ST.

PAUL’S SCHOOL, AJMER

QUADRATIC EQUATION

Practice Time
1. The value of 6 + 6 + 6 + ... is
7
(a) 4 (b) 3 (c) –2 (d)
2

2. If 2 is the root of the equation x2 + bx + 12 = 0 and the equation x2 + bx + q = 0 has equal roots
then q =
(a) 8 (b) 16 (c) –8 (d) –16

3.If a and b can take values 1, 2, 3, 4. Then the number of the equations of the form ax2+ bx + c =
0 having real roots is
(a) 6 (b) 7 (c) 10 (d) 12

4. The number of quadratic equations having real roots and which do not change by squaring
their roots is
(a) 4 (b) 3 (c) 2 (d) 1
2 2 2 2 2
5.If the equation (a + b )x – 2(ac + bd)x + c + d = 0 has equal roots then
(a) ab = cd (b) ad = bc (c) ad = bc (d) ab = cd

6. If one of the roots of the quadratic equation (k2 + 4)x2 + 13x + 4k is reciprocal of the other then
k=
(a) 2 (b) 1 (c) –1 (d) – 2

1 1
7. If α, β are the roots of the quadratic equation 4x2 + 3x + 7 = 0, then +
α β
7 −7 3 −3
(a) (b) (c) (d)
3 3 7 7

8. If α, β are the roots of the quadratic equation x2 –p(x + 1) – c = 0, then (α + 1)(β + 1) =

(a) c – 1 (b) 1 – c (c) c (d) 1 + c

9. Find the values of k for which the quadratic equation 2x2 + kx + 3 = 0 has real equal roots.
(a) 2 6 (b) 2 6 (c) 0 (d)  2

10. Find the values of k for which the quadratic equation kx(x – 3) + 9 = 0 has real equal roots.
(a) k = 0 or k = 4 (b) k = 1 or k = 4 (c) k = –3 or k = 3 (d) k = –4 or k = 4

11. Find the values of k for which the quadratic equation 4x2 – 3kx + 1 = 0 has real and equal
roots.
4 2
(a)  (b)  (c)  2 (d) none of these
3 3
 ST. PAUL’S SCHOOL, AJMER
QUADRATIC EQUATION

Practice Time
12. Find the values of k for which the quadratic equation (k – 12)x2 + 2(k – 12)x + 2 = 0 has real
and equal roots.
(a) k = 0 or k = 14 (b) k = 12 or k = 24 (c) k = 14 or k = 12 (d) k = 1 or k = 12

PRACTICE SHEET -2
1. Find the values of k for which the quadratic equation k2x2 – 2(k – 1)x + 4 = 0 has real and equal
roots.
1 1 1 1
(a) k = 0 or k = (b) k = 1 or k = (c) k = –1 or k = (d) k = –3 or k =
3 3 3 3

2. If –4 is a root of the equation x2 + px – 4 = 0 and the equation x2 + px + q = 0 has equal roots, find
the value of p and q.
4 9
(a) p = 3, q = 9 (b) p = 9, q = 3 (c) p = 3, q = (d) p = 3, q =
9 4

3. If the roots of the equation (a – b)x2 + (b – c)x + (c – a) = 0 are equal, then b + c =

(a) 2a (b) 2bc (c) 2c (d) none of these

4. Find the positive value of k for which the equations x2 + kx + 64 = 0 and x2 – 8x + k = 0 will have
real roots.
(a) 8 (b) 16 (c) –8 (d) –16

5. Find the positive value of k for which the equation kx2 – 6x – 2 = 0 has real roots
−9 −9 −9 −9
(a) k  (b) k  (c) k  (d) k 
2 2 2 2

6. Find the positive value of k for which the equation 3x2 + 2x + k = 0 has real roots
1 1 1 1
(a) k  (b) k  (c) k  (d) k 
3 3 3 3

7. Find the positive value of k for which the equation 2x2 + kx + 2 = 0 has real roots
(a) k  4 (b) k  −4 (c) both (a) and (c) (d) none of these

10
8. The sum of a number and its reciprocal is . Find the number.
3
1
(a) 3 (b) (c) both (a) and (c) (d) none of these
3

9. Divide 12 into two parts such that the sum of their squares is 74.
(a) 7 and 5 (b) 8 and 4 (c) 10 and 2 (d) none of these

10. The sum of the squares of two consecutive natural numbers is 421. Find the numbers.
 ST. PAUL’S SCHOOL, AJMER
QUADRATIC EQUATION

Practice Time
(a) 14 and 5 (b) 14 and 15 (c) 10 and 5 (d) none of these

3
11. The sum of two numbers is 15 and the sum of their reciprocals is . Find the numbers.
10
(a) 14 and 5 (b) 14 and 15 (c) 10 and 5 (d) none of these

12. Divide 12 into two parts such that their product is 32.
(a) 7 and 5 (b) 8 and 4 (c) 10 and 2 (d) none of these

PRACTICE SHEET -3

FACTORISATION METHOD & METHOD OF QUADRATIC FORMULA

Solve the following quadratic equation:

1 40 + 3x – x2 = 0 x + 3 3x – 7
12 =
x + 2 2x – 3
2 7x2 + 49x + 84 = 0
x –1 x – 3 1
13 + = 3 (x  2, 4)
x–2 x–4 3
3 m2 + 17mn – 84n2 = 0
1 1 1 1
14 = + + , [x  0, – (a + b)]
4 18x2 + 3x – 10 = 0 a+b+x a b x

5 8a2 – 27ab + 9b2 = 0  2x – 1   x+3  1

15 2  – 3  = 5, x  –3,
 x+3   2x – 1  2
6 5x2 + 33xy – 14y2 = 0 16 5(x+1) + 5(2–x) = 53 + 1

7 36x2 – 12ax + (a2 – b2) = 0 17 5x –

35
= 18, x  0
x
8. 2x 2 + 7x + 5 2 = 0 18. 4x 2 – 2(a 2 + b 2 )x + a 2b 2 = 0
3
9. 2x – =1 19. 9x 2 – 9(a + b)x + (2a 2 + 5ab + 2b 2 ) = 0
x

4 5 –3 20. 4x 2 – 4a 2 x + (a4 – b 4 ) = 0
10. –3= , x  0,
x 2x + 3 2
a+b a 
21. x2 +  +  x +1 = 0
x x + 1 34  a a + b
11 + = , x  –1 and x  0
x +1 x 15
22. x 2 + x – (a + 1)(a + 2) = 0
 ST. PAUL’S SCHOOL, AJMER
QUADRATIC EQUATION

Practice Time
23. x 2 + 3x – (a 2 + a – 2) = 0 24. a 2b 2 x 2 + b 2 x – a 2 x – 1 = 0

PRACTICE SHEET -4
1. Find the value of k for which the quadratic equation 2x2 + kx + 3 = 0 has two real equal roots.

6. Find the value of k for which the quadratic equation (k – 12)x2 + 2(k – 12)x + 2 = 0 has two real
equal roots..
8. If the roots of the equation (a – b)x2 + (b – c)x + (c – a) = 0 are equal, prove that b + c = 2a.
13. Find the value of k for which the quadratic equation 2x2 + kx + 2 = 0 has two real roots.
18. If the equation (1 + m2)x2 + 2mcx + (c2 – a2) = 0 has equal roots, prove that c2 = a2(1 + m2).
20. Find the value of k for which the quadratic equation 9x2 + 8kx + 16 = 0 has two real equal roots.
29. Find the value of k for which the quadratic equation (4 – k)x2 + (2k + 4)x + (8k + 1) = 0 has real
and equal roots.

30. Find the value of k for which the quadratic equation (2k + 1)x2 + 2(k + 3)x + (k + 5) = 0 has real
and equal roots.

PRACTICE SHEET -5

1
1. The difference of two numbers is 5 and the difference of their reciprocals is . Find the
10
numbers.
2. The sum of the squares of the two positive integers is 208. If the square of the larger number is 18
times the smaller number, find the numbers.
3. The denominator of a fraction is one more than twice the numerator. The sum of the fraction and
16
its reciprocal is 2 . Find the fraction.
21
1
4. The sum of two numbers is 16 and the sum of their reciprocals is . Find the numbers.
3
5. Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50
times their difference.
6. A two digit number is such that the product of its digits is 12. When 36 is added to the number,
the digits are reversed. Find the number.
7. A two digit number is such that the product of its digits is 35. When 18 is added to the number,
the digits interchange their places. Find the number.
8 The sum of ages of a father and his son is 45 years. Five years ago, the product of their ages in
years was 124. Find their present ages.
 ST. PAUL’S SCHOOL, AJMER
QUADRATIC EQUATION

Practice Time
9. Seven years ago Varun’s age was five times the square of Swati’s age. Three years hence Swati’s
age will be two fifth of Varun’s age. Find their present ages.

10. The product of Rohit’s age five years ago with his age 9 years later is 15 in years. Find his present
age.
11. The sum of the ages of a boy and his brother is 12 years and the sum of the square of their ages is
4 in years. Find their ages.

12. A boy is one year older than his friend. If the sum of the square of their ages is 421, find their
ages.
13. The sum of the ages of a boy and his brother is 57 years and the product of their ages in years is 7
82. Find their ages.
14 A motor boat whose speed is 18 km/hr in still water takes 1 hour more to go 24 upstream than to
return to the same point. Find the speed of the stream.
15 In a flight for 3000 km, an aircraft was slowed down due to bad weather. Its average speed for
the trip was reduced by 100 km/hr and consequently time of flight increased by one hour. Find
the original duration of flight.

16 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 train is 11km/h more than that of the passenger train, find the
average speed of the two trains.
1
17 The time taken by a man to cover 300 km on a scooter was 1 hours more than the time taken by
2
him during the return journey. If the speed in returning be 10 km/hr more than the speed in
going, find its speed in each direction.
18 An aeroplane left 30 minutes later than it schedule time and in order to reach its destination
1500 km away in time, it had to increase its speed by 250 km/hr from its usual speed. Determine
its usual speed.
19 The hypotenuse of a right triangle is 3 10 cm. If the smaller side is tripled and the longer sides
doubled, new hypotenuse will be 9 5 cm. How long are the sides of the triangle?
20 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.
21. 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 breadth of the rectangle. Their areas being equal, find their dimensions.
3
22 Two water taps together can fill a tank in 9 hours. The tap of larger diameter takes 10 hours
8
less than the smaller one to fill the tank separately. Find the time in which each tap can
separately fill the tank.

23. A takes 6 days less than the time taken by B to finish a piece of work. If both A and B together
can finish it in 4 days, find the time taken by B to finish the work.
24. 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?
 ST. PAUL’S SCHOOL, AJMER
QUADRATIC EQUATION

Practice Time
25. In a class test, the sum of Ranjitha’s marks in mathematics and English is 40. Had she got 3
marks more in mathematics and 4 marks less in English, the product of the marks would have
been 360. Find her marks in two subjects separately.
26. A teacher 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 1 student, he found that he
was short of 25 students. Find the number of students.
27. John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of
the number of marble they now have is 124. We would like to find out how many marbles they
had to start with.
28. 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

29. 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.
30. A shopkeeper buys a number of books for Rs. 80. If he had bought 4 more books for the same
amount, each book would have cost Rs. 1 less. How many books did he buy?