Author Archives: Swati

Chapter 31 Probability Exercise Ex. 31.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 31 Probability Exercise Ex. 31.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is probability that both drawn balls are black?Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 31 Probability Exercise Ex. 31.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 5(iv)

If A and B are two events such that

Solution 5(iv)

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that

(i) the youngest is a girl

(ii) at least one is girl.Solution 27

(i) Let ‘A’ be the event that both the children born are girls.

Let ‘B’ be the event that the youngest is a girl.

We have to find conditional probability P(A/B).

(ii) Let ‘A’ be the event that both the children born are girls.

Let ‘B’ be the event that at least one is a girl.

We have to find the conditional probability P(A/B).

Chapter 31 Probability Exercise Ex. 31.4

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Given that the events ‘A coming in time’ and ‘B coming in time’ are independent.

The advantage of coming to school in time is that you will not miss any part of the lecture and will be able to learn more.Question 24

Two dice are thrown together and the total score is noted. The event E, F and G are “a total 4”, “a total of 9 or more”, and “a total divisible by 5”, respectively. Calculate P (E), P(F) and P(G) and decide which pairs of events, if any, are independent.Solution 24

Question 25

Let A and B be two independent events such that P (A) = p₁ and P (B) = p₂. Describe in words the events whose probabilities are:

(i) p₁p₂ (ii) (1 – p₁)p₂ (iii) 1-(1- p₁) (1 – p₂) (iv) p₁ + p₂ = 2p₁p₂Solution 25

Chapter 31 Probability Exercise Ex. 31.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the refree asked both the captains to throw a die alternately and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the refree was fair or not.Solution 35

Chapter 31 Probability Exercise Ex. 31.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

There machines E₁, E₂, E₃ in a certain factory produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the tubes produced one each of machines E₁ and E₂ are defective, and that 5% of those produced on E₃ are defective. If one tube is picked up at random from a day’s production, calculate the probability that it is defective.Solution 13

Chapter 31 Probability Exercise Ex. 31.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3,4, 5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3, 4, 5 or 6 with the die?Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

An item is manufactured by three machine A, B and C. out of the total number of items manufactured during a specified period, 50% are manufacture on machine A 30% on B and 20% on C. 2% of the items produced on A and 2% of items produced on B are defective and 3% of these produced on C are defective. All the items stored at one godown. One items is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?Solution 14

Question 15

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Chapter 30 Linear programming Exercise Ex. 30.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Chapter 30 Linear programming Exercise Ex. 30.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13 Old

Solution 13 Old

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below:

2x + 4y £ 8

3x + y £ 6

X + y £ 4

X ³ 0, y ³ 0Solution 26

Converting the inequations into equations, we obtain the lines

2x + 4y = 8, 3x + y = 6, x + y = 4, x = 0, y = 0.

These lines are drawn on a suitable scale and the feasible region of the LPP is shaded in the graph.

From the graph we can see the corner points as (0, 2) and (2, 0).

Chapter 30 Linear programming Exercise Ex. 30.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. food p costs Rs. 60 kg and Food Q costs Rs. 80 kg. Food P contains 3 units / kg of Vitamin A and 5 units/ kg of Vitamin B while food Q contains 4 unit / kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.Solution 10

Question 11

One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Solution 11

Question 12

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240units of calcium, atleasy 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?Solution 12

Question 13

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 unit of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture pre bag?Solution 13

Note: Answer given in the book is incorrect.Question 14

A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:

Food	Vitamin A	Vitamin B	Vitamin C
X	1	2	3
Y	2	2	1

One kg of food X costs Rs. 16 and one kg of food costs Rs. 20. Find the least cost of the mixture which will produce the required diet?Solution 14

Question 15

A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, acid potash and chlorine in a bag of each brand are given in the Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

Kg per bag
	Brand P	Brand Q
Nitrogen	3	3.5
Phosphoric acid	1	2
Potash	3	1.5
Chlorine	1.5	2

If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?Solution 15

Chapter 30 Linear programming Exercise Ex. 30.4

Question 1

If a young man drives his vehicle at 25km/hr, he has to spend Rs 2per km on petrol. if he drives it as a fast of 40 km/hr, the petrol cost increase to Rs 5 per km. He has Rs 100 to speed on petrol and travel a maximum distance in one hour time with less polution . Express this problem as an LPP and solve it graphically. What value do you find hear?Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and half an hour to make an item B. The maximum time available per day is 16 hours. The profit on an item of A is Rs 300 and on one item of B is Rs 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

A manufacturer makes two products, A and B. Product A sells at Rs. 200 each and takes ½ hour to make. Product B sells at Rs.300 each and takes 1 hour to make. There is a permanent order for 14 units of product A and 16 units of product B. A working week consist of 40 hours of production and the weekly turn over must not be less than Rs. 1000. If the profit on each of product A is Rs. 20 and an product B is Rs. 30, then how many of each should be produced so that the profit is maximum? Also find the maximum profit.Solution 35

Question 36

If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2 per km on petrol. If he drives is at a faster speed of 40 km/hr, the petrol cost increases to Rs. 5/ per km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an LPP solve the same.Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

A factory makes tennis rackets and cricket bats A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hour of machine time and 24 hours of craftman’s time. If the profit on racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.    Solution 40

Question 41

A merchant plans to sell two types of personal computers-a desktop model and a portable model that will cost Rs 25,000 and Rs 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and his profit on the desktop model is Rs 4500 and on the portable model is RS 5000. Make an LPP and solve it graphically.    Solution 41

Question 42

A cooperative society of formers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so at to maximise the total profit of the society?Solution 42

Question 43

A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 8000 on each piece of model A and Rs. 1200 on each piece of Model B. How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?Solution 43

Question 44

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftrnan’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

1. What number of rackets and bats must be made if the factory is to Work at full capacity?
2. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.

Solution 44

Question 45

A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000.Solution 45

Question 46

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?Solution 46

Question 47

There are two types of fertilisers F₁ and F₂. F₁ consists of 10% nitrogen and 6% phosphoric acid and F₂ consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F₁ costs Rs. 6/kg and F₂ costs Rs. 5 /kg, determine how much of each type of fertiliser should be used so that nutrient requirement are met at a minimum cost. What is the minimum cost?Solution 47

Question 48

A manufacturer has three machines I, II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines.

The number of hours required for producing 1 unit of each of M and N on the three machines are given in the following table:

Items	Number of hours required on machines
	I	II	III
M	1	2	1
N	2	1	1.25

She makes a profit of Rs. 600 and Rs. 400 on items M and N respectively. How many of each item should she produce so as to maximize her profit assuming that she can sell all the items that she produced? What will be the maximum profit?Solution 48

Question 49

There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:

To/from	Cost (in Rs.)
	A	B	C
P	160	100	150
Q	100	120	100

How many units should be transported from each factory to each depot in order that the transportation cost is minimum. What will be the minimum transportation cost?Solution 49

Let x and y units of commodity be transported from factory P to the depots at A and B respectively.

Then (8 – x – y) units will be transported to depot at C.

The flow is shown below.

Question 50

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

Types of Toys	Machines
	I	II	III
A	12	18	6
B	6	0	9

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs. 7.50 and that on each toy of type B is Rs.5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.Solution 50

Question 51

An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?Solution 51

Question 52

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources to maximize the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?Solution 52

Chapter 30 – Linear programming Exercise Ex. 30.5

Question 1

Solution 1

Question 2

Solution 2

Chapter 29 The plane Exercise Ex. 29.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Chapter 29 The plane Exercise Ex. 29.2

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 29 The plane Exercise Ex. 29.3

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3

Solution 3

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13(i)

Solution 13(i)

Question 13(ii)

Solution 13(ii)

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Find the vector and Cartesian equations of the plane which passes through the point (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1.Solution 18

Question 19

If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.Solution 19

Question 20

If O is the origin and the coordinates of A are (a, b, c). Find the direction cosines of OA and the equation of the plane through A at right angles to OA.Solution 20

Chapter 29 The plane Exercise Ex. 29.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Find the distance of the plane 2x – 3y + 4z – 6 = 0 from the origin.Solution 11

Chapter 29 The plane Exercise Ex. 29.5

Question 1

Solution 1

Question 2

Find the vector equation of the plane passing through the points P(2, 5, -3), Q(-2, -3, 5) and R(5, 3, -3).Solution 2

Question 3

Solution 3

Question 4

Find the vector equation of the plane passing through the points (1, 1, -1), (6, 4, -5) and (-4, -2, 3).Solution 4

Question 5

Solution 5

Chapter 29 The plane Exercise Ex. 29.6

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Find the angle between the planes:

2x + y – 2z = 5 and 3x – 6y – 2z = 7Solution 2(v)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.Solution 11

Question 12

Find the equation of the plane that contains the point (1, -1, 2) and is perpendicular to each of the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8.Solution 12

Question 13

Solution 13

Question 14

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.Solution 14

Question 15

Find the vector equation of the plane through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x – 2y + 4z = 10.Solution 15

Chapter 29 The plane Exercise Ex. 29.7

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Chapter 29 The plane Exercise Ex. 29.8

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Find the equation of the plane through the intersection of the planes 3x – y 2z = 4 and x + y + z = 2 and the point (2, 2, 1).Solution 15

Question 16

Find the vector equation of the plane through the line of intersection of the plane x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.Solution 16

Question 17

Find the equation of the plane passing through (a, b, c) and parallel to the plane

Solution 17

Chapter 29 The plane Exercise Ex. 29.9

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Find the distance between the point (7, 2, 4) and the plane determined by the points A (2, 5, -3), B (-2, -3, 5) and C (5, 3, -3)Solution 11

Question 12

A plane makes intercepts -6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.Solution 12

Chapter 29 The plane Exercise Ex. 29.10

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 29 The plane Exercise Ex. 29.11

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Find the equation of the plane passing through the points

(-1, 2, 0),(2, 2, -1) and parallel to the line

Solution 25

Chapter 29 The plane Exercise Ex. 29.12

Question 1(i)

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the yz-plane.Solution 1(i)

Question 1(ii)

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the zx-plane.Solution 1(ii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Find the distance of the point P (-1, -5, -10) from the point of intersection of the line joining the points A (2, -1, 2) and B (5, 3, 4) with the plane x – y + z = 5.Solution 5

Question 6

Find the distance of the point P(3, 4,4) from the point, where the line joining the points A(3, -4, -5) and B (2, -3, 1) intersects the plane 2x + y + z =7.Solution 6

Chapter 29 The plane Exercise Ex. 29.13

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line

Solution 13

Chapter 29 The plane Exercise Ex. 29.14

Question 1

Solution 1

Question 2

Solution 2

Question 3

Find the shortest distance between the lines

Solution 3

Chapter 29 The plane Exercise Ex. 29.15

Question 1

Solution 1

Question 2

Solution 2

Question 3

Hence or otherwise deduce the length of the perpendicular.Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Find the length and the foot of perpendicular from the point (1, 3/2, 2) to the plane 2x – 2y + 4z + 5 = 0.Solution 14

Chapter 28 Straight line in space Exercise Ex. 28.1

Question 1

Solution 1

Question 2

Find the vector equation of the line passing through the points (-1, 0, 2) and (3, 4, 6).Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

.

Question 17

Solution 17

Chapter 28 Straight line in space Exercise Ex. 28.2

Question 1

Solution 1

Question 2

Show that the line through the points (1, -1, 2) and (3, 4, -2) is perpendicular to the line through points (0, 3, 2) and (3, 5, 6).Solution 2

Question 3

Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (-1, -2, 1) and (1, 2, 5).Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1) and (4, 3, -1).Solution 6

Question 7

Find the equation of a line parallel to x-axis and passing through the origin.Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 9(iv)

Solution 9(iv)

Question 9(v)

Solution 9(v)

Question 9(vi)

find the angle between the following pairs of line :

Solution 9(vi)

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 10(iii)

Solution 10(iii)

Question 10(iv)

find the angle between the pairs of lines with directions ratios proposal to a, b, c and b – c, c – a, a – b.Solution 10(iv)

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Chapter 28 Straight line in space Exercise Ex. 28.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 7

Solution 7

Chapter 28 Straight line in space Exercise Ex. 28.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

..Question 4

Solution 4

Question 5

Find the foot of perpendicular from the point (2, 3, 4) to the line . Also find the perpendicular distance from the given point to the line.Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Find the coordinates of the foot of perpendicular drawn from the point A (1, 8, 4) to the line joining the points B (0, -1, 3) and C (2, -3, -1).Solution 14

Chapter 28 Straight line in space Exercise Ex. 28.5

Question 1(i)

Find the shortest distance between the pair of lines whose vector equation are

Solution 1(i)

Question 1(ii)

Find the shortest distance between the pair of lines whose vector equations are:

Solution 1(ii)

Question 1(iii)

Find the shortest distance between the pair of lines whose vector equations are:

Solution 1(iii)

Question 1(iv)

Find the shortest distance between the lines whose vector equations are:

Solution 1(iv)

Question 1(v)

Find the shortest distance between the pair of lines whose vector equations are:

Solution 1(v)

Question 1(vi)

Find the shortest distance between the pair of lines whose vector equations are:

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 2(i)

Find the shortest distance between the pair of lines whose cartesian equations are:

.Solution 2(i)

Question 2(ii)

Find the shortest distance between the pair of lines whose cartesian equations are:

.Solution 2(ii)

Question 2(iii)

Find the shortest distance between the pair of lines whose cartesian equations are:

.Solution 2(iii)

Question 2(iv)

Find the shortest distance between the pair of lines whose Cartesian equations are:

.Solution 2(iv)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 5

Solution 5

Question 6

Solution 6

Question 7(i)

Find the shortest distance between the lines

Solution 7(i)

Question 7(ii)

Find the shortest distance between the lines

Solution 7(ii)

Question 7(iii)

Find the shortest distance between the lines

Solution 7(iii)

Question 7(iv)

Find the shortest distance between the lines

Solution 7(iv)

Question 8

Find the distance between the lines l₁ and l₂ given by

Solution 8

Chapter 27 Direction Cosines and Direction Ratios Exercise Ex. 27.1

Question 1

If a line makes angles of 90°, 60° and 30° with the positive direction of x,y and z-axis respectively, find its direction cosines.Solution 1

Let l, m and n be the direction cosines of a line.

l = cos 90° = 0

Question 2

If a line has direction ratios 2, -1, -2, determine its direction cosines.Solution 2

Question 3

Find the direction cosines of the line passing through two points (-2, 4, -5) and (1, 2, 3).Solution 3

Question 4

Using direction ratios show that the points A (2, 3, -4), B (1, -2, 3) and C (3, 8, -11) are collinear.Solution 4

Question 5

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2)Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2.Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16(i)

Solution 16(i)

Question 16(ii)

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 16(iv)

Find the angle between the lines whose direction cosines are given by equations

2l + 2m – n = 0, mn + ln + lm = 0Solution 16(iv)

Chapter 26 Scalar Triple Product Exercise Ex. 26.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 5(iv)

Solution 5(iv)

Question 6

Solution 6

Question 7

Solution 7

Question 8

Show that the four points whose position vectors are  Coplanar.Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12(i)

Solution 12(i)

Question 12(ii)

Solution 12(ii)

Question 13

Solution 13

Chapter 25 Vector or Cross Product Exercise Ex. 25.1

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 8(iv)

Solution 8(iv)

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 9(iv)

Solution 9(iv)

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27(i)

Solution 27(i)

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Using Vectors, find the area of the triangle with vertices: (i) A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5) (ii) A (1, 2, 3), B(2, -1, 4) and C (4, 5, -1).Solution 34

Question 35

Solution 35

Chapter 24 Scalar Or Dot Product Exercise Ex. 24.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5 (i)

Solution 5 (i)

Question 5 (ii)

Solution 5 (ii)

Question 5 (iii)

Solution 5 (iii)

Question 5 (iv)

Solution 5 (iv)

Question 5 (v)

Solution 5 (v)

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 8 (i)

Solution 8 (i)

Question 8 (ii)

Solution 8 (ii)

Question 9

Solution 9

Question 10

Solution 10

Given that  are mutually perpendicular, so,

Question 11

Solution 11

Question 12

Show that the vector  is equally inclined with the coordinate axes.Solution 12

Question 13

Show that the vectors  are mutually perpendicualr unit vectors.Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

If two vector  are such that , then find the value of (3a – 5b) . (2a + 7b).Solution 29

Question 30(i)

Solution 30(i)

Question 30(ii)

Solution 30(ii)

Question 31(i)

Solution 31(i)

Question 31(ii)

Solution 31(ii)

Question 31(iii)

Solution 31(iii)

Question 32(i)

Solution 32(i)

Question 32(ii)

Solution 32(ii)

Question 32(iii)

Solution 32(iii)

Question 33(i)

Solution 33(i)

Question 33(ii)

Solution 33(ii)

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Chapter 24 Scalar Or Dot Product Exercise Ex. 24.2

Question 1

Solution 1

Question 2

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.Solution 2

Question 3

(Pythagoras’s Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.Solution 3

Question 4

Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.Solution 4

Question 5

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.Solution 5

Question 6

Prove that the diagonals of a rhombus are perpendicular bisectors of each other.Solution 6

Question 7

Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.Solution 7

Question 8

If AD is the median of D ABC, using vectors, prove that

AB² + AC² = 2 (AD² + CD²).Solution 8

Question 9

If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.Solution 9

Question 10

In a quadrilateral ABCD, prove that AB² + BC² + CD²+ DA² = AC² + BD² + 4 PQ², where P and Q are middle points of diagonals AC and BD. Solution 10

Chapter 23 Algebra of Vectors Exercise Ex. 23.1

Question 1(i)

Represent graphically a dispacement of 40 km, 30° east of north.Solution 1(i)

Question 1(ii)

Represent graphically a displacement of 50 km, south-eastSolution 1(ii)

Here, vector  represents the displacement of 50 km, south-east.Question 1(iii)

Represent graphically a displacement of 70 km, 40° north of west.Solution 1(iii)

Here, vector  represents the displacement of 70 km, 40° north of west.Question 2

Classify the following measures as scalars and vectors.

(i) 15 kg

(ii) 20 kg weight

(iii) 45°

(iv) 10 metres south-east

(v) 50 m/s²Solution 2

(i) 15 kg is a scalar quantity because it involves only

(ii) 20 kg weight is a vector quantity as it involves both magnitude and direction.

(iii) 45° is a scalar quantity as it involves only magnitude.

(iv) 10 metres south-east is a vector quantity as it involve direction.

(v) 50 m/s² is a scalar quantity as it involves magnitude of acceleration.Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 23 Algebra of Vectors Exercise Ex. 23.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 23 Algebra of Vectors Exercise Ex. 23.3

Question 1

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors

Solution 1

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Chapter 23 – Algebra of Vectors Exercise Ex. 23.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.Solution 6

Chapter 23 – Algebra of Vectors Exercise Ex. 23.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Find |AB| in each case.Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 12

Solution 12

Chapter 23 Algebra of Vectors Exercise Ex. 23.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Chapter 23 Algebra of Vectors Exercise Ex. 23.7

Question 1

Solution 1

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Using vectors, find the value of λ such that the points

 (λ, – 10, 3), (1 -1, 3) and (3, 5, 3) are collinear.Solution 13

Chapter 23 Algebra of Vectors Exercise Ex. 23.8

Question 1

Solution 1

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

Question 2 (iii)

Solution 2 (iii)

Question 2 (iv)

Solution 2 (iv)

Question 3 (i)

Solution 3 (i)

Question 3 (ii)

Solution 3 (ii)

Question 4

Solution 4

Question 5 (i)

Solution 5 (i)

Question 5 (ii)

Solution 5 (ii)

Question 6 (i)

Solution 6 (i)

Question 6 (ii)

Solution 6 (ii)

Question 7 (i)

Solution 7 (i)

Question 7 (ii)

Solution 7 (ii)

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 23 Algebra of Vectors Exercise Ex. 23.9

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7 (i)

Solution 7 (i)

Question 7 (ii)

Solution 7 (ii)

Question 7 (iii)

Solution 7 (iii)

Question 8

Solution 8

Question 9

Solution 9

Question 10

If a unit vector  makes and angles  and an acute angle θ with , then find θ and hence, the components of .Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 22 Differential Equations Exercise Ex. 22.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Determine the order and degree of the following differential equations. State also whether they are linear or non-linear.

Solution 27

The order of a differential equation is the order of the highest order derivative appearing in the equation.

The degree of a differential equation is the degree of the highest order derivative.

Consider the given differential equation

In the above equation, the order of the highest order derivative is 1.

So the differential equation is of order 1.

In the above differential equation, the power of the highest order derivative is 3.

Hence, it is a differential equation of degree 3.

Since the degree of the above differential equation is 3, more than one, it is a non-linear differential equation.

Chapter 22 – Differential Equations Exercise Ex. 22.2

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Form the differential equation having y = (sin^-1x)² + A cos ^-1 x + B, where A and B are arbitrary constants, as its general solution.Solution 14

Question 15(i)

Solution 15(i)

Question 15(ii)

Solution 15(ii)

Question 15(iii)

Solution 15(iii)

Question 16(i)

Solution 16(i)

Question 16(ii)

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 16(iv)

Represent the following family of curves by forming the corresponding differential equation (a,b being parameters):

x² + (y – b)² = 1Solution 16(iv)

Question 16(v)

Solution 16(v)

Question 16(vi)

Solution 16(vi)

Question 16(vii)

Solution 16(vii)

Question 16(viii)

Solution 16(viii)

Question 16(ix)

Solution 16(ix)

Question 16(x)

Solution 16(x)

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Chapter 22 – Differential Equations Exercise Ex. 22.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

show that y = ae²x + be^-x is a solution of the differential equation Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Verify that y =  + b is a solution of the differential equation Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Show that y = e^x(A cos x + B sin x) is the solution of the differential equation

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Show that y = e^-x + ax + b is solution of the differential equation Solution 20

Question 21(i)

For the following differential equation verify that the accompanying function is a solution in the mentioned domain (a, b are parameters) Solution 21(i)

Question 21(ii)

Solution 21(ii)

Question 21(iii)

Solution 21(iii)

Question 21(iv)

Solution 21(iv)

Question 21(v)

Solution 21(v)

Chapter 22 – Differential Equations Exercise Ex. 22.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

For the following initial value problem verify that the accompanying function is a solution:

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 22 – Differential Equations Exercise Ex. 22.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solve the following differential equation:

(sin x + cos x)dy + (cos x – sin x) dx = 0Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solve the following differential equation:

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solve the following differential equation

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

solve the following differential equation

Solution 26

Chapter 22 – Differential Equations Exercise Ex. 22.6

Question 1

Solve the following differential equation:

Solution 1

Question 2

Solve the following differential equation:

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 22 – Differential Equations Exercise Ex. 22.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solve the following differential equation:

Solution 35

Question 36

Solve the following differential equation:

Solution 36

Question 37(i)

Solution 37(i)

Question 37(ii)

Solve the following differential equation:

Solution 37(ii)

Question 38(i)

Solution 38(i)

Question 38(ii)

Solution 38(ii)

Question 38(iii)

ye^x/y dx = (xe^x/y + y²) dy, y ¹ 0Solution 38(iii)

Question 38(iv)

(1 + y²) tan^-1 x dx + 2y (1 + x²)dy = 0Solution 38(iv)

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45(i)

Solution 45(i)

Question 45(ii)

Solution 45(ii)

Question 45(iii)

Solution 45(iii)

Question 45(iv)

Solution 45(iv)

Question 45(v)

Solution 45(v)

Question 45(vi)

Solve the following initial value problem

=1 + x² + y² + x²y², y(0) = 1Solution 45(vi)

Question 45(vii)

Solve the following initial value problem

Solution 45(vii)

Question 45(viii)

Solution 45(viii)

Question 45(ix)

Solution 45(ix)

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Find the particular solution of e= x + 1, given that y = 3 when x = 0.Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Find the equation of a curve passing through the point (0,0) and whose differential equation is Solution 52

Question 53

Solution 53

Question 54

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after after t seconds.Solution 54

Question 55

in a bank,principal increases continuously at the rate of r% per  year. Find The value of r if Rs 100 doubles itself in 10 years (log_e 2 = 0.6931).Solution 55

Let p, t and represent the principal, time, and rate of interest respectively.

It is given that the principal increases continuously at the rate of r% per year.

Integrating both side, we get:

Question 56

Solution 56

Question 57

Solution 57

..Question 58

Solution 58

Chapter 22 – Differential Equations Exercise Ex. 22.8

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solve the following differential equation.

Solution 11

Chapter 22 – Differential Equations Exercise Ex. 22.9

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solve the following differential equation:

Solution 4

Question 5

Solve the following differential equation:

Solution 5

Question 6

Solve the following initial value problem

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solve the following initial value problem

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solve the following initial value poblem

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solve the following differential equation:

Solution 35

Question 36(i)

Solution 36(i)

Question 36(ii)

Solution 36(ii)

Question 36(iii)

Solve the following initial value problem

Solution 36(iii)

Question 36(iv)

Solution 36(iv)

Question 36(v)

Solution 36(v)

Question 36(vi)

Solution 36(vi)

Question 36(vii)

Solution 36(vii)

Question 36(viii)

Solution 36(viii)

Question 36(ix)

Solve the following initial value problem

Solution 36(ix)

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Chapter 22 – Differential Equations Exercise Ex. 22.10

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solve the following differential equation:

Solution 30

Question 31

Solve the following differential equation:

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36(i)

Solution 36(i)

Question 36(ii)

Solution 36(ii)

Question 36(iii)

Solution 36(iii)

Question 36(iv)

Solution 36(iv)

Question 36(v)

Solution 36(v)

Question 36(vi)

Solution 36(vi)

Question 36(vii)

Solution 36(vii)

Question 36(viii)

Solution 36(viii)

Question 36(ix)

Solution 36(ix)

Question 36(x)

Solution 36(x)

Question 36(xi)

Solution 36(xi)

Question 36(xii)

Solution 36(xii)

Question 37(i)

Solution 37(i)

Question 37(ii)

Solution 37(ii)

Question 37(iii)

Solution 37(iii)

Question 37(iv)

Solution 37(iv)

Question 37(v)

Solve the following initial value problem:

Solution 37(v)

Question 37(vi)

Solution 37(vi)

Question 37(vii)

Solution 37(vii)

Question 37(viii)

Solve the following initial value problem

Solution 37(viii)

Question 37(ix)

Solution 37(ix)

Question 37(x)

Solution 37(x)

Question 37(xi)

Solution 37(xi)

Question 37(xii)

dy = cos x (2 – y cosec x) dxSolution 37(xii)

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solve the differential equation

Solution 40

Question 41

Solution 41

Chapter 22 – Differential Equations Exercise Ex. 22.11

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Chapter 22 – Differential Equations Exercise Ex. 22RE

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solve the following differential equation:

Solution 46

Question 47

Solution 47

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63

Question 64(i)

Solution 64(i)

Question 64(ii)

Solution 64(ii)

Question 64(iii)

Solution 64(iii)

Question 64(iv)

Solution 64(iv)

Question 64(v)

Solution 64(v)

Question 64(vi)

Solution 64(vi)

Question 65(i)

Solution 65(i)

Question 65(ii)

Solution 65(ii)

Question 65(iii)

Solution 65(iii)

Question 66(i)

Solution 66(i)

Question 66(ii)

Solution 66(ii)

Question 66(iii)

Solution 66(iii)

Question 66(iv)

Solution 66(iv)

Question 66(v)

Solution 66(v)

Question 66(vi)

Solution 66(vi)

Question 66(vii)

Solution 66(vii)

Question 66(viii)

Solution 66(viii)

Question 66(ix)

Solution 66(ix)

Question 66(x)

Solution 66(x)

Question 66(xi)

Solution 66(xi)

Question 66(xii)

Solution 66(xii)

Question 66(xiii)

Solution 66(xiii)

Question 66(xiv)

Solution 66(xiv)

Question 66(xv)

Solution 66(xv)

Question 67(i)

Solution 67(i)

Question 67(ii)

Solution 67(ii)

Question 67(iii)

Solution 67(iii)

Question 68

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Question 73

Solution 73

Question 74

Solution 74

Question 75

Solution 75

Question 76

Solution 76

Question 77

Solution 77

Question 78

Solution 78

Question 79

Solution 79