# 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
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, –$$\frac{1}{\mathbf{p}}$$
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
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
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
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
A) -1

Question 7.
The number of real roots of 2x4 + 5x2 – 3 = 0 is …………….. .
A) 1
B) 2
C) 3
D) 4
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
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) $$\frac{a^2+c^2}{b^2}$$ = $$\frac{b^2+a^2}{d^2}$$
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) $$\frac{1}{5}$$
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
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, $$\frac{1}{\mathrm{b}^2}$$
C) $$\frac{1}{\mathrm{a}^2}$$, b2

Question 13.
If ax2 + bx + c = 0isa perfect square, then b = …………….. .
A) ac
B) $$\sqrt{4 \mathrm{ac}}$$
C) 4ac
D) 2ac
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
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
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) $$\frac{r}{4 q}$$
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
A) 16

Question 19.
The value of $$\sqrt{6+\sqrt{6+\sqrt{6+\ldots \ldots}}}$$ =
A) 2
B) 1
C) 3
D) 4
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
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
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) –$$\frac{1}{2}$$
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
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
A) 3

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
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
C) x2 – 5x + 6

Question 27.
Observe the given rectangular figure, . then its area in polynomial function is ………………. .

A) A(x) = x2 + 7x + 30
B) A(x) = -x2 + 7x + 30
C) A(x) = x2 – 7x + 30
D) A(x) = -x2 – 7x + 30
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
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
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
C) x2 – $$\frac{5 x-1}{6}$$

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
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
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
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
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
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
B) 4

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) $$\frac{-b+\sqrt{b^2-4 a c}}{2 a} ; \frac{-b-\sqrt{b^2-4 a c}}{2 a}$$
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
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.
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
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
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
D) x2 + 3x + 2 = 0

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.
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
C) 2x2 – 7x + 6 = 0

Fill in the blanks :

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

3. If ax2 + bx + c = 0 is a perfect square, then a = __________ .
$$\frac{b^2}{4 c}$$

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

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

6. If x = 1 is common root of the equations px2 + px + 3 = 0 and x2 + x + q = 0, then
pq = __________ .
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 __________ .
10

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

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

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

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

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

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

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

15. The values of a for the equation x2 + ax – 1 = 0 has real roots are __________ .
All real values

16. The product of two consecutive integers is 306, then the numbers are ________ .
17 and 18

17. Roots of x2 – 9 = 0 are __________ .
x = +3 (or) -3

18. Factors of 10x – $$\frac{1}{x}$$ = 3 are __________ .
$$\frac{1}{2}$$, –$$\frac{1}{5}$$

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

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

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

22. k2x2 – 2 (2k – 1)x + 4 = 0 has real and equal roots, then k = __________ .
$$\frac{1}{4}$$
23. If the roots of ax2+ bx + c = 0 and bx2 – 2$$\sqrt{\mathrm{ac}}$$ x + b = 0 are real, then ac = __________ .