Ch.13.Pythagoras' Theoram

Question 1:

In the figures below, find the value of 'x'.

ANSWER:

(i)
  
In the right angled triangle LMN, ∠M = 90^∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
l(LN)² = l(LM)² + l(MN)²
⇒(x)² = (7)² + (24)²
⇒x² = 49 + 576
⇒x² = 625
⇒x² = (25)²
⇒x = 25
∴ the value of x is 25.

(ii)

In the right angled triangle PQR, ∠Q = 90^∘. Hence, side PR is the hypotenuse.
According to Pythagoras' theorem,
l(PR)² = l(QR)² + l(PQ)²
⇒(41)² = (x)² + (9)²
⇒1681 = x² + 81
⇒x² = 1681 − 81
⇒x² = 1600
⇒x² = (40)²
⇒x = 40
∴ the value of x is 40.

(iii)

In the right angled triangle EDF, ∠D = 90^∘. Hence, side EF is the hypotenuse.
According to Pythagoras' theorem,
l(EF)² = l(ED)² + l(DF)²
⇒(17)² = (x)² + (8)²
⇒289 = x² + 64
⇒x² = 289 − 64
⇒x² = 225
⇒x² = (15)²
⇒x = 15
∴ the value of x is 15.

Page No 90:

Question 2:

In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.

ANSWER:

In the right angled triangle PQR, ∠P = 90^∘. Hence, side QR is the hypotenuse.
According to Pythagoras' theorem,
l(QR)² = l(PQ)² + l(PR)²
⇒l(QR)² = (24)² + (10)²
⇒l(QR)² = 576 + 100
⇒l(QR)² = 676
⇒l(QR)² = (26)²
⇒l(QR) = 26
∴ Length of seg QR = 26 cm.

Page No 90:

Question 3:

In the right-angled ∆LMN, ∠ M = 90°. If l(LM) = 12 cm and l(LN) = 20 cm, find the length of seg MN.

ANSWER:

In the right angled triangle LMN, ∠M = 90^∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
l(LN)² = l(MN)² + l(LM)²
⇒(20)² = l(MN)² + (12)²
⇒400 = l(MN)² + 144
⇒l(MN)² = 400 − 144
⇒l(MN)² = 256
⇒l(MN)² = (16)²
⇒l(MN) = 16
∴ Length of seg MN = 16 cm.

Page No 90:

Question 4:

The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder ?

ANSWER:

Let LN be ladder of length 15 m that is resting against a wall. Let M be the base of the wall and L be the position of the window.
The window is 9 m above the ground. Now, MN is the distance between base of the wall and that of the ladder.
In the right angled triangle LMN, ∠M = 90^∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
l(LN)² = l(MN)² + l(LM)²
⇒(15)² = l(MN)² + (9)²
⇒225 = l(MN)² + 81
⇒l(MN)² = 225 − 81
⇒l(MN)² = 144
⇒l(MN)² = (12)²
⇒l(MN) = 12
∴ Length of seg MN = 12 m.
Hence, the distance between base of the wall and that of the ladder is 12 m.

Page No 90:

Question 1:

Find the Pythagorean triplets from among the following sets of numbers.
 (i) 3, 4, 5
(ii) 2, 4, 5
(iii) 4, 5, 6
(iv) 2, 6, 7
(v) 9, 40, 41
(vi) 4, 7, 8

ANSWER:

It is known that, if in a triplet of natural numbers, the square of the biggest number is equal to sum of the squares of the other two numbers, then the three numbers form a Pythgorean triplet.

(i) The given set of numbers is (3, 4, 5).
The biggest number among the given set is 5.
5² = 25; 4² = 16; 3² = 9
Now, 16 + 9 = 25
∴ 4² + 3² = 5²
Thus, (3, 4, 5) forms a Pythagorean triplet.

(ii) The given set of numbers is (2, 4, 5).
The biggest number among the given set is 5.
5² = 25; 4² = 16; 2² = 4
Now, 16 + 4 = 20 ≠ 25
∴ 4² + 2² ≠  5²
Thus, (2, 4, 5) does not form a Pythagorean triplet.

(iii) The given set of numbers is (4, 5, 6).
The biggest number among the given set is 6.
6² = 36; 5² = 25; 4² = 16
Now, 25 + 16 = 41 ≠ 36
∴ 5² + 4² ≠  6²
Thus, (4, 5, 6) does not form a Pythagorean triplet.

(iv) The given set of numbers is (2, 6, 7).
The biggest number among the given set is 7.
7² = 49; 6² = 36; 2² = 4
Now, 4 + 36 = 40 ≠ 49
∴ 2² + 6² ≠  7²
Thus, (2, 6, 7) does not form a Pythagorean triplet.

(v) The given set of numbers is (9, 40, 41).
The biggest number among the given set is 41.
9² = 81; 40² = 1600; 41² = 1681
Now, 81 + 1600 = 1681
∴ 9² + 40² = 41²
Thus, (9, 40, 41) forms a Pythagorean triplet.

(vi) The given set of numbers is (4, 7, 8).
The biggest number among the given set is 8.
8² = 64; 7² = 49; 4² = 16
Now, 16 + 49 = 65 ≠ 64
∴ 4² + 7² ≠  8²

Page No 90:

Question 2:

The sides of some triangles are given below. Find out which ones are right-angled triangles?
(i) 8, 15, 17
(ii) 11, 12, 15
(iii) 11, 60, 61
(iv) 1.5, 1.6, 1.7
(v) 40, 20, 30

ANSWER:

It is known that, if in a triplet of natural numbers, the square of the biggest number is equal to sum of the squares of the other two numbers, then the three numbers form a Pythgorean triplet.

If the lengths of the sides of a triangle form such a triplet, then the triangle is right angled triangle.

(i) The sides of the given triangle is 8, 15 and 17.
Let us check whether the given set (8, 15, 17) forms a Pythagorean triplet or not.
The biggest number among the given set is 17.
(17)² = 289; 

(15)² = 225; (8)² = 64
Now, 225 + 64 = 289
∴ (15)² + (8)² = (17)²
Thus, (8, 15, 17) forms a Pythagorean triplet.
Hence, the given triangle with sides 8, 15 and 17 is a right-angled triangle.

(ii) The sides of the given triangle is 11, 12 and 15.
Let us check whether the given set (11, 12, 15) forms a Pythagorean triplet or not.
The biggest number among the given set is 15.
(15)² = 225; (11)² = 121; (12)² = 144
Now, 121 + 144 = 265 ≠ 225
∴ (11)² + (12)² ≠ (15)²
Thus, (11, 12, 15) does not form a Pythagorean triplet.
Hence, the given triangle with sides 8, 15 and 17 is not a right-angled triangle.

(iii) The sides of the given triangle is 11, 60 and 61.
Let us check whether the given set (11, 60, 61) forms a Pythagorean triplet or not.
The biggest number among the given set is 61.
(61)² = 3721; (11)² = 121; (60)² = 3600
Now, 121 + 3600 = 3721
∴ (11)² + (60)² = (61)²
Thus, (11, 60, 61) forms a Pythagorean triplet.
Hence, the given triangle with sides 11, 60 and 61 is a right-angled triangle.

(iv) The sides of the given triangle is 1.5, 1.6 and 1.7.
Let us check whether the given set (1.5, 1.6, 1.7) forms a Pythagorean triplet or not.
The biggest number among the given set is 1.7.
(1.7)² = 2.89; (1.5)² = 2.25; (1.6)² = 2.56
Now, 2.25 + 2.56  = 4.81 ≠ 2.89
∴ (1.5)² + (1.6)² ≠ (1.7)²
Thus, (1.5, 1.6, 1.7) does not form a Pythagorean triplet.
Hence, the given triangle with sides 1.5, 1.6 and 1.7 is not a right-angled triangle.

(v) The sides of the given triangle is 40, 20 and 30.
Let us check whether the given set (40, 20, 30) forms a Pythagorean triplet or not.
The biggest number among the given set is 40.
(40)² = 1600; (20)² = 400; (30)² = 900
Now, 400 + 900 = 1300 ≠ 1600
∴ (20)² + (30)² ≠ (40)²
Thus, (40, 20, 30) does not form a Pythagorean triplet.
Hence, the given triangle with sides 40, 20 and 30 is not a right-angled triangle.

Work and Energy.

Question 1:

Fill in each blank with the appropriate term from the brackets.
(a) A bucketful of water is to be drawn from a well. ....... will be done when a ......... is applied to do this, because there will be a ........... of water.
(displacement, work, force)
(b) If a ball is dropped on the sloping roof of a house, it acquires ...... and falls on the ground. That is, transformation of ........ energy into .......... energy takes place.
(kinetic, potential, motion)
(c) You might have seen some beautiful fireworks during Diwali. It is an example of transformation of ....... energy into....... energy.
(light, atom, chemical, solar)
(d) The solar cooker is an application of the ...... energy of the sun, while solar cells, solar lamps are applications of the ...... energy of the sun.
(light, chemical, heat)
(e) One labourer carried four pans of road metal through a distance of 100 metres. If he carries two pans of road metal through a 200 metre distance ......... work will be done.
(equal, more, less)
(f) The capacity that an object has for doing work is called ....... .
(energy, displacement, force)

ANSWER:

(a) A bucketful of water is to be drawn from a well. Work will be done when a force is applied to do this, because there will be a displacement of water.
(b) If a ball is dropped on the sloping roof of a house, it acquires motion and falls on the ground. That is, transformation of potential energy into kinetic energy takes place.
(c) You might have seen some beautiful fireworks during Diwali. It is an example of transformation of chemical energy into light energy.
(d) The solar cooker is an application of the heat energy of the sun, while solar cells, solar lamps are applications of the light energy of the sun.
(e) One labourer carried four pans of road metal through a distance of 100 metres. If he carries two pans of road metal through a 200 metre distance equal work will be done.
(f) The capacity that an object has for doing work is called energy.

Page No 83:

Question 2:

Match the pairs.

Group 'A'	Group 'B'
(1) Rolling object	(a) Heat energy
(2) Food	(b) Atomic energy
(3) Stretched bow	(c) Kinetic energy
(4) Sunlight	(d) Potential energy
(5) Uranium	(e) Chemical energy

ANSWER:

Group 'A'	Group 'B'
(1) Rolling object	(c) Kinetic energy
(2) Food	(e) Chemical energy
(3) Stretched bow	(d) Potential energy
(4) Sunlight	(a) Heat energy
(5) Uranium	(b) Atomic energy

Page No 83:

Question 3:

Can you tell ?
(a) When can we say that displacement has taken place?
(b) What should be taken into account for measuring work?
(c) What are the various forms of energy ?
(d) Describe one natural chain of transformation of energy.
(e) Why should we save energy ?
(f) What is ‘green energy’ ?
(g) What are the non-conventional energy resources ?
(h) Which forms of energy from the sun are used in solar energy devices ?
(i) Why should we maximise the use of non-conventional energy resources ?

ANSWER:

(a) We can say that displacement has taken place when the initial and final position of an object is different i.e. there is some distance between the initial and final position of an object.

(b) While measuring work, the magnitude of force applied on an object and the displacement of an object should be taken into account.

(c) Various forms of energy are:

• Mechanical: Kinetic and potential energy are two types of mechanical energy.

• Light

• Heat

• Muscular

• Sound

• Magnetic

• Electrical

• Chemical

• Nuclear

(d) When a candle is lit, the chemical energy stored in its wax gets converted to heat and light energy.

(e) We should save energy because of the following:

• To save money: Wasting energy is same as wasting money. We all know that electricity we use at our homes isn't free but costs us a lot. So, saving energy will save money too. 
• To preserve conventional sources: Most of the energy we use in our daily life is generated using conventional energy resources, such as coal, petroleum, etc. These energy resources are non-renewable in nature. So, if we waste energy, we are indirectly wasting these resources and hence a day will come when these resources will get extinct. So to preserve these non-renewable resources, we should save energy.
• To keep environment clean: Energy saving will help us keep the environment clean. Less energy usage will result in less energy production using coal, petroleum and other conventional sources which pollutes the environment when burnt.

(f) The forms of energy which do not produce smoke and carbon gases such as carbon dioxide or carbon monoxide are known as green energy.

(g) Non-conventional energy resources are inexhaustible in nature and can be used in various forms again and again. For example, wind energy, tidal energy, solar energy, etc. are non-conventional energy resources. 

(h) Heat and light energy of Sun are used in solar energy devices.

(i) Because of increase in population and usage of energy resources, there is a potential danger of limited reserves of conventional energy resources, such as coal, petrol, diesel, crude oil and natural gas, getting exhausted in future. Therefore, it will be better to maximise the use of non-conventional energy resources to remove the danger of extinction of conventional energy resources.

Page No 83:

Question 4:

Who is the odd-one-out ?
(1) Diesel, crude oil, natural gas, wind
(2) A running car, hauling a log, a book kept on a table, picking up the school bag.
(3) Sunlight, wind, waves, petrol
(4) Leaving the fan on in a vacant room, leaving the TV on while working, using AC during winter, putting off the light when going out.

ANSWER:

(1) Wind is the odd one out as it is a non-conventional energy resource. The rest are conventional energy resource.

(2) A book kept on a table is the odd one out as no work is being done in this case. In rest of the cases, work is being done.

(3) Petrol is the odd one out as it is a conventional energy resource. The rest are non-conventional energy resource.

(4) Putting off the light when going out is the odd one out as it is one of the measures of saving energy. The rest involves wastage of energy.

Page No 83:

Question 5:

Find out the types of energy from the following puzzle.

ANSWER:

Force and Types of force.

Question 1:

Choose the term to fill in the blanks.
(a) ...... has to be applied to change the ..... of a ...... object.
(moving, direction, force)
(b) When an elephant drags a wooden log over the land, the forces that are applied on the log are ...... , ....... and ........ .
(muscular force, mechanical force, gravitational force, frictional force)
(c) A ball was set rolling on a large table. If its ...... is to be changed, a ........ will have to be applied on it.
(force, motion, gravitation)
(d) The force of friction always acts ........ the motion.
(along, against)

ANSWER:

(a) Force has to be applied to change the direction of a moving object.
(b) When an elephant drags a wooden log over the land, the forces that are applied on the log are muscular force , gravitational force and frictional force.
(c) A ball was set rolling on a large table. If its motion is to be changed, a force will have to be applied on it.
(d) The force of friction always acts against the motion.

Page No 74:

Question 2:

Match the following :

Group ‘A’	Group ‘B’
(i) An ox pulling a cart	(a) Magnetic force
(ii) Lifting a heavy iron object with a crane	(b) Electrostatic force
(iii) Weighing with a spring balance	(c) Muscular force
(iv) Applying brakes to a bicycle	(d) Gravitational force
(v) Picking up pieces of paper with a plastic scale	(e) Frictional force

ANSWER:

Group ‘A’	Group ‘B’
(i) An ox pulling a cart	(c) Muscular force
(ii) Lifting a heavy iron object with a crane	(a) Magnetic force
(iii) Weighing with a spring balance	(d) Gravitational force
(iv) Applying brakes to a bicycle	(e) Frictional force
(v) Picking up pieces of paper with a plastic scale	(b) Electrostatic force

Page No 74:

Question 3:

One or more forces are acting in the following examples. Name them.
(a) An object falling from a tall building ....
(b) An aeroplane flying in sky ....
(c) Squeezing sugarcane juice with a squeezer ........ .
(d) Winnowing foodgrain ....... .

ANSWER:

(a) An object falling from a tall building: Gravitational force
(b) An aeroplane flying in sky: Mechanical and Gravitational force
(c) Squeezing sugarcane juice with a squeezer: Muscular force
(d) Winnowing foodgrain: Muscular and Gravitational force

Page No 75:

Question 4:

Explain in your own words giving one example each.
Muscular force, gravitational force, mechanical force, electrostatic force, the force of friction and magnetic force.

ANSWER:

Muscular force: The force applied by the action of muscles in our body is termed as a muscular force. For example, when you pick up a book placed on the table using your hands, you apply muscular force.

Gravitational force: It is the force that is exerted by the Earth on every object, which is near or on its surface. For example, an apple falling from a tree branch towards the ground is due to gravitational force.

Mechanical force: The force generated by the means of a machine is known as mechanical force. For example, when a car gets started, its engine creates a mechanical force on the tyres that help the car to accelerate. So here, the movement of car occurs due to the force generated by the machine on the tyres.

Electrostatic force: Electrostatic force is the force that exists either between the two charged bodies, or between a charged and an uncharged body. For example, the charged scale attracts the pieces of paper by a non-contact force known as electrostatic force.

The force of friction: The force which acts opposite to the direction of motion of a body is known as the force of friction. For example, a ball rolling on a ground stops after sometime because of frictional force acting between the ground and the ball.

Magnetic force: The force exerted by a magnet is known as magnetic force. For example, separation of iron stuffs from junk is done with the help of magnetic force.

Page No 75:

Question 5:

Why ?
(a) Machines are oiled from time to time.
(b) An object thrown upwards comes down after reaching a point.
(c) Powder is sprinkled on a carrom board.
(d) The ramp at a railway station has a rough surface.

ANSWER:

(a) A machine is oiled from time to time to reduce friction between its body parts. By doing so, life span of a machine increases.

(b) An object thrown upwards comes down after reaching a point. This is because of the Earth's gravitational pull.

(c) Powder is sprinkled on a carom board to reduce friction between the striker/carom coins and the carom board. The powder smoothens the surface of the board and thus the striker and coins can move on it easily.

(d) The ramp at a railway station has a rough surface so that the friction between our feet and the ground is enough for us to walk comfortably and without tripping.

Page No 75:

Question 6:

In what way are we different?
(a) Muscular force and mechanical force.
(b) The force of friction and gravitational force.

ANSWER:

(a) 

Muscular force	Mechanical force
The force applied by the action of muscles in our body is termed as a muscular force.	The force generated by the means of a machine is known as mechanical force.
For example, when you pick up a book placed on the table using your hands, you apply muscular force.	For example, when a car gets started, its engine creates a mechanical force on the tyres that help the car to accelerate. So here, the movement of car occurs due to the force generated by the machine on the tyres.

(b) 

The force of friction	Gravitational force
The force which acts opposite to the direction of motion of a body is known as the force of friction.	It is the force that is exerted by the earth on every object, which is near or on its surface.
It is a contact type of force.	It is non-contact type of force.
For example, a ball rolling on a ground stops after sometime because of frictional force acting between the ground and the ball.	For example, an apple falling from a tree branch towards the ground is due to gravitational force.

Page No 75:

Question 7:

Write answers to the following questions in your own words.
(a) What are the things that can be done by applying force?
(b) What is meant by weight?
(c) Which machines run on muscular force?

ANSWER:

(a) Following things can be done by applying force:

(i) It can move a body initially at rest.

(ii) It can bring a moving body to rest.

(iii) It can change the direction of a moving body.

(iv) It can change the speed of a moving body.

(v) It can change the shape of a body.

(vi) It can change the size of a body.

(b) Weight is the force exerted on a body due to the gravitational pull of Earth.

(c) Machines which run on muscular force are:

• Bicycle
• Rickshaw
• Hand pump
• Stitching machine
• Hand cart

Page No 75:

Question 8:

Solve the following crossword puzzle.

Down :
(1) ....... force is to be applied to push a scooter that has failed.
(2) ........... force can be used to pick up scattered pins.
Across :
(1) A .......... pulls an iron nail towards itself.
(3) .......... force was used when the farm was ploughed with a tractor.
(4) It is due to the force of ............ that raindrops fall to the ground

ANSWER:

Nutrition and Diet

Question 1:

Fill in the blanks.
(a) The process of intake of food and utilising it for all life processes is called .......... .
(b) All the substances in our food which are useful for various bodily processes are called .......... .
(c) Carbohydrates and .......... provide .......... to our body.
(d) In a balanced diet, all the nutrients are present in the ....... proportion.
(e) In the food pyramid, cereals are given the maximum space because they fulfil our .......... requirement.
(f) Intake of more food than necessary causes .......... .

ANSWER:

(a) The process of intake of food and utilising it for all life processes is called nutrition.
(b) All the substances in our food which are useful for various bodily processes are called nutrients.
(c) Carbohydrates and fats provide energy to our body.
(d) In a balanced diet, all the nutrients are present in the right proportion.
(e) In the food pyramid, cereals are given the maximum space because they fulfil our nutritional requirement.
(f) Intake of more food than necessary causes overnutrition.

Page No 56:

Question 2:

Spot the following in the table of vitamins and minerals.
(a) The nutrient present in citrous fruits.
(b) Vitamins / minerals present in milk.
(c) Causes and symptoms of night blindness, scurvy, rickets, beriberi.
(d) Foods required to prevent the above diseases.
(e) Causes of anaemia.
(f) Essential mineral for healthy bones and teeth.
(g) Sensory organ affected due to the deficiency of Vitamin A

ANSWER:

(a) The nutrient present in citrous fruits- Vitamin C
(b) Vitamins / minerals present in milk- Vitamin B₁₂ and Vitamin D.
(c) Causes and symptoms of night blindness, scurvy, rickets, beriberi
 

Disease	Causes	Symptoms
Night blindness	Deficiency of Vitamin A	Inability to see in dim light
Scurvy	Deficiency of Vitamin C	Bleeding of gums
Rickets	Deficiency of Vitamin D	Softening of pones causing pain and fractures
Beri beri	Deficiency of Vitamin B1	Nerve disorders

(d) Foods required to prevent the above diseases.
 

Disease	Foods to avoid these diseases
Night blindness	Carrot, milk, butter, dark green vegetables, sweet potato
Scurvy	Amla, kiwi, oranges, other citrus fruits, tomato, green leafy vegetables
Rickets	Milk, fish, eggs and butter
Beri beri	Milk, fish, meat, cereals, nuts, pulses

(e) Causes of anaemia- defeciency of iron can result in anaemia as it is required for the formation of red blood cells.
(f) Essential mineral for healthy bones and teeth- Vitamin D
(g) Sensory organ affected due to the deficiency of Vitamin A- Eyes

Page No 56:

Question 3:

Choose the correct alternative.

(a) Pulses are a very good source of .......... .

(1) carbohydrates	(2) proteins
(3) fats	(4) minerals

(b) .......... provide maximum energy to the body.

(1) Cereals	(2) Leafy vegetables
(3) Water	(4) Amla

(c) Goitre is caused by a deficiency of the mineral .......... .

(1) iron	(2) calcium
(3) iodine	(4) potassium

(d) .......... is a type of junk food.

(1) Orange	(2) Milk
(3) Bhakri	(4) Chocolate

ANSWER:

(a) Pulses are a very good source of proteins.
(b)Cereals provide maximum energy to the body.
(c) Goitre is caused by a deficiency of the mineral iodine.
(d) Chocolate is a type of junk food.

Page No 56:

Question 4:

Use the food pyramid to select food items of your choice for three days.
Conditions :
(1) The diet for all three days should be balanced.
(2) There should be variety in the items chosen for the three days.

ANSWER:

Day 1-
Cereal like cornflakes and milk, apple, roti, sabzi and dal

Day 2
Bread and butter with milk, rice/ roti and chicken, curd, any one type of fruit, salad

Day 3
Poha, green vegetable and dal with rice or roti, raita, one fruit, vegetable sandwich.

Substances in Daily Use

Substances in Daily Use

Question 1:

Fill in the blanks using proper words.
(a) Rubber made by vulcanization is a ...... material.
(b) Man-made materials are made by .......... natural materials.
(c) ............. thread was developed simultaneously in New York and London.
(d) Rayon is also known as ............

ANSWER:

(a) Rubber made by vulcanization is a man-made material.
(b) Man-made materials are made by processing natural materials.
(c) Nylon thread was developed simultaneously in New York and London.
(d) Rayon is also known as synthetic silk.

Question 2:

Answer the following questions.

ANSWER:

(a) Why did the need for man-made materials arise?

a. For several years, humans have been dependant on natural materials to meet their demands. However, there has been a shift towards the use of man made materials as well. Increasing population, deforestation and limited natural resources are some of the reasons due to which the need for man made materials arised. Another reason which led to the increase in the use of man made material was to get a comfortable lifestyle.

(b) Which are the natural materials obtained from plants and animals?

b. 

Sources	Natural material	Objects
Plants	Wood	Furniture
	Rubber	Ballon, tyres etc.
	Cotton	Clothes
Animals	Fur	Hats, Sweaters
	Leather	Handbags, shoes
	Silk	Sarees

(c) What is vulcanization?
c. Vulcanization is the process carried out for the production of rubber. In this process, raw rubber is heated with a mixture of sulphur and an appropriate additive, at a temperature range of 61.85${\mathrm{<merror><mtext>Unknown\; node\; type:\; sup</mtext></merror>}}^{}$ C to 141.85⁰C for three to four hours. Sulphur is added to give toughness to rubber.

(d) Which natural materials are used to obtain fibres?

d. Natural materials obtained from plants and animals are used to obtain fibers. For example, wool and silk are natural materials of animal origin which are used for obtaining fibers. Wool is obtained from the fleece (hair) of sheep or yak and silk is obtained from the cocoons of a silk moth. Cotton and jute are examples of natural materials of plant origin which are used for obtaining fibers.

Question 3:

What are we used for ?
(a) Soil
(b) Wood
(c) Nylon
(d) Paper
(e) Rubber

ANSWER:

a. Soil- It is used in construction, pottery, medicine and cosmetics.
b. Wood- It is used in making furnitures, as a source of fuel, utensils, hand tools, musical instruments, wooden toys etc.
c. Nylon- It is used in making canopies of parachutes, tents, sleeping bags, sails, rope, tennis strings, fishing poles and lines, cookware etc.
d. Paper- It is used in making bags, envelopes, notebooks, packaging purpose etc.
e. Rubber- Balloons, erasers, rubber toys, rubber balls etc.

Question 4:

How is paper manufactured? Write in your own words.

ANSWER:

Paper is manufactured by the following process:

• Logs of wood are made to pass through a debarker, where the barks of the tree are separated.
• Then the wood is chopped into small pieces and is mixed with some chemicals and water in digetser for the formation of the pulp.
• The obtained pulp is washed refined and sometimes even bleached.
• The pulp is made to travel down the screen to remove excess of water and then to the rollers where it is being dried and squeezed between large rollers to form rolls of paper. 

Question 5:

Give scientific reasons.

ANSWER:

(a) We must use cotton clothes in summer.

a. We must use cotton clothes in summers because they protect us from the heat of the sun. They absorb sweat and help keep our bodies cool. The pores in cotton clothes allow the free movement of air, which helps cool our bodies.

(b) We must observe economy in the use of materials.
b. We must observe economy in the use of materials because they directly or indirectly affect our environment. Most of the natural materials which we obtain from our environment are exhaustible/non renewable resources i.e. once these resources are finished they cannot be renewed again or even if they can, it would take several years to do so. For example, our forests from where we obtain wood are precious resources because once a forest is destroyed, it would take several years to renew that forest. Also, destroying the forests for our needs affects the habitat of several other species of organisms and plants. The man made materials which we use are largely non biodegradable for example plastic etc. Increased use of man made materials pose a threat to the environment because their disposal is not easy and leads to environmental pollution. These examples state the importance of judicious use of both natural and man made materials.

(c) Saving paper is the need of the hour.

c. We very well know that paper is obtained from trees. Trees are important resources because they not only provide us with wood or paper but are also a habitat for various organisms. They are important in maintaining the climate as well. The more paper we use or waste, the more no. of trees will have to be cut down for this purpose, thus leading to v arious kinds of harmful effects on our environment. We should make a genuine effort in saving paper by adopting small steps which can make a huge difference.

(d) Man-made materials have more demand.

d. Man made materials have more demand because of their characteristics like easy to use, low cost of production, cheaper products and variety of things which can be made from them.

(e) Humus is a natural material.

e. Humus is a natural material because its formation is a natural process. It is the dark-coloured layer of top soil that is rich in nutrients. The decomposers convert the dead plants and animals into humus that is utilised by the plants. It increases the fertility of the soil.

Question 6:

Find out.
(1) How is lac obtained from nature?

ANSWER:

a. Lac is a natural resin and is a product of animal origin. It is secreted by an insect, known as lac-insect. In order to obtain lac, these insects are cultured and the technique is called lac-culture. Lac culture requires proper care of host plants, regular pruning of host plants, propagation, collection and processing of lac. The lac insect lives as a parasite, feeding on the sap of certain trees and shrubs. Lac secretion begins as soon as the larva of the insect settles on the plant. Intially, lac has a shining appearance but later on it turns harder on coming in contact wih air.
This lac is then collected and used for making jewellery etc.


(2) How are pearls obtained?
b. Pearls are precious jewels and the only gems created by a living animal. They are formed when a microscopic foreign object finds its way inside a mussel or oyster.

The animal coats the object with a substance called nacre. With time, several layers of nacre build up over the foreign material thus leading to the formation of a pearl. These oyesters/mussels are collectd from the sea and opened to collect these pearls. 

HCF and LCM

Page No 15:

Question 1:

Which number is neither a prime number nor a composite number?

ANSWER:

1 is neither a prime number nor a composite number.

Page No 15:

Question 2:

Which of the following are pairs of co-primes?
(i) 8, 14
(ii) 4, 5
(iii) 17, 19
(iv) 27, 15
 

ANSWER:

Two numbers which have only 1 as a common factor are said to be co-prime or relatively prime or mutually prime numbers.
We can write 17 as 17 × 1 and 19 as 17 × 1.
Hence, 17 and 19 is a pair of co-prime numbers.

Page No 15:

Question 3:

List the prime numbers from 25 to 100 and say how many they are.

ANSWER:

There are a total of 16 prime numbers between 25 and 100 which are 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Page No 15:

Question 4:

Write all the twin prime numbers from 51 to 100.

ANSWER:

If the difference between two co-prime numbers is 2, the numbers are said to be twin prime numbers.
Hence, the twin prime numbers between 51 and 100 are 59 and 61, 71 and 73.

Page No 15:

Question 5:

Write 5 pairs of twin prime numbers from 1 to 50.

ANSWER:

If the difference between two co-prime numbers is 2 then, the numbers are said to be twin prime numbers.
Hence, the twin prime numbers from 1 to 50 are (2,3), (5,7), (11,12), (17,19) and (29,30).

Page No 15:

Question 6:

Which are the even prime numbers?

ANSWER:

There is only even prime number which is 2.

Page No 17:

Question 1:

Factorise the following numbers into primes.
(i)    32
(ii)   57
(iii)  23
(iv)  150
(v)   216
(vi)  208
(vii) 765
(viii) 342
(ix)   377
(x)    559

ANSWER:

(i)
32 = 2 × 16
= 2 × 2 × 8
= 2 × 2 × 2 × 4
​= 2 × 2 × 2 × 2 × 2
(ii)
57 = 3 × 19
(iii)
23 = 23 × 1
(iv)
150 = 2 × 75
= 2 × 3 × 25
​= 2 × 3 × 5 × 5
(v)
216 =  2 × 108
 =  2 × 2 × 54
 =  2 × 2 × 2 × 27
 =  2 × 2 × 2 × 3 × 9
 =  2 × 2 × 2 × 3 × 3 × 3
(vi)
208 = 2 × 104
= 2 × 2 × 52
= 2 × 2 × 2 × 26
= 2 × 2 × 2 × 2 × 13
(vii)
765 = 3 × 255
= 3 × 3 × 85
= 3 × 3 × 5 × 17
(viii)
342 = 2 × 171
= 2 × 3 × 57
= 2 × 3 × 3 × 19
(ix)
377 = 13 × 29
(x)
559 = 13 × 43

Page No 19:

Question 1:

Find the HCF.
(i)     25, 40
(ii)    56, 32
(iii)   40, 60, 75
(iv)   16, 27
(v)    18, 32, 48
(vi)   105, 154
(vii)  42, 45, 48
(viii) 57, 75, 102
(ix)   56, 57
(x)    777, 315, 588

ANSWER:

(i)
 
HCF = 5

(ii)
 
HCF = 2 × 2 × 2= 8

(iii)

HCF = 5

(iv)
 
HCF = 1

(v)
 
HCF = 2

(vi)
 
HCF = 7

(vii)
 
HCF = 3

(viii)
 
HCF = 3

(ix)
 
HCF = 1

(x)
 
HCF = 3 × 7 = 21

Page No 19:

Question 2:

Find the HCF by the division method and reduce to the simplest form.

(i) 275525$\frac{275}{525}$

(ii) 76133$\frac{76}{133}$

(iii) 16169$\frac{161}{69}$

ANSWER:

(i) 

HCF = 25
 ∴275/525=275÷25 / 525÷25=11/21
(ii)

HCF = 19
 ∴76/133=76÷19/133÷19=4/7
(iii) 

HCF = 23
∴161/69=161÷23/69÷23=7/3

Page No 21:

Question 1:

Find the LCF.
(i)     12, 15
(ii)    6, 8, 10
(iii)   18, 32
(iv)   10, 15, 20
(v)    45, 86
(vi)   15, 30, 90
(vii)  105, 195
(viii) 12, 15, 45
(ix)   63, 81
(x)    18, 36, 27

ANSWER:

(i)

LCM = 3 × 5 × 4 = 60

(ii)

LCM = 2 × 3 × 4 × 5​ = 120

(iii)

LCM = 2 × 9 × 16 ​= 288

(iv)

LCM = 2 × 2 × 3 × 5 = 60

(v)

LCM = 45 × 86 = 3870

(vi)

LCM = 2 × 3 × 3 × 5 = 90

(vii)

LCM = 3 × 5 × 7 × 13 = 1365

(viii)

LCM = 2 × 2 × 3 × 3 × 5 = 180

(ix)

LCM = 3 × 3 × 3 × 3 × 5 = 567

(x)

​LCM = 2 × 2 × 3 × 3 × 3 = 108

Page No 21:

Question 2:

Find the HCF and LCM of the numbers given below. Verify that their product is equal to the product of the given numbers.
(i) 32, 37
(ii) 46, 51
(iii) 15, 60
(iv) 18, 63
(v) 78, 104

ANSWER:

(i) 

HCF = 1
LCM = 32 × 37 = 1184
Product of two numbers = 32 × 37 = 1184
Product of HCF and LCM = 1 × 1184 = 1184

(ii) 

HCF = 1
LCM = 46 × 51 = 2346
Product of two numbers = 46 × 51= 2346
Product of HCF and LCM  = 1 × 2346 = 2346

(iii) 

HCF = 3 × 5 = 15
LCM = 3 × 5 × 4 = 60
Product of two numbers = 15 × 60 = 900
Product of HCF and LCM = 15 × 60 = 900

(iv) 

HCF = 3 × 3 = 9
LCM = 3 × 3 × 2 × 7 = 126
Product of two numbers = 18 × 63 = 1134
Product of HCF and LCM = 9 × 126 = 1134
(v) 

HCF = 2 × 13 = 26
LCM = 2 × 13 × 3 × 4 = 312
Product of two numbers =  78 × 104 = 8112
Product of HCF and LCM  = 26 × 312 = 8112

Page No 23:

Question 1:

Choose the right option.
(i) The HCF of 120 and 150 is ................... .
(1) 30
(2) 45
(3) 20
(4) 120

(ii) The HCF of this pair of numbers is not 1.
(1) 13, 17
(2) 29, 20
(3) 40, 20
(4) 14, 15

ANSWER:

(i)
 
HCF = 2 × 3 × 5 = 30
Hence, the correct answer is option (1).
(ii)
40 = 2 × 2 × 2 × 5
20 = 2 × 2 × 5
The HCF of 20 and 40 is 2 × 2 × 5 or 20.
Hence, the correct answer is option (3).

Page No 23:

Question 2:

Find the HCF and LCM.
(i) 14, 28
(ii) 32, 16
(iii) 17, 102, 170
(iv) 23, 69
(v) 21, 49, 84

ANSWER:

(i) 

HCF = 2 × ​7 = 14
LCM = 2 × ​7 × 2 = 28

(ii) 

HCF = 2 × ​2 × ​2 × 2 = 16
LCM = 2 × ​2 × ​2 × 2 × 2 = 32

(iii) 


HCF = 17
LCM = ​17 × ​2 × ​3 × 5 = 510

(iv) 

HCF = 23
LCM = ​23 × 3 = 69

(v) 

HCF = 7
LCM = ​3 × ​4 × ​7 × 7 = 588

Page No 23:

Question 3:

Find the LCM.
(i) 36, 42
(ii) 15, 25, 30
(iii) 18, 42, 48
(iv) 4, 12, 20
(v) 24, 40, 80, 120

ANSWER:

(i) 

LCM = 2 × ​2 × 3 × 3 × 7 = 252

(ii) 

LCM = 2 × ​3 × 5 × 5 = 150

(iii) 

LCM = 2 × ​2 × 2 × 2 × 3 × 3 × 7 = 1008

(iv) 

LCM = 2 × ​2 × 3 × 5 = 60

(v) 

LCM = 2 × ​2 × 2 × 2 × 3 × 5 = 240

Page No 23:

Question 4:

Find the smallest number which when divided by 8, 9, 10, 15, 20 gives a remainder of 5 every time.

ANSWER:

LCM of  8, 9, 10, 15, 20 is given by

LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360
Hence, 365 is the smallest number which when divided by 8, 9, 10, 15, 20 gives a remainder of 5 every time.

Page No 23:

Question 5:

Reduce the fractions 348319$\frac{348}{319}$ , 221247$\frac{221}{247}$ , 437551$\frac{437}{551}$ to the lowest terms.

ANSWER:

348319=348÷29319÷29=1211$$\begin{gathered} \frac{348}{319}=\frac{348÷29}{319÷29} \\ =\frac{12}{11} \end{gathered}$$
221247=221÷13247÷13=1719$$\begin{gathered} \frac{221}{247}=\frac{221÷13}{247÷13} \\ =\frac{17}{19} \end{gathered}$$
437551=437÷19551÷19=2329$$\begin{gathered} \frac{437}{551}=\frac{437÷19}{551÷19} \\ =\frac{23}{29} \end{gathered}$$ 

Page No 23:

Question 6:

The LCM and HCF of two numbers are 432 and 72 respectively. If one of the numbers is 216, what is the other ?

ANSWER:

Let the other number be x.
Now, HCF × LCM = Product of two numbers
⇒ 72 × 432 = x × 216
⇒x=72×432216$\Rightarrow x=\frac{72\times 432}{216}$
= 144
​Hence, the other number is 144.

Page No 23:

Question 7:

The product of two two-digit numbers is 765 and their HCF is 3. What is their LCM?

ANSWER:

HCF × LCM = Product of two numbers
⇒ 3 × LCM = 765
⇒LCM=7653$\Rightarrow \mathrm{LCM}=\frac{765}{3}$
= 255
Hence, the LCM of the two numbers is 255.

Page No 23:

Question 8:

A trader has three bundles of string 392 m, 308 m and 490 m long. What is the great est length of string that the bundles can be cut up into without any left over string?

ANSWER:

The greatest length of string that the bundles can be cut up into without any left over string is given by the HCF of 392, 308 and 490.

HCF = 2 × 7 = 14 
Hence, the greatest length of string that the bundles can be cut up into without any left over string is 14 m.

Page No 23:

Question 9:

Which two consecutive even numbers have an LCM of 180?

ANSWER:

Let us suppose the two consecutive even numbers be 2x and 2x + 2.
Now, product of two numbers = HCF × LCM
⇒ (2x)(2x + 2) = 2 × 180            (HCM of two even number is 2)
⇒ (x)(2x + 2) = 180  
⇒ 2x² + 2x = 180   
⇒ 2x² + 2x − 180 = 0
⇒ x² + x − 90 = 0
⇒ (x − 9)(x + 10) = 0
⇒ x − 9 = 0 or x + 10 = 0
⇒ x = 9 or x =  −10 (Neglecting)
Hence, the two consecutive even numbers are18 and 20.