Class 10 maths Quadratic Equations Hots Questions

Q 11 :

If one root of the equation $5 x ² - 13 x + k = 0$ is reciprocal of the other, then k = ?

0
5
6
1/6

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2

3

4

(2)

Given quadratic equation: $5 x ² - 13 x + k = 0$ . Since one root is reciprocal of the other,

Product of roots $= 1 \Rightarrow c / a = 1 \Rightarrow c = a \Rightarrow k = 5$

Q 12 :

If one root of the equation $2 x ² - 10 x + p = 0 i s 2$ , then the value of p is:

-3
-6
9
12

1

2

3

4

(4)

$S i n c e 2 i s a r o o t, 2 (2) ² - 10 (2) + p = 0 \Rightarrow 8 - 20 + p = 0 \Rightarrow p = 12$

Q 13 :

The roots of the quadratic equation $x ² + 5 x - (a + 1) (a + 6) = 0$ , where a is a constant, are:

(i) (a + 1) (ii) –(a + 6) (iii) (a + 6) (iv) –(a + 1)

Choose the correct option from the following:

(i) and (iii)
(i) and (ii)
(iii) and (iv)
(ii) and (iv)

1

2

3

4

(2)

As observed, the sum of roots is –5 and product of roots is –(a + 1)(a + 6). This is possible only when the roots are (a + 1) and –(a + 6).

Q 14 :

If x = –2 and 3/4 are the roots of the quadratic equation $A x ² + B x - 6 = 0$ , based on this what can you say about the nature of values A and B?

(i) Both are positive numbers.
(ii) One is positive and the other is negative.
(iii) Both are odd.
(iv) Sum of A and B is an odd number.

Choose the correct option from the following

Only (ii) and (iii) are correct
Only (i) and (ii) are correct
Only (i) and (iv) are correct
Only (iii) and (iv) are correct

1

2

3

4

(3)

Given, x = –2 and 3/4 are roots of the quadratic equation $A x ² + B x - 6 = 0 .$

Product of roots = $- 6 / A \Rightarrow (- 2 \times 3 / 4) = - 6 / A \Rightarrow A = 4$
Sum of roots = $- B / A \Rightarrow (- 2 + 3 / 4) = - B / 4 \Rightarrow - 5 / 4 = - B / 4 \Rightarrow B = 5$

Both A and B are positive numbers and sum of A and B is an odd number.

Q 15 :

If the roots of equation $a x ² + b x + c = 0, a \neq 0$ are real and equal, then which of the following relation is true?

$a = 2 b ² / c$
$b ² = 2 a c$
$a c = b ² / 4$
$c = 4 b ² / a$

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2

3

4

(3)

Given equation is $a x ² + b x + c = 0, a \neq 0 .$
For real and equal roots, D = 0.

⇒ $b ² - 4 a c = 0 \Rightarrow b ² = 4 a c \Rightarrow b ² / 4 = a c$

Q 16 :

If the quadratic equation $a x ² + b x + c = 0, a \neq 0$ has two real and equal roots, then ‘c’ is equal to

−b/2a
b/2a
$- b ² / 4 a$
$b ² / 4 a$

1

2

3

4

(4)

Given quadratic equation $a x ² + b x + c = 0, a \neq 0 .$
For real and equal roots, discriminant = 0.

⇒ $b ² - 4 a c = 0 \Rightarrow b ² = 4 a c \Rightarrow c = b ² / 4 a$

Q 17 :

The quadratic equation $x ² - 4 x + k = 0$ has distinct real roots if:

k = 4
k > 4
k = 16
k < 4

1

2

3

4

(4)

$G i v e n x ² - 4 x + k = 0 . F o r d i s t i n c t r e a l r o o t s, D > 0 .$

$\Rightarrow (- 4) ² - 4 \times 1 \times k > 0 \Rightarrow 16 - 4 k > 0 \Rightarrow 16 > 4 k \Rightarrow k < 4 .$

Q 18 :

The value(s) of k for which the quadratic equation $2 x ² + k x + 2 = 0$ has equal roots, is/are

4
$\pm 4$
-4
0

1

2

3

4

(2)

Given quadratic equation is $2 x ² + k x + 2 = 0 .$
For equal roots, discriminant = 0.

$\Rightarrow k ² - 4 \times 2 \times 2 = 0 \Rightarrow k ² - 16 = 0 \Rightarrow k = \pm 4$

Q 19 :

$(x ² + 1) ² - x ² = 0 h a s$

Four real roots
Two real roots
No real roots
One real root

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2

3

4

(3)

Given equation: $(x ² + 1) ² - x ² = 0 \Rightarrow (x ² + 1 - x) (x ² + 1 + x) = 0 .$

$\Rightarrow (x ² - x + 1) (x ² + x + 1) = 0 o r x ² - x + 1 = 0,$
$I t s d i s c r i m i n a n t D = (- 1) ² - 4 \times 1 \times 1 = - 3 < 0$
$I t s d i s c r i m i n a n t D = 1 ² - 4 \times 1 \times 1 = - 3 < 0$

So, there are no real roots.

Q 20 :

The possible values of k for which the quadratic equation $9 x ² - 3 k x + k = 0$ has real and distinct roots:

(i) 2 (ii) 0 (iii) 5 (iv) 6

Choose the correct option from the following

(i) and (ii)
(iii) and (iv)
(i), (ii) and (iii)
(i), (iii) and (iv)

1

2

3

4

(2)

For real and distinct roots, the discriminant (D) of the equation 9 $x ² - 3 k x + k = 0$ must be greater than zero.

$D = (- 3 k) ² - 4 \times 9 \times k > 0 \Rightarrow 9 k ² - 36 k > 0 \Rightarrow 9 k (k - 4) > 0$

This inequality implies that k < 0 or k > 4.
Therefore, the possible values of k are (iii) and (iv).

Topic Question Set

Q 11 :

If one root of the equation $5 x ² - 13 x + k = 0$ is reciprocal of the other, then k = ?

Q 12 :

If one root of the equation $2 x ² - 10 x + p = 0 i s 2$ , then the value of p is:

Q 13 :

The roots of the quadratic equation $x ² + 5 x - (a + 1) (a + 6) = 0$ , where a is a constant, are:

(i) (a + 1) (ii) –(a + 6) (iii) (a + 6) (iv) –(a + 1)

Choose the correct option from the following:

Q 15 :

If the roots of equation $a x ² + b x + c = 0, a \neq 0$ are real and equal, then which of the following relation is true?

Q 16 :

If the quadratic equation $a x ² + b x + c = 0, a \neq 0$ has two real and equal roots, then ‘c’ is equal to

Q 17 :

The quadratic equation $x ² - 4 x + k = 0$ has distinct real roots if:

Q 18 :

The value(s) of k for which the quadratic equation $2 x ² + k x + 2 = 0$ has equal roots, is/are

Q 19 :

$(x ² + 1) ² - x ² = 0 h a s$

Q 20 :

The possible values of k for which the quadratic equation $9 x ² - 3 k x + k = 0$ has real and distinct roots:

(i) 2 (ii) 0 (iii) 5 (iv) 6

Choose the correct option from the following

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Topic Question Set

Q 12 : If one root of the equation 2x² – 10x + p = 0 is 2, then the value of p is: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 17 : The quadratic equation x² − 4x + k = 0 has distinct real roots if: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 19 : (x² + 1)² − x² = 0 has .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

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Q 12 :

If one root of the equation $2 x ² - 10 x + p = 0 i s 2$ , then the value of p is:

Q 17 :

The quadratic equation $x ² - 4 x + k = 0$ has distinct real roots if:

Q 19 :

$(x ² + 1) ² - x ² = 0 h a s$