AP 10th Class Maths 4th Chapter Quadratic Equations Bits

Chapter 4 Quadratic Equations Bits for 10th Class AP Board

Multiple Choice Questions (MCQs) :

Question 1.
The roots of 8x2 – 8ax + (a2 – b2) = 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 x2 + (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 5x2 + 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)x2 + (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 2x2 – 3x – 6 = 0, then whose roots are α2 + 2, β2 + 2 is ……………… .
A) 5x2 + 49x + 59 = 0
B) 4x2 + 49x – 118 = 0
C) 4x2 – 49x + 118 = 0
D) 4x2 + 59x – 59 = 0
Answer:
C) 4x2 – 49x + 118 = 0

Question 6.
If the product of the roots of ax2 + bx + a2 + 4 = 0 is -4, then a is …………. .
A) -1
B) -4
C) -3
D) -2
Answer:
A) -1

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

Question 7.
The number of real roots of 2x4 + 5x2 – 3 = 0 is …………….. .
A) 1
B) 2
C) 3
D) 4
Answer:
D) 4

Question 8.
x2 + x + a = 0, bx2 + 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 (a2 + b2) x2 – 2 (ac + bd) + (c2 + d2) = 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 ax2 – 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 x2 – 8x + k = 0 is 40 ?
A) -12
B) 14
C) 12
D) -14
Answer:
C) 12

Question 12.
Roots of a2x2 – (a2b2 + 1)x + b2 = 0 are ………………. .
A) a, \(\frac{1}{b}\)
B) a2, \(\frac{1}{b}\)
C) \(\frac{1}{\mathrm{a}^2}\), b2
D) a2, \(\frac{1}{\mathrm{b}^2}\)
Answer:
C) \(\frac{1}{\mathrm{a}^2}\), b2

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

Question 13.
If ax2 + 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 x2 – 7x + 12 = 0, then the equation whose roots are 3α, 3β is ………………. .
A) x2 – 21x + 108 = 0
B) x2 + 7x – 108
C) 9x2 – 21x + 108 = 0
D) 9x2 – 7x – 108
Answer:
A) x2 – 21x + 108 = 0

Question 16.
If the equation x2 – 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 px2 + 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 x2 + ax + 64 = 0 and x2 – 8x + a = 0 both have real roots is ……………. .
A) 16
B) 8
C) 4
D) 2
Answer:
A) 16

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

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 x2 – 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 x2 – 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 x2 – 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 x2 + 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 x2 + px + q = 0, then p – q = …………….. .
A) 3
B) -1
C) 2
D) 1
Answer:
A) 3

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

Question 25.
If one root of the ax2 + bx + c = 0 is 2 times the other, then b2: 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) x2 – 5x – 6
B) x2 + 5x – 6
C) x2 – 5x + 6
D) x2 + 5x + 6
Answer:
C) x2 – 5x + 6

Question 27.
Observe the given rectangular figure, . then its area in polynomial function is ………………. .
AP 10th Class Maths 4th Chapter Quadratic Equations Bits 1
A) A(x) = x2 + 7x + 30
B) A(x) = -x2 + 7x + 30
C) A(x) = x2 – 7x + 30
D) A(x) = -x2 – 7x + 30
Answer:
B) A(x) = -x2 + 7x + 30

Question 28.
Which of the following is a quadra tic equation ?
A) x3 – 6x2 + 2x – 1 = 0
B) x2 + \(\frac{1}{x^2}\) = 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) x2 – 5 = 0
B) x2 – 10x + 25 = 0
C) x2 + 5x + 6 = 0
D) x2 – 1 = 0
Answer:
B) x2 – 10x + 25 = 0

Question 30.
The quadratic polynomial having \(\frac{1}{3}\) and \(\frac{1}{2}\) as its zeroes, is ………………. .
A) x2 + \(\frac{5 x+1}{6}\)
B) -6x2 – 5x + 1
C) x2 – \(\frac{5 x-1}{6}\)
D) 6x2 – 5x – 1
Answer:
C) x2 – \(\frac{5 x-1}{6}\)

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

Question 31.
If x2 – px + q = 0 (p, q ∈ R and p ≠ 0, q ≠ 0) has distinct real roots, then ………….. .
A) p2 < 4q
B) p2 > 4q
C) p2 = 4q
D) p2 + 4q = 0
Answer:
B) p2 > 4q

Question 32.
In a quadratic equation ax2 + bx + c = 0, if b2 – 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
x2 – 4 = 0 and x2 + px – 4 = 0, then p =
A) 2
B) 0
C) 4
D) 1
Answer:
B) 0

Question 35.
The sum of the roots of 6x2 = 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

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

Question 37.
The roots of a quadratic equation ax2 – 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 x2 – 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 x2 + 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) x2 + 4x + 1 = 0
B) x2 + 4x – 1 = 0
C) x2 – 4x + 1 = 0
D) x2 – 4x – 1 = 0
Answer:
C) x2 – 4x + 1 = 0

Question 42.
The Quadratic equation, whose sum of the roots is -3 and product of the roots is 2.
A) x2 + 6x + 5 = 0
B) x2 – x – 6 = 0
C) x2 – 3x + 2 = 0
D) x2 + 3x + 2 = 0
Answer:
D) x2 + 3x + 2 = 0

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

Question 43.
The roots of a quadratic equation 2x2 + 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) x2 – 4x + 5 = 0
B) x2 + 3x – 12 = 0
C) 2x2 – 7x + 6 = 0
D) 3x2 – 6x – 2 = 0
Answer:
C) 2x2 – 7x + 6 = 0

Fill in the blanks :

1. The number of real roots of 2x4 + 5x2 + 3 = 0 is __________ .
Answer:
0

2. If a and b are such that the quadratic equation ax2 – 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 ax2 + bx + c = 0 is a perfect square, then a = __________ .
Answer:
\(\frac{b^2}{4 c}\)

4. If the x2 + 4x + a = 0 has real and distinct, then k < ___________ .
Answer:
4

5. If x2 + px + 12 = 0 and the equation x2 + px + q = 0 has equal root is 2, then q = __________ .
Answer:
12

6. If x = 1 is common root of the equations px2 + px + 3 = 0 and x2 + 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 px2 + qx + 1 = 0 having read roots is __________ .
Answer:
10

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

8. If the sum and product of the roots of ax2 + 6x + 4a = 0 are equal, then a = __________ .
Answer:
\(\frac{-3}{2}\)

9. If p and q are roots of x2 + px + q = 0, then p + q = _________ .
Answer:
-1

10. If one root of ax2 + bx + c = 0 is 3 times other, then b2 : ac = __________ .
Answer:
16 : 3

11. If one root of x2 + px + 3 = 0 is 1, then other root is __________ .
Answer:
3

12. The value of ’a’ for which x2 + 4x + a is a perfect square __________ .
Answer:
4

13. The condition for the ax2 + bx + c = 0 and bx2 – 2√a x + b = 0 have equal roots is __________ .
Answer:
b2 = ac

14. The discriminant of 3√3x2 + 10x + √3 = 0 is __________ .
Answer:
64

15. The values of a for the equation x2 + 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

AP 10th Class Maths 4th Chapter Quadratic Equations Bits

17. Roots of x2 – 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 x2 – (√2 + 1)x+ √2 =0 are _________ .
Answer:
–\(\frac{1}{\sqrt{2}}\), 2√2

20. Discriminant of √3x2 + 2√2x – 2√3 = 0 is __________ .
Answer:
32

21. The value of k if kx2 – 5x + k = 0 has real and equal roots is __________ .
Answer:
± \(\frac{5}{2}\)

22. k2x2 – 2 (2k – 1)x + 4 = 0 has real and equal roots, then k = __________ .
Answer:
\(\frac{1}{4}\)

23. If the roots of ax2+ bx + c = 0 and bx2 – 2\(\sqrt{\mathrm{ac}}\) x + b = 0 are real, then ac = __________ .
Answer:
b2

24. The sum of the squares of two consecutive natural numbers is 313, then the numbers are __________ .
Answer:
12 and 13

10th Class Maths Bits

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