# Solution of Chapter 10. Quadratic Equations (RS Aggarwal - Mathematics Book)

## Exercise 10A

1

Which of the following are quadratic equations in x?

x2 - x + 3 = 0

1

Which of the following are quadratic equations in x? 1

Which of the following are quadratic equations in x? 1

Which of the following are quadratic equations in x? 1

Which of the following are quadratic equations in x? 1

Which of the following are quadratic equations in x? 1

Which of the following are quadratic equations in x? 1

Which of the following are quadratic equations in x?

(x + 2)3 = x3 – 8

1

Which of the following are quadratic equations in x?

2x + 3)(3x + 2) = 6(x - 1)(x - 2)

1

Which of the following are quadratic equations in x? 2

Which of the following are the roots of 3x2 + 2x - 1 = 0 ?

(i) - 1 (ii) 1/3 (iii) –1/2

3

Find the value of k for which x = 1 is a root of the equation x2 + kx + 3 = 0. Also, find the other root.

4

Find the values of a or b for which x = 3/4 or x = - 2 are the roots of the equation

ax2 + bx – 6 = 0

5

Solve each of the following quadratic equations:

(2x-3) (3x + 1) = 0

6

Solve each of the following quadratic equations:

4x2 + 5x = 0

7

Solve each of the following quadratic equations:

3x2 – 243 = 0

8

Solve each of the following quadratic equations:

2x2 + x - 6 = 0

9

Solve each of the following quadratic equations:

x2 + 6x + 5 = 0

10

Solve each of the following quadratic equations:

9x2 - 3x - 2 = 0

11

Solve each of the following quadratic equations:

x2 + 12x + 35 = 0

12

Solve each of the following quadratic equations:

x2 = 18x – 77

13

Solve each of the following quadratic equations:

6x2 + 11x + 3 = 0

14

Solve each of the following quadratic equations:

6x2 + x - 12 = 0

15

Solve each of the following quadratic equations:

3x2 - 2x - 1 = 0

16

Solve each of the following quadratic equations:

4x2 - 9x = 100

17

Solve each of the following quadratic equations:

15x2 - 28 = x

18

Solve each of the following quadratic equations:

4 - 11x = 3x2

19

Solve each of the following quadratic equations:

48x2 - 13x - 1 = 0

20

Solve each of the following quadratic equations:

x2 + 2√2 x – 6 = 0

21

Solve each of the following quadratic equations:

√3x2 + 10x + 7√3 = 0

22

Solve each of the following quadratic equations:

√3x2 + 11x + 6√3 = 0

23

Solve each of the following quadratic equations:

3√7x2 + 4x – √7 = 0

24

Solve each of the following quadratic equations:

√7x2 – 6x – 13√7 = 0

25

Solve each of the following quadratic equations:

4√6x2 – 13 x – 2√6 = 0

26

Solve each of the following quadratic equations:

3x2 – 2√6x + 2 = 0

27

Solve each of the following quadratic equations:

√3x2 – 2√2x – 2√3 = 0

28

Solve each of the following quadratic equations:

x2 – 3√5x + 10 = 0

29

Solve each of the following quadratic equations:

x2 – (√3 + 1) x + √3 = 0

30

Solve each of the following quadratic equations:

x2 + 3√3x – 30 = 0

31

Solve each of the following quadratic equations:

√2x2 + 7x + 5√2 = 0

32

Solve each of the following quadratic equations:

5x2 + 13x + 8 = 0

33

Solve each of the following quadratic equations:

x2 – (1+ √2)x + √2 = 0

34

Solve each of the following quadratic equations:

9x2 + 6x + 1 = 0

35

Solve each of the following quadratic equations:

100x2 - 20x + 1 = 0

36

Solve each of the following quadratic equations: 37

Solve each of the following quadratic equations: 38

Solve each of the following quadratic equations: 39

Solve each of the following quadratic equations:

2x2 + ax - a2 = 0

40

Solve each of the following quadratic equations:

4x2 + 4bx - (a2 - b2) = 0

41

Solve each of the following quadratic equations:

4x2 - 4a2x + (a4 - b4) = 0

42

Solve each of the following quadratic equations:

x2 + 5x - (a2 + a - 6) = 0

43

Solve each of the following quadratic equations:

x2 - 2ax - (4b2 - a2) = 0

44

Solve each of the following quadratic equations:

x2 - (2b - 1)x + (b2 - b - 20) = 0

45

Solve each of the following quadratic equations:

x2 + 6x - (a2 + 2a - 8) = 0

46

Solve each of the following quadratic equations:

abx2 + (b2 - ac)x - bc = 0

47

Solve each of the following quadratic equations:

x2 - 4ax - b2 + 4a2 = 0

48

Solve each of the following quadratic equations:

4x2 - 2 (a2 + b2) x + a2b2 = 0

49

Solve each of the following quadratic equations:

12abx2 - (9a2 - 8b2)x - 6ab = 0

50

Solve each of the following quadratic equations:

a2b2x2 + b2x - a2x - 1 = 0

51

Solve each of the following quadratic equations:

9x2 - 9 (a + b)x + (2 a2 + 5ab + 2b2) = 0

52

Solve each of the following quadratic equations: 53

Solve each of the following quadratic equations: 54

Solve each of the following quadratic equations: 55

Solve each of the following quadratic equations: 56

Solve each of the following quadratic equations: 57

Solve each of the following quadratic equations: 58

Solve each of the following quadratic equations: 59

Solve each of the following quadratic equations: 60

Solve each of the following quadratic equations: 61

Solve each of the following quadratic equations: 62

Solve each of the following quadratic equations: 63

Solve each of the following quadratic equations: 64

Solve each of the following quadratic equations: 65

Solve each of the following quadratic equations: 66

Solve each of the following quadratic equations: 67

Solve each of the following quadratic equations: 68

Solve each of the following quadratic equations: 69

Solve each of the following quadratic equations: 70

Solve each of the following quadratic equations: 71

Solve each of the following quadratic equations:

3(x + 2) + 3 - x = 10

72

Solve each of the following quadratic equations:

4(x + 1) + 4(1 - x) = 10

73

Solve each of the following quadratic equations:

22x - 3.2(x + 2) + 32 = 0

## Exercise 10B

1

Solve each of the following equations by using the method of completing the square:

x2 - 6x + 3 = 0

2

Solve each of the following equations by using the method of completing the square:

x2 - 4x + 1 = 0

3

Solve each of the following equations by using the method of completing the square:

x2 + 8x - 2 = 0

4

Solve each of the following equations by using the method of completing the square: 5

Solve each of the following equations by using the method of completing the square:

2x2 + 5x - 3 = 0

6

Solve each of the following equations by using the method of completing the square:

3x2 - x - 2 = 0

7

Solve each of the following equations by using the method of completing the square:

8x2 - 14x - 15 = 0

8

Solve each of the following equations by using the method of completing the square:

7x2 + 3x - 4 = 0

9

Solve each of the following equations by using the method of completing the square:

3x2 - 2x - 1 = 0

10

Solve each of the following equations by using the method of completing the square:

5x2 - 6x - 2 = 0

11

Solve each of the following equations by using the method of completing the square: 12

Solve each of the following equations by using the method of completing the square:

4x2 + 4bx - (a2 - b2) = 0

14

Solve each of the following equations by using the method of completing the square: 15

Solve each of the following equations by using the method of completing the square:

√3x2 + 10 x + 7√3 = 0

16

By using the method of completing the square, show that the equation 2x2 + x + 4 = 0 has no real roots.

## Exercise 10C

1

Find the discriminant of each of the following equations:

2x2 – 7x + 6 = 0

1

Find the discriminant of each of the following equations:

3x2 – 2x + 8 = 0

1

Find the discriminant of each of the following equations:

2x2 – 5√x + 4 = 0

1

Find the discriminant of each of the following equations:

√3x2 + 2√2x – 2√3 = 0

1

Find the discriminant of each of the following equations:

(x – 1)(2x – 1) = 0

1

Find the discriminant of each of the following equations:

1 – x = 2x2

2

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 – 4x – 1 = 0

3

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 – 6x + 4 = 0

4

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

2x2 + x – 4 = 0

5

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

25x2 + 30x + 7 = 0

6

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

16x2 = 24x + 1

7

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

15x2 – 28 = x

8

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

2x2 – 2√2x + 1 = 0

9

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

√2x2 + 7x + 5√2 = 0

10

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

√3x2 + 10x – 8√3 = 0

11

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

√3x2 – 2√2x –2√3 = 0

12

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

2x2 + 6√3x – 60 = 0

13

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

4√3x2 + 5x – 2√3 = 0

14

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

3x2 –2√6x + 2 = 0

15

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

2√3x2 – 5x + √3 = 0

16

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 + x + 2 = 0

17

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

2x2 + ax – a2 = 0

18

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 – (√3 + 1) x + √3 = 0

19

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

2x2 + 5√3x + 6 = 0

20

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

3x2 – 2x + 2 = 0

21

Find the roots of each of the following equations, if they exist, by applying the quadratic formula: 22

Find the roots of each of the following equations, if they exist, by applying the quadratic formula: 23

Find the roots of each of the following equations, if they exist, by applying the quadratic formula: 24

Find the roots of each of the following equations, if they exist, by applying the quadratic formula: 25

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

36x2 – 12ax + (a2 – b2) = 0

26

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 – 2ax + (a2 – b2) = 0

27

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 – 2ax – (4b2 – a2) = 0

28

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 + 6x – (a2 + b2 – 8) = 0

29

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 + 5x – (a2 + a – 6) = 0

30

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 – 4ax – b2 + 4a2 = 0

31

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

4 x2 – 4a2x + (a4 – b4) = 0

32

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

4 x2 + 4bx – (a2 – b2) = 0

33

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

x2 – (2b – 1)x + (b2 – b – 20) = 0

34

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

3a2x2 + 8abx + 4b2 = 0, a ≠ 0

35

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

a2b2x2 – (4b4 – 3a4)x – 12a2b2 = 0, a ≠ 0 and b ≠ 0

36

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

12abx2 – (9a2 – 8b2)x – 6ab = 0, where a ≠ 0 and b ≠ 0

## Exercise 10D

1

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

2x2 – 8x + 5 = 0

1

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

3x2 –2√6x + 2 = 0

1

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

5x2 – 4x + 1 = 0

1

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

5x (x – 2) + 6 = 0

1

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

12x2 – 4√15x + 5 = 0

1

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

x2 – x + 2 = 0

2

If a and b are distinct real numbers, show that the quadratic equation 2 (a2 + b2)x2 + 2(a + b)x + 1 = 0 has no real roots.

3

Show that the roots of the equation x2 + px – q2 = 0 are real for all real values of p and q.

4

For what values of k are the roots of the quadratic equation 3x2 + 2kx + 27 = 0 real and equal?

5

For what value of k are the roots of the quadratic equation kx(x –2√5) + 10 = 0 real and equal?

6

For what values of p are the roots of the equation 4 x2 + px + 3 = 0 real and equal?

7

Find the nonzero value of k for which the roots of the quadratic equation 9x2 – 3kx + k = 0 are real and equal.

8

Find the values of k for which the quadratic equation (3k + 1) x2 + 2(k + 1)x + 1 = 0 has real and equal roots.

9

Find the values of p for which the quadratic equation (2p + 1)x2 – (7p + 2)x + (7p – 3) = 0 has real and equal roots.

10

Find the values of p for which the quadratic equation (p + 1)x2 – 6 (p + 1) x + 3 (p + 9) = 0, p 1 has equal roots. Hence, find the roots of the equation.

11

If – 5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p (x2 + x) + k = 0 has equal roots, find the value of k.

12

If 3 is a root of the quadratic equation x2 – x + k = 0, find the value of p so that the roots of the equation x2 + k (2x + k + 2) + p = 0 are equal.

13

If – 4 is a root of the equation x2 + 2x + 4p = 0, find the value of k for which the quadratic equation x2 + px (1 + 3k) + 7(3 + 2k) = 0 has equal roots.

14

If the quadratic equation (1 + m2)x2 + 2mcx + c2 – a2 = 0 has equal roots, prove that c2 = a2(1 + m2) .

15

If the roots of the equation (c2 – ab)x2 – 2(a2 – bc)x + (b2 – ac) = 0 are real and equal, show that either a = 0 or (a3 + b3 + c3) = 3abc.

16

Find the values of p for which the quadratic equation 2x2 + px + 8 = 0 has real roots.

17

Find the value of a for which the equation (a – 12)x2 + 2(a – 12)x + 2 = 0 has equal roots.

18

Find the value of k for which the roots of 9x2 + 8kx + 16 = 0 are real and equal.

19

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

20

If a and b are real and a b then show that the roots of the equation (a – b)x2 + 5(a + b)x – 2(a – b) = 0 are real and unequal.

21

If the roots of the equation (a2 + b2)x2 – 2 (ac + bd)x + (c2 + d2) = 0 are a c equal, prove that 22

If the roots of the equations ax2 + 2bx + c = 0 and are simultaneously real then prove that b2 = ac.

## Exercise 10E

1

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

2

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

3

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

4

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

5

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

6

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

7

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

8

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

9

Find two consecutive multiples of 3 whose product is 648.

10

Find two consecutive positive odd integers whose product is 483.

11

Find two consecutive positive even integers whose product is 288.

12

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

13

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

14

The difference of two natural numbers is 3 and the difference of their 3 reciprocals is 3/28.Find the numbers.

15

The difference of two natural numbers is 5 and the difference of their reciprocals is 5/14. Find the numbers.

16

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

17

The sum of a natural number and its reciprocal is 65/8. Find the number.

18

Divide 57 into two parts whose product is 680.

19

Divide 27 into two parts such that the sum of their reciprocals is 3/20.

20

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

21

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

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.

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.

24

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

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.

26

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

27

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by . Find the fraction.

28

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

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 that he was short of 25 students. Find the number of students.

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.

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.

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?

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.

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.

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.

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?

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.

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.

39

The sum of the reciprocals of Meena's ages (in years) 3 years ago and 5 years hence is 1/3. Find her present age.

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.

41

The product of Tanvy's age (in years) 5 years ago and her age 8 years later is 30. Find her present age.

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.

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.

44

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/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?

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.

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?

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.

48

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

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.

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 km/hr.

51

A motor boat whose speed in still water is 18 km/hr, takes 1 hour more to go 24 km upstream than to return to the same spot. Find the speed of the stream.

52

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.

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.

54

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.

55

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

56

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.

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.

58

The length of a rectangle is twice its breadth and its area is 288 cm2. Find the dimensions of the rectangle.

59

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

60

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

61

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

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.

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.

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.

65

A farmer prepares a rectangular vegetable garden of area 180 sq metres. 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.

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.

67

The area of a right – angled triangle is 96 sq metres. If the base is three times the altitude, find the base.

68

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

69

The hypotenuse of a right – angled triangle is 20 meters. If the difference between the lengths of the other side’s be 4 meters, find the other sides.

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.

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.

## Exercise 10F

1

Which of the following is a quadratic equation?

2

Which of the following is a quadratic equation?

3

Which of the following is not a quadratic equation?

4

If x = 3 is a solution of the equation 3x2 + (k – 1)x + 9 = 0 then k = ?

5

If one root of the equation 2x2 + ax + 6 = 0 is 2 then a = ?

6

The sum of the roots of the equation x2 – 6x + 2 = 0 is

7

If the product of the roots of the equation x2 – 3x + k = 10 is - 2 then the value of k is

8

The ratio of the sum and product of the roots of the equation

7x2 - 12x + 18 = 0 is

9

If one root of the equation 3x2 - 10x + 3 = 0 is 1/3 then the other root is

10

If one root of 5x2 + 13x + k = 0 be the reciprocal of the other root then the value of k is

11

If the sum of the roots of the equation kx2 + 2x + 3k = 0 is equal to their product then the value of k is

12

The roots of a quadratic equation are 5 and - 2. Then, the equation is

13

If the sum of the roots of a quadratic equation is 6 and their product is 6, the equation is

14

If α and β are the roots of the equation 3x2 + 8x + 2 = 0 then = ?

15

The roots of the equation ax2 + bx + c = 0 will be reciprocal of each other if

16

If the roots of the equation ax2 + bx + c = 0 are equal then c = ?

17

If the equation 9x2 + 6kx + 4 = 0 has equal roots then k = ?

18

If the equation x2 + 2(k + 2)x + 9k = 0 has equal roots then k = ?

19

If the equation 4x2 - 3kx + 1 = 0 has equal roots then k = ?

20

The roots of ax2 + bx + c = 0, a 0 are real and unequal, if (b2 - 4ac) is

21

In the equation ax2 + bx + c = 0, it is given that D = (b2 - 4ac) > 0. Then, the roots of the equation are

22

The roots of the equation 2x2 - 6x + 7 = 0 are

23

The roots of the equation 2x2 - 6x + 3 = 0 are

24

If the roots of 5x2 - kx + 1 = 0 are real and distinct then

25

If the equation x2 + 5kx + 16 = 0 has no real roots then

26

If the equation x2 - kx + 1 = 0 has no real roots then

27

For what values of k, the equation kx2 - 6x - 2 = 0 has real roots?

28

The sum of a number and its reciprocal is . The number is

29

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

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

31

The roots of the quadratic equation 2x2 - x - 6 = 0 are

32

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

33

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

34

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

35

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

36

Find the roots of the quadratic equation 2x2 - x - 6 = 0.

37

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

38

If the roots of the quadratic equation 2x2 + 8x + k = 0 are equal then find the value of k.

39

If the quadratic equation px2 –2√5px + 15 = 0 has two equal roots then find the value of p.

40

If 1 is a root of the equation ay2 + ay + 3 = 0 and y2 + y + b = 0 then find the value of ab.

41

If one zero of the polynomial x2 - 4x + 1 is (2 + √3 ), write the other zero.

42

If one root of the quadratic equation 3x2 - 10x + k = 0 is reciprocal of the other, find the value of k.

43

If the roots of the quadratic equation px(x – 2) + 6 = 0 are equal, find the value of p.

44

Find the values of k so that the quadratic equation x2 – 4kx + k = 0 has equal roots.

45

Find the values of k for which the quadratic equation 9x2 – 3kx + k = 0 has equal roots.

46

Solve: 47

Solve: 2x2 + ax – a2 = 0

48

Solve: 3x2 + 5√5x – 10 = 0

49

Solve: √3x2 + 10x – 8√3 = 0.

50

Solve: √3x2 – 2√2x – 2√3 = 0

51

Solve: 4√3x2 + 5x –2√3 = 0

52

Solve: 4x2 + 4bx – (a2 – b2) = 0.

53

Solve: x2 + 5x – (a2 + a – 6) = 0