CHAPTER 4: QUADRATIC EQUATIONS

1 Which one of the following is not a quadratic equation?

(a) (x + 2)2 = 2(x + 3) (b) x2 + 3x = (–1) (1 – 3x)2
(c) (x + 2) (x – 1) = x2 – 2x – 3 (d) x3 – x2 + 2x + 1 = (x + 1)3

2 Which of the following equations has 2 as a root?

(a) x2 – 4x + 5 = 0 (b) x2 + 3x – 12 = 0
(c) 2x2 – 7x + 6 = 0 (d) 3x2 – 6x – 2 = 0

3 If is a root of the equation x2 + kx – = 0, then the value of k is

(a) 2 (b) -2 (c) (d)

4 Which of the following equations has the sum of its roots as 3?

(a) 2x2 – 3x + 6 = 0 (b) –x2 + 3x – 3 = 0

(c) +1=0 (d) 3x2 – 3x + 3 = 0

5 Values of k for which the quadratic equation 2x2 – kx + k = 0 has equal roots is
(a) 0 Only (b) 4 Only (c) 8 Only (d) 0 and 8

6 The quadratic equation 2x2 – x + 1 = 0 has

(a) two distinct real roots (b) two equal real roots
(c) no real roots (d) more than 2 real roots

7 Which of the following equations has two distinct real roots?

(a) 2x2 –3 x+ =0 (b) x2 + x – 5 = 0

(c) x2 + 3x + 2 =0 (d) 5x2 – 3x + 1 = 0

8 Which of the following equations has no real roots?

(a) x2 – 4x + 3 =0 (b) x2 + 4x - 3 =0
(c) x2 – 4x - 3 =0 (d) 3x2 + 4 x+4=0
9 The discriminant of the quadratic equation 3 x2 + 10x + = 0 is
(a) 8 (b) 64 (c) (d) -

10 A sum of ₹4000 was divided among x persons. Had there been 10 more persons, each
would have got ₹80 less. Which of the following represents the above situation?
(a) x2 + 10x – 500 = 0 (b) 8x2 + 10x – 400 = 0
(c) x2 + 10x + 500 = 0 (d) 8x2 + 10x + 400 = 0
11 The product of two consecutive integers is equal to 6 times the sum of the two integers. If
the smaller integer is x, which of the following equations represent the above situation?
(a) x2 + 11x + 6 = 0 (b) x2 - 11x - 6 = 0
(c) x2 + 11x - 6 = 0 (d) x2 - 11x + 6 = 0

12 Consider the equation kx2 + 2x = c (2x2 + b)

For the equation to be quadratic, which of these cannot be the value of k?
(a) c (b) 2c (c) 3c (d) 2c + 2b

13 What is the smallest positive integer value of k such that the roots of the equation x2 - 9x +
18 + k = 0 can be calculated by factoring the equation?
(a) 1 (b) 2 (c) 3 (d) 4

14 Rahul follows the below steps to find the roots of the equation
3x2 – 11x - 20 = 0, by splitting the middle term.
Step 1: 3x2 – 11x - 20 = 0
Step 2: 3x2 – 15x + 4x - 20 = 0
Step 3: 3x (x - 5) + 4(x - 5) = 0
Step 4: (3x - 4) (x - 5) = 0
Step 5: x = 𝑎𝑛𝑑 5

In which step did Rahul make the first error?

(a) Step 2 (b) Step 3 (c) Step 4 (d) Step 5

15 The roots of ax2 + bx + c = 0, a ≠ 0 are real and unequal. Which of these is true about the
value of discriminant, D?
(a) 𝐷 < 0 (b) 𝐷 > 0 (c) 𝐷 = 0 (d) 𝐷 ≤ 0
16 Consider the equation px2 + qx + r = 0. Which conditions are sufficient to conclude that the
equation have real roots?
(a) p>0, r<0 (b) p>0, r>0 (c) p>0, q>0 (d) p>0, q<0

17 For what value of k, the quadratic equation 3x2 + 2kx + 27 = 0 has equal real roots?
(a) 𝑘 = ±3 (b) 𝑘 = ±9 (c) 𝑘 = ±6 (d) 𝑘 = ±4

18 If the equation x2 -mx + 1 = 0 does not possess real roots, then

(a) -3 < 𝑚 < 3 (b) -2 < 𝑚 < 2 (c) 𝑚 > 2 (d) 𝑚 < -2

19 If  and  are the roots of x2 + 7x + 10 = 0, find the value of

(a) 29 (b) 69 (c) 49 (d) 20

20 If ,  are the roots of the equation 2x2 – x -1 = 0, then find the value of .

(a) 1 (b) -1 (c) (d)

21 If one root of the equation 2y2 – ay + 64 = 0 is twice the other, then find the values of a.
(a) a = ±8 (b) a = ±16 (c) a = ±24 (d) a = ±4

22 If one root of the equation 3x2 + kx + 81 = 0 (having real roots) is the square of the other,
then value of k
(a) k = 27 (b) k = -27 (c) k = 36 (d) k = -36

23 A quadratic equation, the sum of whose roots is 0 and one root is 4, is

(a) 𝑥2 - 16 (b) 𝑥2 + 16 (c) 𝑥2 + 4 (d) 𝑥2 – 4

24 If the quadratic equation 𝑥2 - 8𝑥 + 𝑘 = 0 has real roots, then

(a) 𝑘 < 16 (b) 𝑘 ≤ 16 (c) 𝑘 > 16 (d) 𝑘 ≥ 16

25 If x = 3 is one of the roots of the quadratic equation x2 – 2kx – 6 = 0, then the value of k is

(a) (b) (c) 3 (d) 2

 Assertion-Reason Questions
DIRECTION: In the question number 26 and 30, a statement of assertion (A) is followed by
a statement of Reason (R). Choose the correct option
(a) Both assertion (A) and reason (R) are true and reason (R) is the
correct explanation of assertion (A)
(b) Both assertion (A) and reason (R) are true and reason (R) is not the
correct explanation of assertion (A)
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
26 Assertion(A): If one root of the quadratic equation 6x2 – x – k = 0 is , then the value of k

is 2.
Reason(R): The quadratic equation ax2 + bx + c = 0, a ≠ 0 has almost two roots.
27 Assertion(A): The roots of the quadratic equation x2 + 2x + 2 = 0 are real
Reason(R): If discriminant D = b2 – 4ac < 0 then the roots of quadratic equation ax2 + bx +
c = 0 are not real.
28 Assertion: (2x – 1)2 – 4x2 + 5 = 0 is not a quadratic equation.
Reason: An equation of the form ax2 + bx + c = 0, (a ≠ 0, where a, b and c are real
numbers) is called a quadratic equation.
29 Assertion: 3x2 – 6x + 3 = 0 has equal real roots.
Reason: The quadratic equation ax2 + bx + c = 0 have equal real roots if discriminant D >
0.
30 Assertion(A): The equation 9x² + 3kx + 4 = 0 has equal roots for
k = 9.
Reason (R): If discriminant 'D' of a quadratic equation is equal to zero,
then roots of equation are real and equal.
 ANSWERS
1 (c) (x + 2) (x – 1) = x2 – 2x – 3 16 (a) p>0, r<0
2 (c) 2x2 – 7x + 6 = 0 17 (b) 𝑘 = ±9
3 (a) 2 18 (b) -2 < 𝑚 < 2
4 (b) –x2 + 3x – 3 = 0 19 (a) 29
5 (c) 8 Only 20 (b) -1
6 (c) no real roots 21 (c) a = ±24
7 (b) x2 + x – 5 = 0 22 (d) k = -36
8 (a) x2 – 4x + 3 =0 23 (a) 𝑥2 - 16
9 (b) 64 24 (b) 𝑘 ≤ 16
10 (a) x2 + 10x – 500 = 0 25 (b)

11 (b) x2 - 11x - 6 = 0 26 (b)

12 (b) 2c 27 (d)
13 (b) 2 28 (a)
14 (c) Step 4 29 (c)
15 (b) 𝐷 > 0 30 (d)