## Chapter 4 Quadratic Equations Bits for 10th Class AP Board

Multiple Choice Questions (MCQs) :

Question 1.

The roots of 8x^{2} – 8ax + (a^{2} – b^{2}) = 0 are ……………… .

A) \(\frac{a+b}{8}, \frac{a-b}{8}\)

B) \(\frac{\mathrm{a}+\mathrm{b}}{4}, \frac{\mathrm{a}-\mathrm{b}}{4}\)

C) \(\frac{a+b}{2}, \frac{a-b}{2}\)

D) a + b, a – b

Answer:

B) \(\frac{\mathrm{a}+\mathrm{b}}{4}, \frac{\mathrm{a}-\mathrm{b}}{4}\)

Question 2.

The roots of x^{2} + (p + \(\frac{1}{\mathbf{p}}\)) x + 1 = 0 are …………….. where p ≠ 0

A) -p, p

B) -p, –\(\frac{1}{\mathbf{p}}\)

C) p, \(\frac{1}{\mathbf{p}}\)

D) p, –\(\frac{1}{\mathbf{p}}\)

Answer:

B) -p, –\(\frac{1}{\mathbf{p}}\)

Question 3.

If one root of the 5x^{2} + 13x + a = 0 is reciprocal of the other root, then a = ……………… .

A) 13

B) -13

C) 5

D) -5

Answer:

C) 5

Question 4.

If the roots of (q – r)x^{2} + (r – p)x + (p – q) = 0 are equal, then which of the following is correct ?

A) p., r, q are in AP

B) p, q, r in AP

C) r, q, p are in AP

D) q, r, p in AP

Answer:

B) p, q, r in AP

Question 5.

If α, β are roots of 2x^{2} – 3x – 6 = 0, then whose roots are α^{2} + 2, β^{2} + 2 is ……………… .

A) 5x^{2} + 49x + 59 = 0

B) 4x^{2} + 49x – 118 = 0

C) 4x^{2} – 49x + 118 = 0

D) 4x^{2} + 59x – 59 = 0

Answer:

C) 4x^{2} – 49x + 118 = 0

Question 6.

If the product of the roots of ax^{2} + bx + a^{2} + 4 = 0 is -4, then a is …………. .

A) -1

B) -4

C) -3

D) -2

Answer:

A) -1

Question 7.

The number of real roots of 2x^{4} + 5x^{2} – 3 = 0 is …………….. .

A) 1

B) 2

C) 3

D) 4

Answer:

D) 4

Question 8.

x^{2} + x + a = 0, bx^{2} + bx + 3 = 0 has one root as 1, then a.b = …………….. .

A) 1

B) 2

C) 3

D) 4

Answer:

C) 3

Question 9.

If the roots of the equation (a^{2} + b^{2}) x^{2} – 2 (ac + bd) + (c^{2} + d^{2}) = 0 are equal, then ……………….. .

A) \(\frac{a}{b}=\frac{-c}{d}\)

B) \(\frac{-\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}\)

C) \(\frac{a}{b}=\frac{c}{d}\)

D) \(\frac{a^2+c^2}{b^2}\) = \(\frac{b^2+a^2}{d^2}\)

Answer:

C) \(\frac{a}{b}=\frac{c}{d}\)

Question 10.

If a and b are such that the quadratic equation ax^{2} – 5x + b = 0 has 15 as the sum of the roots and also as the product of roots, then the value of a = ……………… .

A) 5

B) \(\frac{1}{2}\)

C) \(\frac{1}{3}\)

D) \(\frac{1}{5}\)

Answer:

C) \(\frac{1}{3}\)

Question 11.

What is the value of k such that the sum of the squares of the roots of x^{2} – 8x + k = 0 is 40 ?

A) -12

B) 14

C) 12

D) -14

Answer:

C) 12

Question 12.

Roots of a^{2}x^{2} – (a^{2}b^{2} + 1)x + b^{2} = 0 are ………………. .

A) a, \(\frac{1}{b}\)

B) a^{2}, \(\frac{1}{b}\)

C) \(\frac{1}{\mathrm{a}^2}\), b^{2}

D) a^{2}, \(\frac{1}{\mathrm{b}^2}\)

Answer:

C) \(\frac{1}{\mathrm{a}^2}\), b^{2}

Question 13.

If ax^{2} + bx + c = 0isa perfect square, then b = …………….. .

A) ac

B) \(\sqrt{4 \mathrm{ac}}\)

C) 4ac

D) 2ac

Answer:

B) \(\sqrt{4 \mathrm{ac}}\)

Question 14.

The value of \(\sqrt{30+\sqrt{30+\sqrt{30+\ldots+\infty}}}\) is

A) 30

B) -30

C) 6

D) -5

Answer:

C) 6

Question 15.

If α, β are roots of x^{2} – 7x + 12 = 0, then the equation whose roots are 3α, 3β is ………………. .

A) x^{2} – 21x + 108 = 0

B) x^{2} + 7x – 108

C) 9x^{2} – 21x + 108 = 0

D) 9x^{2} – 7x – 108

Answer:

A) x^{2} – 21x + 108 = 0

Question 16.

If the equation x^{2} – bx + 1 = 0 has two distinct roots, then ………………. .

A) |b| > 2

B) |b| < 2 C) |b| = 2 D) |b| = \(\frac{1}{2}\) Answer: A) |b| > 2

Question 17.

If px^{2} + qx + r = 0 has equal roots, then P = ………….. .

A) \(\frac{q}{4 r}\)

B) \(\frac{q^2}{4 r}\)

C) \(\frac{r^2}{4 q}\)

D) \(\frac{r}{4 q}\)

Answer:

B) \(\frac{q^2}{4 r}\)

Question 18.

The positive value of ‘a’ for which x^{2} + ax + 64 = 0 and x^{2} – 8x + a = 0 both have real roots is ……………. .

A) 16

B) 8

C) 4

D) 2

Answer:

A) 16

Question 19.

The value of \(\sqrt{6+\sqrt{6+\sqrt{6+\ldots \ldots}}}\) =

A) 2

B) 1

C) 3

D) 4

Answer:

C) 3

Question 20.

If the equation x^{2} – px + 1 = 0 does not possess real roots, then …………….. .

A) – 1 < p < 1

B) – 3 < p < 3

C) – 2 < p < 3

D) – 2 < p > 2

Answer:

C) – 2 < p < 3

Question 21.

If a and b are the roots of the x^{2} – ax + b = 0 then ……………….. .

A) a = – 1, b = 1

B) a = 1, b = -2

C) a = 0, b = -1

D) a = 1, b = 2

Answer:

B) a = 1, b = -2

Question 22.

If the sum of the roots of the equation x^{2} – x = p (2x – 1) is zero, then p = ……………. .

A) 1

B) \(\frac{1}{2}\)

C) -2

D) –\(\frac{1}{2}\)

Answer:

D) –\(\frac{1}{2}\)

Question 23.

If x^{2} + a (4x + a – 1) + 2 = 0 has equal roots, then a = …………….. .

A) \(\frac{2}{3}\), -1

B) –\(\frac{2}{3}\), 1

C) \(\frac{3}{2}\), -1

D) –\(\frac{3}{2}\), -1

Answer:

A) \(\frac{2}{3}\), -1

Question 24.

The p and q are roots of x^{2} + px + q = 0, then p – q = …………….. .

A) 3

B) -1

C) 2

D) 1

Answer:

A) 3

Question 25.

If one root of the ax^{2} + bx + c = 0 is 2 times the other, then b^{2}: ac = ……………….. .

A) 2 : 9

B) 9 : 4

C) 9 : 2

D) 4 : 9

Answer:

C) 9 : 2

Question 26.

The quadratic polynomial, whose zeros are 2 and 3, is ………………. .

A) x^{2} – 5x – 6

B) x^{2} + 5x – 6

C) x^{2} – 5x + 6

D) x^{2} + 5x + 6

Answer:

C) x^{2} – 5x + 6

Question 27.

Observe the given rectangular figure, . then its area in polynomial function is ………………. .

A) A(x) = x^{2} + 7x + 30

B) A(x) = -x^{2} + 7x + 30

C) A(x) = x^{2} – 7x + 30

D) A(x) = -x^{2} – 7x + 30

Answer:

B) A(x) = -x^{2} + 7x + 30

Question 28.

Which of the following is a quadra tic equation ?

A) x^{3} – 6x^{2} + 2x – 1 = 0

B) x^{2} + \(\frac{1}{x^2}\) = 2

C) x + \(\frac{1}{x}\) = 3

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

Answer:

C) x + \(\frac{1}{x}\) = 3

Question 29.

Which of the following quadratic equations the roots are equal ?

A) x^{2} – 5 = 0

B) x^{2} – 10x + 25 = 0

C) x^{2} + 5x + 6 = 0

D) x^{2} – 1 = 0

Answer:

B) x^{2} – 10x + 25 = 0

Question 30.

The quadratic polynomial having \(\frac{1}{3}\) and \(\frac{1}{2}\) as its zeroes, is ………………. .

A) x^{2} + \(\frac{5 x+1}{6}\)

B) -6x^{2} – 5x + 1

C) x^{2} – \(\frac{5 x-1}{6}\)

D) 6x^{2} – 5x – 1

Answer:

C) x^{2} – \(\frac{5 x-1}{6}\)

Question 31.

If x^{2} – px + q = 0 (p, q ∈ R and p ≠ 0, q ≠ 0) has distinct real roots, then ………….. .

A) p^{2} < 4q

B) p^{2} > 4q

C) p^{2} = 4q

D) p^{2} + 4q = 0

Answer:

B) p^{2} > 4q

Question 32.

In a quadratic equation ax^{2} + bx + c = 0, if b^{2} – 4ac > 0, then their roots are …………… .

A) real and distirlct

B) real and equal

C) imaginary

D) none

Answer:

A) real and distirlct

Question 33.

If a number is 132 smaller than its square, then the number is

A) 11

B) 8

C) 9

D) 12

Answer:

D) 12

Question 34.

If both roots are common to the Qua-dratic equations

x^{2} – 4 = 0 and x^{2} + px – 4 = 0, then p =

A) 2

B) 0

C) 4

D) 1

Answer:

B) 0

Question 35.

The sum of the roots of 6x^{2} = 1 is

A) 0

B) \(\frac{1}{6}\)

C) – \(\frac{1}{6}\)

D) 6

Answer:

A) 0

Question 36.

The sum of a number and reciprocal is \(\frac{17}{4}\), then the number is

A) 3

B) 4

C) 5

D) 17

Answer:

B) 4

Question 37.

The roots of a quadratic equation ax^{2} – bx + c = 0, a ≠ 0 are

A) \(\frac{-b+\sqrt{b^2-4 a c}}{2 a} ; \frac{-b+\sqrt{b^2+4 a c}}{2 a}\)

B) \(\frac{-b+\sqrt{b^2-4 a c}}{2 a} ; \frac{-b-\sqrt{b^2+4 a c}}{2 a}\)

C) \(\frac{b+\sqrt{b^2-4 a c}}{2 a} ; \frac{b-\sqrt{b^2-4 a c}}{2 a}\)

D) \(\frac{-b+\sqrt{b^2-4 a c}}{2 a} ; \frac{-b-\sqrt{b^2-4 a c}}{2 a}\)

Answer:

C) \(\frac{b+\sqrt{b^2-4 a c}}{2 a} ; \frac{b-\sqrt{b^2-4 a c}}{2 a}\)

Question 38.

If one root of the Quadratic equation x^{2} – kx + 36 = 0 is 4, then the value of ‘k’ is …………….. .

A) 12

B) 17

C) 18

D) 13

Answer:

D) 13

Question 39.

The nature of roots of the Quadratic equation x^{2} + 6x + 9 = 0 is …………… .

A) Real and distinct.

B) Real and equal.

C) No real roots.

D) One is positive and the other is negative.

Answer:

B) Real and equal.

Question 40.

‘The product of two consecutive positive integers is 30″. This can be expressed algebraically as ………………… .

A) x(x + 2) = 30

B) x(x – 2) = 30

C) x(x – 3) = 30

D) x(x + 1) = 30

Answer:

D) x(x + 1) = 30

Question 41.

The quadratic equation with roots 2 + √ 3 and 2 – √ 3 is ……………….. .

A) x^{2} + 4x + 1 = 0

B) x^{2} + 4x – 1 = 0

C) x^{2} – 4x + 1 = 0

D) x^{2} – 4x – 1 = 0

Answer:

C) x^{2} – 4x + 1 = 0

Question 42.

The Quadratic equation, whose sum of the roots is -3 and product of the roots is 2.

A) x^{2} + 6x + 5 = 0

B) x^{2} – x – 6 = 0

C) x^{2} – 3x + 2 = 0

D) x^{2} + 3x + 2 = 0

Answer:

D) x^{2} + 3x + 2 = 0

Question 43.

The roots of a quadratic equation 2x^{2} + x + 4 = 0 are ………………. .

A) One is positive and other is negative.

B) Both are positive.

C) Both are negative.

D) No real roots.

Answer:

D) No real roots.

Question 44.

Which of the following equation has ‘2’ as a root ?

A) x^{2} – 4x + 5 = 0

B) x^{2} + 3x – 12 = 0

C) 2x^{2} – 7x + 6 = 0

D) 3x^{2} – 6x – 2 = 0

Answer:

C) 2x^{2} – 7x + 6 = 0

Fill in the blanks :

1. The number of real roots of 2x^{4} + 5x^{2} + 3 = 0 is __________ .

Answer:

0

2. If a and b are such that the quadratic equation ax^{2} – 5x + b = 0 has 20 as the sum of the roots and also as the products of the roots, then the value of b is __________ .

Answer:

5

3. If ax^{2} + bx + c = 0 is a perfect square, then a = __________ .

Answer:

\(\frac{b^2}{4 c}\)

4. If the x^{2} + 4x + a = 0 has real and distinct, then k < ___________ .

Answer:

4

5. If x^{2} + px + 12 = 0 and the equation x^{2} + px + q = 0 has equal root is 2, then q = __________ .

Answer:

12

6. If x = 1 is common root of the equations px^{2} + px + 3 = 0 and x^{2} + x + q = 0, then

pq = __________ .

Answer:

3

7. If p and q can take values 1, 2, 3, 4. Then the number of equations of the form px^{2} + qx + 1 = 0 having read roots is __________ .

Answer:

10

8. If the sum and product of the roots of ax^{2} + 6x + 4a = 0 are equal, then a = __________ .

Answer:

\(\frac{-3}{2}\)

9. If p and q are roots of x^{2} + px + q = 0, then p + q = _________ .

Answer:

-1

10. If one root of ax^{2} + bx + c = 0 is 3 times other, then b^{2} : ac = __________ .

Answer:

16 : 3

11. If one root of x^{2} + px + 3 = 0 is 1, then other root is __________ .

Answer:

3

12. The value of ’a’ for which x^{2} + 4x + a is a perfect square __________ .

Answer:

4

13. The condition for the ax^{2} + bx + c = 0 and bx^{2} – 2√a x + b = 0 have equal roots is __________ .

Answer:

b^{2} = ac

14. The discriminant of 3√3x^{2} + 10x + √3 = 0 is __________ .

Answer:

64

15. The values of a for the equation x^{2} + ax – 1 = 0 has real roots are __________ .

Answer:

All real values

16. The product of two consecutive integers is 306, then the numbers are ________ .

Answer:

17 and 18

17. Roots of x^{2} – 9 = 0 are __________ .

Answer:

x = +3 (or) -3

18. Factors of 10x – \(\frac{1}{x}\) = 3 are __________ .

Answer:

\(\frac{1}{2}\), –\(\frac{1}{5}\)

19. Roots of x^{2} – (√2 + 1)x+ √2 =0 are _________ .

Answer:

–\(\frac{1}{\sqrt{2}}\), 2√2

20. Discriminant of √3x^{2} + 2√2x – 2√3 = 0 is __________ .

Answer:

32

21. The value of k if kx^{2} – 5x + k = 0 has real and equal roots is __________ .

Answer:

± \(\frac{5}{2}\)

22. k^{2}x^{2} – 2 (2k – 1)x + 4 = 0 has real and equal roots, then k = __________ .

Answer:

\(\frac{1}{4}\)

23. If the roots of ax^{2}+ bx + c = 0 and bx^{2} – 2\(\sqrt{\mathrm{ac}}\) x + b = 0 are real, then ac = __________ .

Answer:

b^{2}

24. The sum of the squares of two consecutive natural numbers is 313, then the numbers are __________ .

Answer:

12 and 13