RS Agarwal Solution | Class 10th | Chapter-4 | Quadratic Equations | Edugrown

Exercise Ex. 4A

Question 19

Solve the following quadratic equation:

Solution 19

Hence, are the roots of Question 1(i)

Which of the following are quadratic equations in x?

(i)Solution 1(i)

(i)is a quadratic polynomial

= 0 is a quadratic equationQuestion 1(ii)

Which of the following are quadratic equations in x?

Solution 1(ii)

Clearly  is a quadratic polynomial

is a quadratic equation.Question 1(iii)

Which of the following are quadratic equations in x?

Solution 1(iii)

 is a quadratic polynomial

 = 0 is a quadratic equationQuestion 1(iv)

Solution 1(iv)

Clearly, is a quadratic equation

is a quadratic equationQuestion 1(v)

Solution 1(v)

is not a quadratic polynomial since it contains in which power  of x is not an integer.

is not a quadratic equationQuestion 1(vi)

Solution 1(vi)

And Being a polynomial of degree 2, it is a quadratic polynomial.

Hence, is a quadratic equation.Question 1(vii)

Solution 1(vii)

And being a polynomial of degree 3, it is not a quadratic polynomial

Hence, is not a quadratic equationQuestion 1(viii)

Solution 1(viii)

is not a quadratic equationQuestion 1(ix)

Which of the following are quadratic equations in x?

Solution 1(ix)

Question 1(x)

Which of the following are quadratic equations in x?

Solution 1(x)

Question 1(xi)

Which of the following are quadratic equations in x?

Solution 1(xi)

Question 2

Which of the following are the roots of 

(i)-1

(ii)

(iii)Solution 2

The given equation is 

(i)On substituting x = -1 in the equation, we get

(ii)On substituting in the equation, we get

(iii)On substituting in the equation , we get

Question 3(i)

Find the value of k for which x = 1 is a root of the equation Solution 3(i)

Since x = 1 is a solution of it must satisfy the equation.

Hence the required value of k = -4Question 3(ii)

Find the value of a and b for which and x = -2 are the roots of the equation Solution 3(ii)

Since is a root of , we have

Again x = -2 being a root of , we have

Multiplying (2) by 4 adding the result from (1), we get

11a = 44 a = 4

Putting a = 4 in (1), we get

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solve each of the following quadratic equations:

3– 243 = 0Solution 7

Hence, 9 and -9 are the roots of the equation Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solve the following quadratic equation:

Solution 13

Hence, are the roots of Question 14

Solve the following quadratic equation:

Solution 14

Hence, are the roots of equationQuestion 15

Solve the following quadratic equation:

Solution 15

Hence, and 1 are the roots of the equation .Question 16

Solve the following quadratic equation:

Solution 16

are the roots of the equation Question 17

Solve the following quadratic equation:

Solution 17

Hence, are the roots of the given equation Question 18

Solve the following quadratic equation:

Solution 18

Hence, are the roots of given equationQuestion 20

Solution 20

Question 21

Solution 21

Question 22

Solve the following quadratic equation:

Solution 22

Hence, are the roots of the given equationQuestion 23

Solve the following quadratic equation:

Solution 23

Hence, are the roots of the given equationQuestion 24

Solve the following quadratic equation:

Solution 24

Hence, are the roots of the given equationQuestion 25

Solve the following quadratic equation:

Solution 25

Hence, are the roots of the given equationQuestion 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

Solve the following quadratic equation:

Solution 33

Hence, 1 and are the roots of the given equationQuestion 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solve the following quadratic equation:

Solution 37

Hence, are the roots of the given equationQuestion 38

Solve the following quadratic equation:

Solution 38

Hence, 2 and are the roots of given equationQuestion 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 quadratic equation:

Solution 46

Hence, are the roots of the given equationQuestion 47

Solution 47

Question 48

Solve the following quadratic equation:

Solution 48

Hence, are the roots of given equationQuestion 49

Solve the following quadratic equation:

Solution 49

Hence, are the roots of given equationQuestion 50

Solve the following quadratic equation:

Solution 50

Hence, are the roots of given equationQuestion 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55(i)

Solution 55(i)

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59(i)

Solution 59(i)

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63

Question 64(i)

Solution 64(i)

Question 65

Solution 65

Question 66

Solution 66

Question 67

Solution 67

Question 68

Solve the following quadratic equation:

Solution 68

Putting the given equation become

Case I:

Case II:

Hence, are the roots of the given equationQuestion 69

Solve the following quadratic equation:

Solution 69

The given equation

Hence, is the roots of the given equationQuestion 70

Solution 70

Question 71

Solve the following quadratic equation:

Solution 71

Hence, -2,0 are the roots of given equationQuestion 72

Solve the following quadratic equation:

Solution 72

Hence, are the roots of given equationQuestion 73

Solve the following quadratic equation:

Solution 73

Hence, 3 and 2 are roots of the given equation

Exercise Ex. 4B

Question 4

Solution 4

Question 5

Solution 5

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 2

Solution 2

Question 3

Solution 3

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

Exercise Ex. 10E

Question 8

Two numbers differ by 3 and their product is 504. Find the numbers.Solution 8

Let the required number be x and x – 3, then

Hence, the required numbers are (24,21) or (-21 and-24)Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

The sum of the squares of two consecutive multiples of 7 is 1225. Find the multiples.Solution 16

Question 17

Solution 17

Question 18

Divide 57 into two parts whose product is 680.Solution 18

Question 19

Solution 19

Question 20

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.Solution 20

Let the smaller part and larger part be x, 16 – x

Then,

-42 is not a positive part

Hence, the larger and smaller parts are 10, 6 respectivelyQuestion 21

Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.Solution 21

Let the required number be x and y, hen

Question 22

The difference of the squares of two natural numbers is 45. The square of the smaller number is four times the larger number. Find the numbers.Solution 22

Let x, y be the two natural numbers and x > y

——(1)

Also, square of smaller number = 4 larger number

———(2)

Putting value of from (1), we get

Thus, the two required numbers are 9 and 6Question 23

Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers.Solution 23

Let the three consecutive numbers be x, x + 1, x + 2

Sum of square of first and product of the other two

Required numbers are 4, 5 and 6Question 24

A two-digit number is 4 times the sum of its digits and twice the product of its digits. Find the numbers.Solution 24

Let the tens digit be x and units digit be y

Hence, the tens digit is 3 and units digit is (2 3) is

Hence the required number is 36Question 25

A two-digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number.Solution 25

Let the tens digit and units digits of the required number be x and y respectively

The ten digit is 2 and unit digit is 7

Hence, the required number is 27Question 26

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is. Find the fraction.Solution 26

Let the numerator and denominator be x, x + 3

Then,

Hence, numerator and denominator are 2 and 5 respectively and fraction is Question 27

Solution 27

Question 28

Solution 28

Question 29

A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left. When he increased the size of the square by one student, he found he was short of 25 students. Find the number of students.Solution 29

Let there be x rows and number of student in each row be x

Then, total number of students = 

Hence total number of student

Total number of students is 600Question 30

300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of students.Solution 30

Let the number of students be x, then

Hence the number of students is 50Question 31

In a class test, the sum of Kamal’s marks in Mathematics and English is 40. Had he got 3 marks more in Mathematics and 4 marks less in English, the product of the marks would have been 360. Find his marks in two subjects separately.Solution 31

Let the marks obtained by Kamal in Mathematics and English be x and y

The marks obtained by Kamal in Mathematics and English respectively are (21,19) or (12,28)Question 38

One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.Solution 38

Let the age of son be x and age of man = y

1 year ago

Question 41

The product of Meena’s age 5 years ago and her age 8 years later is 30. Find her present age.Solution 41

Let the present age of Meena be x

Then,

Hence the present age of Meena is 7 yearsQuestion 43

A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed which is 20 km per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck.Solution 43

Question 44

While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalise the injured and so the plane started late by 30 minutes. To reach the destination, 1500 km away, in time, the pilot increased the speed by 100 km/hour. Find the original speed of the plane.

Do you appreciate the values shown by the pilot, namely promptness in providing help to the injured and his efforts to reach in time?Solution 44

Question 45

A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.Solution 45

Question 46

A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/hr more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?Solution 46

Question 47

A train travels 180 km at a uniform speed. If the speed had been 9 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.Solution 47

Question 48

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 kmph more, it would have taken 30 minutes less for the journey. Find the original speed of the train.Solution 48

Let the original speed of the train be x km.hour

Then speed increases by 15 km/ph = (x + 15)km/hours

Then time taken at original speed = 

Then, time taken at in increased speed = 

Difference between the two lines taken 

Then, original speed of the train = 45km / hQuestion 49

A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find its usual speed.Solution 49

Question 50

The distance between Mumbai and Pune is 192 km. Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speeds of the two trains differ by 20 kmph.Solution 50

Let the speed of the Deccan Queen = x kmph

The, speed of other train = (x – 20)kmph

Then, time taken by Deccan Queen = 

Time taken by other train = 

Difference of time taken by two trains is 

Hence, speed of Deccan Queen = 80km/hQuestion 51

A motorboat whose speed is 18 km per hour in still water takes 1 hour more to go 24 km upstream than to return to the same point. Find the speed of the stream.Solution 51

Let the speed of stream be x km/h

Speed of boat in still stream = 18 km/h

Speed of boat up the stream = 18 – x km/h

Time taken by boat to go up the stream 24 km = 

Time taken by boat to go down the stream = 

Time taken by the boat to go up the stream is 1 hour more that the time taken down the stream

Speed of the stream = 6 km/hQuestion 52

The speed of a boat in still water is 8 kmph. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.Solution 52

Let the speed of the stream be x kmph

Then the speed of boat down stream = (8 + x) kmph

And the speed of boat upstream = (8 – x)kmph

Time taken to cover 15 km upstream = 

Time taken to cover 22 km downstream = 

Total time taken = 5 hours

Hence, the speed of stream is 3 kmphQuestion 53

A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.Solution 53

Let the speed of the stream be = x km/h

Speed of boat in still waters = 9 km/h

Speed of boat down stream = 9 + x

time taken by boat to go 15 km downstream = 

Speed of boat upstream = 9 – x

time taken by boat to go 15 km of stream = 

Question 54

A takes 10 days less than time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work.Solution 54

Question 55

Two pipes running together can fill a cistern inminutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.Solution 55

Let the faster pipe takes x minutes to fill the cistern

Then, the other pipe takes (x + 3) minute

The faster pipe takes 5 minutes to fill the cistern

Then, the other pipe takes (5 + 3) minutes = 8 minutesQuestion 56

Solution 56

Question 57

Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.Solution 57

Question 58

The length of rectangle is twice its breadth and its area is 288 sq.cm. Find the dimensions of the rectangle.Solution 58

Let the breadth of a rectangle = x cm

Then, length of the rectangle = 2x cm

Thus, breadth of rectangle = 12 cm

And length of rectangle = (2 12) = 24 cmQuestion 59

The length of a rectangular field is three times its breadth. If the area of the field be 147 sq.m, find the length of the field.Solution 59

Let the breadth of a rectangle = x meter

Then, length of rectangle = 3x meter

Thus, breadth of rectangle = 7 m

And length of rectangle = (3 7)m = 21 mQuestion 60

The length of a hall is 3 metres more than its breadth. If the area of the hall is 238 sq.m, calculate its length and breadth.Solution 60

Let the breadth of hall = x meters

Then, length of the hall = (x + 3) meters

Area = length breadth = 

Thus, the breadth of hall is 14 cm

And length of the hall is (14 + 3) = 17 cmQuestion 61

The perimeter of a rectangular plot is 62 m and its area is 228 sq. metres. Find the dimensions of the plot.Solution 61

Question 62

A rectangular field is 16 m long and 10 m wide. There is a path of uniform width all around it, having an area of 120 m². Find the width of the path.Solution 62

Let the width of the path be x meters,

Then,

Area of path = 16 10 – (16 – 2x) (10 – 2x) = 120

Hence the required width is 3 meter as x cannot be 10mQuestion 63

The sum of the areas of two squares is 640 m². If the difference in their perimeters be 64 m, find the sides of the two squares.Solution 63

Let x and y  be the lengths of the two square fields.

4x – 4y = 64

x – y = 16——(2)

From (2),

x = y + 16,

Putting value of x in (1)

Sides of two squares are 24m and 8m respectivelyQuestion 64

The length of a rectangle is thrice as long as the side of a square. The side of the square is 4 cm more than the width of the rectangle. Their areas being equal, find their dimensions.Solution 64

Let the side of square be x cm

Then, length of the rectangle = 3x cm

Breadth of the rectangle = (x – 4) cm

Area of rectangle = Area of square x

Thus, side of the square = 6 cm

And length of the rectangle = (3 6) = 18 cm

Then, breadth of the rectangle = (6 – 4) cm = 2 cmQuestion 65

A farmer prepares a rectangular vegetable garden of area 180 sq. m. With 39 metres of barbed wire, he can fence the three sides of the garden, leaving one of the longer sides unfenced. Find the dimensions of the garden.Solution 65

Let the length = x meter

Area = length breadth = 

If ength of the rectangle = 15 m

Also, if length of rectangle = 24 m

Question 66

The area of a right triangle is 600 cm². If the base of the triangle exceeds the altitude by 10 cm, find the dimensions of the triangle.Solution 66

Let the altitude of triangle be x cm

Then, base of triangle is (x + 10) cm

Hence, altitude of triangle is 30 cm and base of triangle 40 cm

Question 67

The area of a right-angled triangle is 96 sq m. If the base is three times the altitude, find the base.Solution 67

Let the altitude of triangle be x meter

Hence, base = 3x meter

Hence, altitude of triangle is 8 cm

And base of triangle = 3x = (3 8) cm = 24 cmQuestion 68

The area of a right-angled triangle is 165 sq m. Determine its base and altitude if the latter exceeds the former by 7 metres.Solution 68

Let the base of triangle be x meter

Then, altitude of triangle = (x + 7) meter

Thus, the base of the triangle = 15 m

And the altitude of triangle = (15 + 7) = 22 mQuestion 69

The hypotenuse of a right-angled triangle is 20 metres. If the difference between the lengths of the other sides be 4 m, find the other sides.Solution 69

Let the other sides of triangle be x and (x -4) meters

By Pythagoras theorem, we have

Thus, height of triangle be = 16 cm

And the base of the triangle = (16 – 4) = 12 cmQuestion 70

The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle.Solution 70

Let the base of the triangle be x

Then, hypotenuse = (x + 2) cm

Thus, base of triangle = 15 cm

Then, hypotenuse of triangle = (15 +2 )= 17 cm

And altitude of triangle = Question 71

The hypotenuse of a right-angled triangle is 1 metre less than twice the shortest side. If the third side is 1 metre more than the shortest side, find the sides of the triangle.Solution 71

Let the shorter side of triangle be x meter

Then, its hypotenuse = (2x – 1)meter

And let the altitude = (x + 1) meter

Exercise Ex. 10F

Question 1

Which of the following is a quadratic equation?

Solution 1

Question 2

Which of the following is a quadratic equation?

Solution 2

Question 3

Which of the following is not a quadratic equation?

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 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

(a) Real and equal

(b) Real and unequal

(c) Imaginary

(d) None of theseSolution 21

Question 22

(a) Real, unequal and rational

(b) Real, unequal and irrational

(c) Real and equal

(d) ImaginarySolution 22

Question 23

(a) Real, unequal and rational

(b) Real, unequal and irrational

(c) Real and equal

(d) ImaginarySolution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

The perimeter of a rectangle is 82 m and its area is 400 m². The breadth of the rectangle is

(a) 25 m

(b) 20 m

(c) 16 m

(d) 9 mSolution 29

Question 30

The length of a rectangular field exceeds its breadth by 8 m and the area of the field is 240 m². The breadth of the field is

(a) 20 m

(b) 30 m

(c) 12 m

(d) 16 mSolution 30

Question 31

Solution 31Question 32

The sum of two natural numbers is 8 and their product is 15. Find the numbers.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

Solution 46

Question 47

Solution 47

Question 48

Solution 48

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 14

If 1/α + 1/β are the roots of the equation

Solution 14

Exercise MCQ

Question 1

Which of the following is a quadratic equation?

Solution 1

Question 2

Which of the following is a quadratic equation?

Solution 2

Question 3

Which of the following is not a quadratic equation?

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

If 1/α + 1/β are the roots of the equation

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

(a) Real and equal

(b) Real and unequal

(c) Imaginary

(d) None of theseSolution 21

Question 22

(a) Real, unequal and rational

(b) Real, unequal and irrational

(c) Real and equal

(d) ImaginarySolution 22

Question 23

(a) Real, unequal and rational

(b) Real, unequal and irrational

(c) Real and equal

(d) ImaginarySolution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

The perimeter of a rectangle is 82 m and its area is 400 m². The breadth of the rectangle is

(a) 25 m

(b) 20 m

(c) 16 m

(d) 9 mSolution 29

Question 30

The length of a rectangular field exceeds its breadth by 8 m and the area of the field is 240 m². The breadth of the field is

(a) 20 m

(b) 30 m

(c) 12 m

(d) 16 mSolution 30

Question 31

Solution 31Question 32

The sum of two natural numbers is 8 and their product is 15. Find the numbers.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

Solution 46

Question 47

Solution 47

Question 48

Solution 48

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

Exercise Ex. 4C

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 2

Solution 2

Question 3

Show that the roots of the equation are real for all real values of p and q.Solution 3

The given equation is 

This is the form of 

Now .

So, the roots of the given equation are real for all real value of p and q.Question 4

For what values of k are the roots of the quadratic equation real and equal.Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8(i)

Solution 8(i)

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

If the equation has equal roots , prove that Solution 14

The given equation is 

For real and equal roots, we must have D = 0

Question 15

If the roots of the equation are real and equal, show that either a = 0 or Solution 15

The given equation is 

For real and equal roots , we must have D =0

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19(i)

Solution 19(i)

Question 19(ii)

Solution 19(ii)

Question 19(iii)

Solution 19(iii)

Question 19(iv)

Solution 19(iv)

Question 20

Solution 20

Question 21

Solution 21

Question 23

Solution 23

Exercise Ex. 4D

Question 1

The sum of a natural number and its square is 156. Find the number.Solution 1

Question 2

The sum of a natural number and its positive square root is 132. Find the number.Solution 2

Question 3

The sum of two natural numbers is 28 and their product is 192. Find the numbers.Solution 3

Question 4

The sum of the squares of two consecutive positive integers is 365. Find the integers.Solution 4

Question 5

The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers.Solution 5

Question 6

The sum of the squares of two consecutive positive even numbers is 452. Find the numbers.Solution 6

Question 7

The product of two consecutive positive integers is 306. Find the integers.Solution 7

Question 8

Two numbers differ by 3 and their product is 504. Find the numbers.Solution 8

Let the required number be x and x – 3, then

Hence, the required numbers are (24,21) or (-21 and-24)Question 9

Find two consecutive multiples of 3 whose product is 648.Solution 9

Question 10

Find two consecutive positive odd integers whose product is 483.Solution 10

Question 11

Find two consecutive positive even integers whose product is 288.Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

The sum of the squares of two consecutive multiples of 7 is 1225. Find the multiples.Solution 16

Question 17

Solution 17

Question 18

Divide 57 into two parts whose product is 680.Solution 18

Question 19

Solution 19

Question 20

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.Solution 20

Let the smaller part and larger part be x, 16 – x

Then,

-42 is not a positive part

Hence, the larger and smaller parts are 10, 6 respectivelyQuestion 21

Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.Solution 21

Let the required number be x and y, hen

Question 22

The difference of the squares of two natural numbers is 45. The square of the smaller number is four times the larger number. Find the numbers.Solution 22

Let x, y be the two natural numbers and x > y

——(1)

Also, square of smaller number = 4 larger number

———(2)

Putting value of from (1), we get

Thus, the two required numbers are 9 and 6Question 23

Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers.Solution 23

Let the three consecutive numbers be x, x + 1, x + 2

Sum of square of first and product of the other two

Required numbers are 4, 5 and 6Question 24

A two-digit number is 4 times the sum of its digits and twice the product of its digits. Find the numbers.Solution 24

Let the tens digit be x and units digit be y

Hence, the tens digit is 3 and units digit is (2 3) is

Hence the required number is 36Question 25

A two-digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number.Solution 25

Let the tens digit and units digits of the required number be x and y respectively

The ten digit is 2 and unit digit is 7

Hence, the required number is 27Question 26

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is. Find the fraction.Solution 26

Let the numerator and denominator be x, x + 3

Then,

Hence, numerator and denominator are 2 and 5 respectively and fraction is Question 27

Solution 27

Question 28

Solution 28

Question 29

A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left. When he increased the size of the square by one student, he found he was short of 25 students. Find the number of students.Solution 29

Let there be x rows and number of student in each row be x

Then, total number of students = 

Hence total number of student

Total number of students is 600Question 30

300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of students.Solution 30

Let the number of students be x, then

Hence the number of students is 50Question 31

In a class test, the sum of Kamal’s marks in Mathematics and English is 40. Had he got 3 marks more in Mathematics and 4 marks less in English, the product of the marks would have been 360. Find his marks in two subjects separately.Solution 31

Let the marks obtained by Kamal in Mathematics and English be x and y

The marks obtained by Kamal in Mathematics and English respectively are (21,19) or (12,28)Question 32

Some students planned a picnic. The total budget for food was Rs.2000. But, 5 students failed to attend the picnic and thus the cost for food for each member increased by Rs.20. How many students attended the picnic and how much did each student pay for the food?Solution 32

Question 33

If the price of a book is reduced by Rs.5, a person can buy 4 more books for Rs.600. Find the original price of the book.Solution 33

Question 34

A person on tour has Rs.10800 for his expenses. If he extends his tour by 4 days, he has to cut down his daily expenses by Rs.90. Find the original duration of the tour.Solution 34

Question 35

In a class test, the sum of the marks obtained by P in mathematics and science is 28. Had he got 3 more marks in mathematics and 4 marks less in science, the product of marks obtained in the two subjects would have been 180. Find the marks obtained by him in the two subjects separately.Solution 35

Question 36

A man buys a number of pens for Rs.180. If he had bought 3 more pens for the same amount, each pen would have cost him Rs.3 less. How many pens did he buy?Solution 36

Question 37

A dealer sells an article for Rs.75 and gains as much per cent as the cost price of the article. Find the cost price of the article.Solution 37

Question 38(i)

One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.Solution 38(i)

Let the age of son be x and age of man = y

1 year ago

Question 39

Solution 39

Question 40

The sum of the ages of a boy and his brother is 25 years and the product of their ages in years is 126.Find their ages.Solution 40

Question 41

The product of Meena’s age 5 years ago and her age 8 years later is 30. Find her present age.Solution 41

Let the present age of Meena be x

Then,

Hence the present age of Meena is 7 yearsQuestion 42

Two years ago, a man’s age was three times the square of his son’s age. In three years time, his age will be four times his son’s age. Find their present ages.Solution 42

Question 43

A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed which is 20 km per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck.Solution 43

Question 44

While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalise the injured and so the plane started late by 30 minutes. To reach the destination, 1500 km away, in time, the pilot increased the speed by 100 km/hour. Find the original speed of the plane.

Do you appreciate the values shown by the pilot, namely promptness in providing help to the injured and his efforts to reach in time?Solution 44

Question 45

A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.Solution 45

Question 46

A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/hr more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?Solution 46

Question 47

A train travels 180 km at a uniform speed. If the speed had been 9 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.Solution 47

Question 48

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 kmph more, it would have taken 30 minutes less for the journey. Find the original speed of the train.Solution 48

Let the original speed of the train be x km.hour

Then speed increases by 15 km/ph = (x + 15)km/hours

Then time taken at original speed = 

Then, time taken at in increased speed = 

Difference between the two lines taken 

Then, original speed of the train = 45km / hQuestion 49

A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find its usual speed.Solution 49

Question 50

The distance between Mumbai and Pune is 192 km. Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speeds of the two trains differ by 20 kmph.Solution 50

Let the speed of the Deccan Queen = x kmph

The, speed of other train = (x – 20)kmph

Then, time taken by Deccan Queen = 

Time taken by other train = 

Difference of time taken by two trains is 

Hence, speed of Deccan Queen = 80km/hQuestion 51

A motorboat whose speed is 18 km per hour in still water takes 1 hour more to go 24 km upstream than to return to the same point. Find the speed of the stream.Solution 51

Let the speed of stream be x km/h

Speed of boat in still stream = 18 km/h

Speed of boat up the stream = 18 – x km/h

Time taken by boat to go up the stream 24 km = 

Time taken by boat to go down the stream = 

Time taken by the boat to go up the stream is 1 hour more that the time taken down the stream

Speed of the stream = 6 km/hQuestion 52

The speed of a boat in still water is 8 kmph. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.Solution 52

Let the speed of the stream be x kmph

Then the speed of boat down stream = (8 + x) kmph

And the speed of boat upstream = (8 – x)kmph

Time taken to cover 15 km upstream = 

Time taken to cover 22 km downstream = 

Total time taken = 5 hours

Hence, the speed of stream is 3 kmphQuestion 53

A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.Solution 53

Let the speed of the stream be = x km/h

Speed of boat in still waters = 9 km/h

Speed of boat down stream = 9 + x

time taken by boat to go 15 km downstream = 

Speed of boat upstream = 9 – x

time taken by boat to go 15 km of stream = 

Question 54

A takes 10 days less than time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work.Solution 54

Question 55

Two pipes running together can fill a cistern inminutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.Solution 55

Let the faster pipe takes x minutes to fill the cistern

Then, the other pipe takes (x + 3) minute

The faster pipe takes 5 minutes to fill the cistern

Then, the other pipe takes (5 + 3) minutes = 8 minutesQuestion 56

Solution 56

Question 57

Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.Solution 57

Question 58

The length of rectangle is twice its breadth and its area is 288 sq.cm. Find the dimensions of the rectangle.Solution 58

Let the breadth of a rectangle = x cm

Then, length of the rectangle = 2x cm

Thus, breadth of rectangle = 12 cm

And length of rectangle = (2 12) = 24 cmQuestion 59

The length of a rectangular field is three times its breadth. If the area of the field be 147 sq.m, find the length of the field.Solution 59

Let the breadth of a rectangle = x meter

Then, length of rectangle = 3x meter

Thus, breadth of rectangle = 7 m

And length of rectangle = (3 7)m = 21 mQuestion 60

The length of a hall is 3 metres more than its breadth. If the area of the hall is 238 sq.m, calculate its length and breadth.Solution 60

Let the breadth of hall = x meters

Then, length of the hall = (x + 3) meters

Area = length breadth = 

Thus, the breadth of hall is 14 cm

And length of the hall is (14 + 3) = 17 cmQuestion 61

The perimeter of a rectangular plot is 62 m and its area is 228 sq. metres. Find the dimensions of the plot.Solution 61

Question 62

A rectangular field is 16 m long and 10 m wide. There is a path of uniform width all around it, having an area of 120 m². Find the width of the path.Solution 62

Let the width of the path be x meters,

Then,

Area of path = 16 10 – (16 – 2x) (10 – 2x) = 120

Hence the required width is 3 meter as x cannot be 10mQuestion 63

The sum of the areas of two squares is 640 m². If the difference in their perimeters be 64 m, find the sides of the two squares.Solution 63

Let x and y  be the lengths of the two square fields.

4x – 4y = 64

x – y = 16——(2)

From (2),

x = y + 16,

Putting value of x in (1)

Sides of two squares are 24m and 8m respectivelyQuestion 64

The length of a rectangle is thrice as long as the side of a square. The side of the square is 4 cm more than the width of the rectangle. Their areas being equal, find their dimensions.Solution 64

Let the side of square be x cm

Then, length of the rectangle = 3x cm

Breadth of the rectangle = (x – 4) cm

Area of rectangle = Area of square x

Thus, side of the square = 6 cm

And length of the rectangle = (3 6) = 18 cm

Then, breadth of the rectangle = (6 – 4) cm = 2 cmQuestion 65

A farmer prepares a rectangular vegetable garden of area 180 sq. m. With 39 metres of barbed wire, he can fence the three sides of the garden, leaving one of the longer sides unfenced. Find the dimensions of the garden.Solution 65

Let the length = x meter

Area = length breadth = 

If ength of the rectangle = 15 m

Also, if length of rectangle = 24 m

Question 66

The area of a right triangle is 600 cm². If the base of the triangle exceeds the altitude by 10 cm, find the dimensions of the triangle.Solution 66

Let the altitude of triangle be x cm

Then, base of triangle is (x + 10) cm

Hence, altitude of triangle is 30 cm and base of triangle 40 cm

Question 67

The area of a right-angled triangle is 96 sq m. If the base is three times the altitude, find the base.Solution 67

Let the altitude of triangle be x meter

Hence, base = 3x meter

Hence, altitude of triangle is 8 cm

And base of triangle = 3x = (3 8) cm = 24 cmQuestion 68

The area of a right-angled triangle is 165 sq m. Determine its base and altitude if the latter exceeds the former by 7 metres.Solution 68

Let the base of triangle be x meter

Then, altitude of triangle = (x + 7) meter

Thus, the base of the triangle = 15 m

And the altitude of triangle = (15 + 7) = 22 mQuestion 69

The hypotenuse of a right-angled triangle is 20 metres. If the difference between the lengths of the other sides be 4 m, find the other sides.Solution 69

Let the other sides of triangle be x and (x -4) meters

By Pythagoras theorem, we have

Thus, height of triangle be = 16 cm

And the base of the triangle = (16 – 4) = 12 cmQuestion 70

The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle.Solution 70

Let the base of the triangle be x

Then, hypotenuse = (x + 2) cm

Thus, base of triangle = 15 cm

Then, hypotenuse of triangle = (15 +2 )= 17 cm

And altitude of triangle = Question 71

The hypotenuse of a right-angled triangle is 1 metre less than twice the shortest side. If the third side is 1 metre more than the shortest side, find the sides of the triangle.Solution 71

Let the shorter side of triangle be x meter

Then, its hypotenuse = (2x – 1)meter

And let the altitude = (x + 1) meter

Exercise 29

Question 29

The perimeter of a rectangle is 82 m and its area is 400 m². The breadth of the rectangle is

(a) 25 m

(b) 20 m

(c) 16 m

(d) 9 mSolution 29