 = 12

Volume = 1/3πr^{2} h

= 22*5*5*12/7*3 = 314.2 Sq cm

QUESTION: 13

Smallest side of a right angled triangle is 6 cm less than the side of a square of perimeter 60 cm. Second largest side of the right angled triangle is 4 cm less than the length of rectangle of area 80 sq. cm and breadth 5 cm. What is the largest side of the right angled triangle?

Solution:

Side of 1^{st} square = 60/4 = 15 cm.

Smallest side of right angled triangle= 15 −6 = 9 cm.

Length of 2^{nd} rectangle = 80/5 = 16 cm.

Second largest side of the 1strectangle = 16−4 = 12 cm.

Largest side = hypotenuse=√9^{2}+12^{2}=15cm

QUESTION: 14

Circumference of a circle-A is 5/4 times perimeter of a square. Area of the square is 961 sq. cm. What is the area of another circle-B whose diameter is half the radius of the circle-A?

Solution:

The side of the square = a=√961=31 cm.

Perimeter = 4 X 31 = 124 cm The circumference of the circle = 5/4 x 124 = 22/7 x 2 x r

Radius = 155*7/44 = 24.6 cm

Half the radius of Circle-A = 12.3 cm.

Radius of Circle-B = 14/2 = 6.15 cm

The area = 22/7 x 6.15 x 6.15 = 54 cm^{2}

QUESTION: 15

A horse is tethered to a peg with a 16 cm long rope at the corner of a 31 cm long and 27cm wide rectangular grass field. What area of the field will the horse graze?

Solution:

Area = [22*16*16/7] / 4

= 5632/7*4 = 201.1 cm^{2}

QUESTION: 16

The biggest possible circle is inscribed in rectangle of length 10 m and breadth 7 m. Then the area of circle is?

Solution:

Radius = Breadth /2 = 7/2 = 3.5

Area = 22*3.5*3.5/7 = 269.5/7 = 38.5 m^{2}

QUESTION: 17

In a swimming pool measuring 80 cm x 30 cm, 120 men take a dip. If the average displacement of water by a man is 5 cm cube, What will be the rise in water level ?

Solution:

Total volume displaced by 8 men= 120×5 cm cube

However volume=lxbxh=80x30xh

80x30xh=120×5

h=120*5/80*30 = 600/2400 = 0.25.

so the water level rises by 0.25 cm = ¼ cm

QUESTION: 18

A cylindrical tank of diameter 14 cm is full of water. If 9 litres of water is drawn off, the water level in the tank will drop by

Solution:

1 litres = 1000 cm

9 litres = 9000 cm

Volume of cylinder = πr^{2}h

22/7*7*7*h = 11000

h = 9000*7*4/22*7*7 = 233.8 cm

QUESTION: 19

A circular wire of diameter 42cm is folded in the shape of a rectangle whose sides are in the ratio 6:5. Find the ratio of the enclosed rectangle ?

Solution:

Radius = 42/2 = 21cm

Circumference = 2*22*21/7 = 132cm

L:b = 6x:5x

Perimeter of the rectangle = 2(6x+5x) = 22x

22x = 132

X = 6

Area = 6x * 5x = 30x*x

=30*6*6 = 1080cm^{2}

QUESTION: 20

A cylindrical tub of radius 10 cm contains water up to a depth of 25cm. A spherical iron ball is dropped into the tub and thus the level of water is raised by 5cm. What is the radius of the iron ball(approximate) ?

Solution:

Volume of the ball = volume of the raised water

4*22*r^{3} /3*7 = 22*100*5

r^{3} = 11000*3*7/88 = 2625

r = 13.79 or 14

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