This set of Mathematics MCQs for Schools focuses on “Quadratic Equation – Determination of Types of Roots”.

1. For the equation x^{2} + 5x – 1, which of the following statements is correct?

a) The roots of the equation are equal

b) The discriminant of the equation is negative

c) The roots of the equation are real, distinct and irrational

d) The discriminant is equal to zero

View Answer

Explanation: Roots are real. ∴ b

^{2}– 4ac ≥ 0

5

^{2}– 4(1)(1)

25 – 4 = 21 which is greater than 0. Hence, the discriminant of the equation is greater than zero, so roots are real.

2. If the roots of the equation ax^{2} + bx + c are real and equal, what will be the relation between a, b, c?

a) b = ±\(\sqrt {ac}\)

b) b = ±\(\sqrt {4c}\)

c) b = ±\(\sqrt {-4ac}\)

d) b = ±\(\sqrt {4ac}\)

View Answer

Explanation: Roots are real and equal. ∴ b

^{2}– 4ac = 0

b

^{2}= 4ac

b = ±\(\sqrt {4ac}\)

3. What will be the value of k, so that the roots of the equation are x^{2} + 2kx + 9 are imaginary?

a) -5 < k < 5

b) -3 < k < 3

c) 3 < k < -3

d) -5 < k < 3

View Answer

Explanation: Roots are imaginary. ∴ b

^{2}– 4ac < 0

(2k)

^{2}– 4(9)(1) < 0

4k

^{2}– 36 < 0

k

^{2}– 9 < 0

k

^{2}< 9

k < ±3

-3 < k < 3

4. What will be the nature of the roots of the quadratic equation 5x^{2} – 11x + 13?

a) Imaginary

b) Real

c) Irrational

d) Equal

View Answer

Explanation: To check the nature of the roots, the discriminant must be either equal to zero, less than zero or greater than zero.

Discriminant = b

^{2}– 4ac = – 11

^{2}– 4 × 5 × 13 = 121 – 260 = – 139

Since discriminant is less than zero, the roots of the equation are imaginary.

5. What will be the nature of the roots of the quadratic equation x^{2} + 10x + 25?

a) Imaginary

b) Real

c) Irrational

d) Equal

View Answer

Explanation: To check the nature of the roots, the discriminant must be either equal to zero, less than zero or greater than zero.

Discriminant = b

^{2}– 4ac = 10

^{2}– 4 × 25 × 1 = 100 – 100 = 0

Since discriminant is equal to zero, the roots of the equation are equal.

6. What will be the nature of the roots of the quadratic equation 2x^{2} + 10x + 9?

a) Imaginary

b) Real

c) Irrational

d) Equal

View Answer

Explanation: To check the nature of the roots, the discriminant must be either equal to zero, less than zero or greater than zero.

Discriminant = b

^{2}– 4ac = 10

^{2}– 4 × 2 × 9 = 100 – 72 = 28

Since discriminant is greater than zero, the roots of the equation are real and distinct.

7. The equation 9x^{2} – 2x + 5 is not true for any real value of x.

a) False

b) True

View Answer

Explanation: To check the nature of the roots, the discriminant must be either equal to zero, less than zero or greater than zero.

Discriminant = b

^{2}– 4ac = -2

^{2}– 4 × 9 × 5 = 4 – 180 = -176

Since discriminant is less than zero, the roots of the equation are imaginary. Hence, for any real value of x the equation is not true.

8. The value of p for which the equation 8x^{2} + 9px + 15 has equal roots is \(\frac {4\sqrt {30}}{9}\).

a) True

b) False

View Answer

Explanation: Roots are equal. ∴ b

^{2}– 4ac = 0

(9p)

^{2}– 4(8)(15) = 0

81p

^{2}– 480 = 0

81p

^{2}= 480

p = ± \(\sqrt {\frac {480}{81}}\) = ± \(\frac {4\sqrt {30}}{9}\)

9. What will be the value of a, for which the equation 5x^{2} + ax + 5 and x^{2} – 12x + a will have real roots?

a) a = 37

b) 10 < a < 36

c) 36 < a < 10

d) a = 9

View Answer

Explanation: The roots of both the equations are real.

Discriminant of 5x

^{2}+ ax + 5 : b

^{2}– 4ac = a

^{2}– 4 × 5 × 5 = a

^{2}– 100

Since, roots are real; discriminant will be greater than 0.

a

^{2}≥ 100

a ≥ ±10

Now, discriminant of x

^{2}– 12x + a : b

^{2}– 4ac = -12

^{2}– 4 × 1 × a = 144 – 4a

Since, roots are real; discriminant will be greater than 0.

144 ≥ 4a

a ≤ \(\frac {144}{4}\) = 36

For both the equations to have real roots the value of a must lie between 36 and 10.

10. What will be the value of k, if the roots of the equation (k – 4)x^{2} – 2kx + (k + 5) = 0 are equal?

a) 18

b) 19

c) 20

d) 21

View Answer

Explanation: Roots are equal. ∴ b

^{2}– 4ac = 0

-(2k)

^{2}– 4(k – 4)(k + 5) = 0

4k

^{2}– 4(k

^{2}– 4k + 5k – 20) = 0

4k

^{2}– 4(k

^{2}+ k – 20) = 0

4k

^{2}– 4k

^{2}– 4k + 80 = 0

-4k = – 80

k = \(\frac {-80}{-4}\) = 20

**Sanfoundry Global Education & Learning Series – Mathematics – Class 10**.

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