# Solution of Chapter 8. Quadratic Equations (RD Sharma - Mathematics Book)

## Exercise 8.1

1

Which of the following are quadratic equations?

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)

2

In each of the following, determine whether the given values are solutions of the given equation or not:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

3

In each of the following, find the value of k for which the given value is a solution of the given equation:

(i)

(ii)

(iii)

(iv)

4

If are the roots of the equation , find the values of a and b.

5

Determine, if 3 is a root of the equation given below:

## Exercise 8.2

1

The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers, if x denoted the smaller integer.

2

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Form the quadratic equation to find how many marbles they had to start with, if john and x marbles.

3

A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of articles produced in a day. On a particular day, the total cost of production was Rs.750. If x denotes the number of toys produced that day, form the quadratic equation to find x.

4

The height of a right TRIANGLE IS 7 CM LESS THAN ITS BASE. If the hypotenuse is 13 cm, form the quadratic equation to find the base of the triangle.

5

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 11 km/hr more than that of the passenger train, form the quadratic equation to find the average speed of express train.

6

A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 4 hour less for the same journey. Form the quadratic equation to find the speed of the train.

## Exercise 8.3

1

Solve the following quadratic equations by factorization:

2

Solve the following quadratic equations by factorization:

3

Solve the following quadratic equations by factorization:

4

Solve the following quadratic equations by factorization:

5

Solve the following quadratic equations by factorization:

6

Solve the following quadratic equations by factorization:

7

Solve the following quadratic equations by factorization:

8

Solve the following quadratic equations by factorization:

9

Solve the following quadratic equations by factorization:

10

Solve the following quadratic equations by factorization:

11

Solve the following quadratic equations by factorization:

12

Solve the following quadratic equations by factorization:

13

Solve the following quadratic equations by factorization:

14

Solve the following quadratic equations by factorization:

15

Solve the following quadratic equations by factorization:

16

Solve the following quadratic equations by factorization:

17

Solve the following quadratic equations by factorization:

18

Solve the following quadratic equations by factorization:

19

Solve the following quadratic equations by factorization:

20

Solve the following quadratic equations by factorization:

21

Solve the following quadratic equations by factorization:

22

Solve the following quadratic equations by factorization:

23

Solve the following quadratic equations by factorization:

24

Solve the following quadratic equations by factorization:

25

Solve the following quadratic equations by factorization:

26

Solve the following quadratic equations by factorization:

27

Solve the following quadratic equations by factorization:

28

Solve the following quadratic equations by factorization:

29

Solve the following quadratic equations by factorization:

30

Solve the following quadratic equations by factorization:

31

Solve the following quadratic equations by factorization:

32

Solve the following quadratic equations by factorization:

33

Solve the following quadratic equations by factorization:

34

Solve the following quadratic equations by factorization:

35

Solve the following quadratic equations by factorization:

36

Solve the following quadratic equations by factorization:

37

Solve the following quadratic equations by factorization:

38

Solve the following quadratic equations by factorization:

39

Solve the following quadratic equations by factorization:

40

Solve the following quadratic equations by factorization:

41

Solve the following quadratic equations by factorization:

42

Solve the following quadratic equations by factorization:

43

Solve the following quadratic equations by factorization:

Solve for x:

44

Solve the following quadratic equations by factorization:

45

Solve the following quadratic equations by factorization:

46

Solve the following quadratic equations by factorization:

47

Solve the following quadratic equations by factorization:

48

Solve the following quadratic equations by factorization:

49

Solve the following quadratic equations by factorization:

50

Solve the following quadratic equations by factorization:

51

Solve the following quadratic equations by factorization:

52

Solve the following quadratic equations by factorization:

53

Solve the following quadratic equations by factorization:

54

Solve the following quadratic equations by factorization:

55

Solve the following quadratic equations by factorization:

56

Solve the following quadratic equations by factorization:

57

Solve the following quadratic equations by factorization:

58

Solve the following quadratic equations by factorization:

59

Solve the following quadratic equations by factorization:

60

Solve the following quadratic equations by factorization:

## Exercise 8.4

1

Find the roots of the following quadratic (if they exist) by the method of completing the square.

2

Find the roots of the following quadratic (if they exist) by the method of completing the square.

3

Find the roots of the following quadratic (if they exist) by the method of completing the square.

4

Find the roots of the following quadratic (if they exist) by the method of completing the square.

5

Find the roots of the following quadratic (if they exist) by the method of completing the square.

6

Find the roots of the following quadratic (if they exist) by the method of completing the square.

7

Find the roots of the following quadratic (if they exist) by the method of completing the square.

8

Find the roots of the following quadratic (if they exist) by the method of completing the square.

9

Find the roots of the following quadratic (if they exist) by the method of completing the square.

10

Find the roots of the following quadratic (if they exist) by the method of completing the square.

## Exercise 8.5

1

Write the discriminant of the following quadratic equations:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

2

In the following determine whether the given quadratic equations have real roots and if so, find the roots:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

3

Solve for x:

(i)

(ii)

(iii)

(iv)

## Exercise 8.6

1

Determine the nature of the roots of the following quadratic equations:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii) ,

(viii)

(ix)

2

Find the values of k for which the roots are real and equal in each of the following equations:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii) x2 – 2kx + 7x + 1/4 = 0

(xiii)

(xiv)

(xv)

(xvi)

(xvii)

(xviii)

(xix)

(xx)

(xxi)

(xxii)

(xxiii)

(xxiv)

(xxv)

(xxvi)

3

In the following, determine the set of values of k for which the given quadratic equation has real roots:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

4

For what value of k, , is a perfect square.

5

Find the least positive value of k for which the equation has real roots.

6

Find the values of k for which the given quadratic equation has real and distinct roots:

(i)

(ii)

(iii)

7

If the roots of the equation are equal, then prove that 2b = a + c.

8

If the roots of the equation (a2 + b2)x2 - 2(ab + cd)x + (c2 + d2) = 0 are equal, prove that .

9

If the roots of the equations and are simultaneously real, then prove that b2 = ac.

10

If p, q are real and p q, then show that the roots of the equation are real and unequal

11

If the roots of the equation are equal, prove that either a = 0 or a3 + b3 + c3 = 3abc.

12

Show that the equation has no real roots, when .

13

Prove that both the roots of the equation are real but they are equal only when .

14

If a, b, c are real numbers such that ac 0, then show that at least one of the equations and has real roots.

15

If the equation has equal roots, prove that .

16

Find the values of k for which the quadratic equation has equal roots. Also, find these roots.

17

Find the values of p for which the quadratic equation has equal roots. Also, find these roots.

18

If -5 is a root of the quadratic equation and the quadratic equation has equal roots, find the value of k.

19

If 2 is a root of the quadratic equation and the quadratic equation has equal roots, find the value of k.

20

If 1 is a root of the quadratic equation and the quadratic equation has equal roots, find the value of b.

21

Find the value of p for which the quadratic equation: (p+1)x2 - 6(p+1)x + 3(p + 9) = 0, where, p≠-1 has equal roots. Hence, find the roots of the equation.

## Exercise 8.7

1

Find two consecutive numbers whose squares have the sum of 85

2

Divide 29 into two parts so that the sum of the squares of the parts is 425.

3

Two squares have sides x cm and (x + 4) cm. The sum of their areas is 656 cm2. Find the sides of the squares.

4

The sum of two numbers is 48 and their product is 432. Find the numbers.

5

If an integer is added to its square, the sum is 90. Find the integer with the help of quadratic equation.

6

Find the whole number which when decreased by 20 is equal to 69 times the reciprocal of the number.

7

Find two consecutive natural numbers whose product is 20.

8

The sum of the squares of two consecutive odd positive integers is 394. Find then.

9

The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.

10

The sum of a number and its positive square root is 6/25. Find the number.

11

The sum of a number and its square is 63/4, find the numbers.

12

There are three consecutive integers such that the square of the first increased by the product of the other two gives 154. What are the integers?

13

The product of two successive integral multiples of 5 is 300. Determine the multiples.

14

The sum of the squares of two numbers is 233 and one of the numbers is 3 less than twice the other number. Find the numbers.

15

Find the consecutive even integers whose squares have the sum 340.

16

The difference of two numbers is 4. If the difference of their reciprocals is , find the numbers.

17

Find two natural numbers which differ by 3 and whose squares have the sum 117.

18

The sum of the squares of three consecutive natural numbers is 149. Find the numbers.

19

The sum of two numbers is 16. The sum of their reciprocals is 1/3. Find the numbers.

20

Determine two consecutive multiples of 3 whose product is 270.

21

The sum of a number and its reciprocal is 17/4. Find the number.

22

A two-digit number is such that the product of its digits is 8. When 18 is subtracted from the number, the digits interchange their places. Find number.

23

A two-digit number is such that the product of the digits is 12. When 36 is added to the number the digits interchange their places. Determine the number.

24

A two-digit number is such that the product of the digits is 16. When 54 is subtracted from the number, the digits are interchanged. Find the number.

25

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

26

Two numbers differ by 4 and their product is 192. Find the numbers.

27

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

28

The difference of the squares of two positive integers is 180. The square of the smaller number is 8 times the larger number, find the numbers.

29

The sum of two numbers is 18. The sum of their reciprocals is 1/4. Find the numbers.

30

The sum of two numbers a and b is 15, and the sum of their reciprocals is 3/10. Find the numbers a and b.

31

The sum of two numbers is 9. The sum of their reciprocals is 1/2. Find the numbers.

32

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

33

The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.

34

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

35

Find two consecutive odd positive integers, sum of whose squares is 970.

36

The difference of two natural numbers is 3 and the difference of their reciprocals is . Find the numbers.

37

The sum of the squares of two consecutive odd numbers is 394. Find the numbers.

38

The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples.

39

The sum of the squares of two consecutive even numbers is 340. Find the numbers.

40

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is , find the original fraction.

## Exercise 8.8

1

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

2

A passenger train takes 3 hours less for a journey of 360 km, if its speed is increased by 10 km / hr from its usual speed. What is the usual speed?

3

A fast train takes one hour less than a slow train for a journey of 200 km. If the speed of the slow train is 10 km / hr less than that of the fast rain, find the speed of the two trains.

4

A passenger train takes one hour less for a journey of 150 km if its speed is increased by

5 km/hr from the usual speed. Find the usual speed of the train.

5

The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km / hr more than the speed of going, what was the speed per hour in each direction?

6

A plane left 40 minutes late due to bad weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km / hr from its usual speed. Find the usual speed of the plane.

7

An areoplane takes 1 hour less for a journey of 1200 km if its speed is increased by 100 km / hr from its usual speed. Find its usual speed.

8

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 the usual speed of the train.

9

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

10

A train travels 360 km at a uniform speed. If the speed had been 5 km / hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.

11

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express trains is 11 km / hr more than that of the passenger train, find the average speeds of the two trains.

12

An aeroplane left 50 minutes later than its schedule time, and in order to reach the destination, 1250 km away, in time, it had to increase its speed b 250 km / hr from its usual speed. Find its usual speed.

13

While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalize 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 / hr. Find the original speed / hour of the plane.

14

A motorboat whose speed in still water is 18 km/hr takes 1 hour more to go 24 km upstream that to return downstream to the same spot. Find the speed of the stream.

## Exercise 8.9

1

Ashu is x years old while his mother Mrs. Veena is x2 years old. Five years hence Mrs. Veena will be three times old as Ashu. Find their present ages.

2

The sum of the ages of a man and his son is 45 years. Five years ago, the product of their ages was four times the man’s age at the time. Find their present ages.

3

The product of Shikha’s age five years ago and her age 8 years later is 30, her age at both times being given in years. Find her present age.

4

The product of Ramu’s age (in years) five years ago and his age (in years) nine years later is 15. Determine Ramu’s present age.

5

Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

6

A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

7

The sum of the reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is 1/3. Find the present age.

## Exercise 8.10

1

The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.

2

The hypotenuse of a right triangle is cm. If the smaller leg is tripled and the longer leg doubled, new hypotenuse will be cm. How long are the legs of the triangle?

3

A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

4

The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field

## Exercise 8.11

1

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

2

The length of a hall is 5 m more than its breadth. If the area of the floor of the hall is 84 m2, what are the length and breadth of the hall?

3

Two squares have sides x cm and (x + 4) cm. The sum of their areas is 656 cm2. Find the sides of the squares.

4

The area of a right angled triangle is 165 m2. Determine its base and altitude if the latter exceeds the former by 7 m.

5

Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m2? If so, find its length and breadth.

6

Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.

7

Sum of the areas of two squares is 640 m2. If the difference of their perimeters is 64 m, find the sides of the two squares.

8

Sum of the areas of two squares is 400 cm2. If the difference of their perimeters is 16 cm, find the sides of two squares.

9

The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one metre more then twice its breadth. Find the length and the breadth of the plot.

## Exercise 8.12

1

A takes 10 days less than the 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.

2

If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours will the second pipe take to fill the reservoir?

3

Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

4

Two pipes running together can fill a tank in minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.

5

To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?

## Exercise 8.13

1

A piece of cloth costs Rs. 35. If the piece were 4 m longer and each metre costs Rs. 1 less, the cost would remain unchanged. How long is the piece?

2

Some students planned a picnic. The budget for food was Rs. 480. But eight of these failed to go and thus the cost of food for each member increased by Rs. 10. How many students attended the picnic?

3

A dealer sells an article for Rs. 24 and gains as much percent as the cost price of the article. Find the cost price of the article.

4

Out of a group of swans, 7/2 times the square root of the total number are playing on the share of a pond. The two remaining ones are swinging in water. Find the total number of swans.

5

If the list price of a toy is reduced by Rs. 2, a person can buy 2 toys more for Rs. 360. Find the original price of the toy.

6

Rs. 9000 were divided equally among a certain number of persons. Had there been 20 more persons, each would have got Rs. 160 less. Find the original number of persons.

7

Some students planned a picnic. The budget for food was Rs. 500. But, 5 of them failed to go and thus the cost of food for each member increased by Rs. 5. How many students attended the picnic?

8

A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

9

In a class test, the sum of the marks obtained by P in Mathematics and science is 28. Had he got 3 marks more in Mathematics and 4 marks less in Science. The product of his marks, would have been 180. Find his marks in the two subjects.

10

In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of her marks would have been 210. Find her marks in two subjects.

11

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs. 90, find the number of articles produced and the cost of each article.

## CCE - Formative Assessment

1

Write the value of k for which the quadratic equation x2 – kx + 4 = 0 has equal roots.

2

What is the nature of roots of the quadratic equation 4x2 – 12x – 9 = 0?

3

If 1 + √2 is a root of a quadratic equation with rational coefficients, write its other root.

4

Write the number of real roots of the equation x2 + 3 |x| + 2 = 0.

5

Write the sum of real roots of the equation x2 + |x| – 6 = 0.

6

Write the set of values of 'a' for which the equation x2 + ax – 1 = 0 has real roots.

7

Is there any real value of 'a' for which the equation

x2 + 2x + (a2 + 1) = 0 has real roots?

8

Write the value of λ for which x2 + 4x + λ, is a perfect square.

9

Write the condition to be satisfied for which equations ax2 + 2bx + c = 0 and have equal roots.

10

Write the set of values of k for which the quadratic equation has 2x2 + kx + 8 = 0 has real roots.

11

Write a quadratic polynomial, sum of whose zeros is 2√3 and their product is 2.

12

Show that x = – 3 is a solution of x2 + 6x + 9 = 0.

13

Show that x = – 2 is a solution of 3x2 + 13x + 14 = 0.

14

Find the discriminant of the quadratic equation 3√3x2 + 10x + √3 = 0

15

If , is a solution of the quadratic equation 3x2 + 2kx – 3 = 0, find the value of k.

1

If the equation x2 + 4x + k = 0 has real and distinct roots, then

2

If the equation x2 – ax + 1 = 0 has two distinct roots, then

3

If the equation 9x2 + 6kx + 4 = o has equal roots, then the roots are both equal to ?

4

If ax2 + bx + c = 0 has equal roots, then c =

5

If the equation ax2 + 2x + a = 0 has two distinct roots, if

6

The positive value of k for which the equation x2 + kx + 64 = 0 and x2 – 8x + k = 0 will both have real roots, is

7

The value of is

8

If 2 is a root of the equation x2 + bx + 12 = 0 and the equation x2 + bx + q = 0 has equal roots, then q =

9

If the equation (a2 + b2) x2 – 2 (ac + bd) x + c2 + d2 = 0 has equal roots, then

10

If the roots of the equation (a2 b2) x2 – 2b (a + c) x + (b2 + c2) = 0 are equal, then

11

If the equation x2 – bx + 1 = 0 does not possess real roots, then

12

If x = 1 is a common root of the equations ax2 + ax + 3 = 0 and x2 + x + b = 0, then ab =

13

If p and q are the roots of the equation x2 + px + q = 0, then

14

If a and b can take values 1, 2, 3, 4. Then the number of the equations of the form ax2 + bx + 1 = 0 having real roots is

15

The number of quadratic equations having real roots and which do not change by squaring their roots is

16

If (a2 + b2) x2 + 2 (ac + bd) x + c2 + d2 = 0 has no real roots, then

17

If the sum of the roots of the equation x2 – x = λ (2x – 1) is zero, then λ =

18

If x = 1 is a common root of ax2 + ax + 2 = 0 and x2 + x + b = 0 then, ab =

19

The value of c for which the equation ax2 + 2bx + c = 0 has equal roots is

20

If x2 + k (4x + k – 1) + 2 = 0 has equal roots, then k =

21

If the sum and product of the roots of the equation kx2 + 6x + 4k = 0 are equal, then k =

22

If sin α and cos α are the roots of the equation ax2 + bx + c = 0, then b2 =

23

If 2 is a root of the equation x2 + ax + 12 = 0 and the quadratic equation x2 + ax + q = 0 has equal roots, then q

24

If the sum of the roots of the equation x2 – (k + 6)x + 2 (2k – 1) = 0 is equal to half of their product, then k =

25

If a and b are roots of the equation x2 + a x + b = 0, then a + b =

26

A quadratic equation whose one root is 2 and the sum of whose roots is zero, is

27

If one root of the equation ax2 + bx + c = 0 is three times the other, then b2 :ac =

28

If one root of the equation 2x2 + kx + 4 = 0 is 2, then the other root is

29

If one root of the equation x2 + ax + 3 = 0 is 1, then its other root is

30

If one root of the equation 4x2 – 2x + (λ – 4) = 0 be the reciprocal of the other, then k