Class 10 maths Quadratic Equations Hots Questions

Q 1 :

Nature of roots of quadratic equation $2 x^{2} - 4 x + 3 = 0$ is

real
equal
not real
none of them

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2

3

4

(3)

$D = b^{2} - 4 a c = 4^{2} - 4 \times 2 \times 3 = 16 - 24 = - 8 < 0$

Since $D < 0$

Hence, roots are not real.

Q 2 :

Let p be a prime number. The quadratic equation having its roots as factors of p is

$x^{2} - p x + p = 0$
$x^{2} - (p + 1) x + p = 0$
$x^{2} + (p + 1) x + p = 0$
$x^{2} - p x + p + 1 = 0$

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(2)

Factors of $p = p \times 1$

$∴$ $Roots are p and 1 .$

The quadratic equation is:

$x^{2} - (sum of roots) x + product of roots = 0$

$\Rightarrow x^{2} - (p + 1) x + p = 0$

Q 3 :

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?

$a = \frac{b^{2}}{c}$
$b^{2} = a c$
$a c = \frac{b^{2}}{4}$
$c = \frac{b^{2}}{a}$

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2

3

4

(3)

If the discriminant is equal to zero, i.e., $b^{2} - 4 a c = 0$ where a, b, c are real numbers and $a \neq 0,$ then roots of the quadratic equation $a x^{2} + b x + c = 0,$ are real and equal $b^{2} - 4 a c = 0 \Rightarrow a c = \frac{b^{2}}{4}$

Q 4 :

If x = 5 is a solution of the quadratic equation $2 x^{2} + (k - 1) x + 10 = 0$ , then the value of k is :

11
– 11
13
– 13

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2

3

4

(2) – 11

Q 5 :

The quadratic equation $x^{2} + x + 1 = 0$ has ................. roots.

real and equal
irrational
real and distinct
not-real

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2

3

4

(4)

Given equation is $x^{2} + x + 1 = 0$

Where $a = 1,$ $b = 1,$ $c = 1$

$D = b^{2} - 4 a c = {(1)}^{2} - 4 \times 1 \times 1$

$\Rightarrow D = - 3$

Where $D < 0$

When $D < 0$ roots are not-real.

Q 6 :

The real roots of the equation $x^{2 / 3} + x^{1 / 3} - 2 = 0$ are:

(i) 1 (ii) -8 (iii) -1 (iv) 8

Choose the correct option from the following:

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

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2

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4

(1)

Given equation is $x^{2 / 3} + x^{1 / 3} - 2 = 0$

⇒ t² + t - 2 = 0 (Take $x^{1 / 3}$ = t)

⇒ t² + 2t - t - 2 = 0

⇒ t(t + 2) - 1(t + 2) = 0 ⇒ (t + 2)(t - 1) = 0

The roots of above equation are t = 1 and t = -2.

If t = 1, then x = 1.

If t = -2, then x = -8.

Q 7 :

If ‘p’ is a root of the quadratic equation $x^{2} - (p + q) x + k = 0,$ then the value of k is:

p
q
p + q
pq

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2

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4

Since p is a root of the quadratic equation

$x^{2} - (p + q) x + k = 0,$
we have
$p^{2} - (p + q) p + k = 0 \Rightarrow p^{2} - p^{2} - p q + k = 0 \Rightarrow k = p q$

Q 8 :

The sum of the roots of $4 x ² - \sqrt 3 x - 5 = 0 i s :$

1/4
$- \sqrt 3 / 4$
$4 / \sqrt 3$
$\sqrt 3 / 4$

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2

3

4

(4)

$G i v e n q u a d r a t i c e q u a t i o n : 4 x ² - \sqrt 3 x - 5 = 0 . S u m o f r o o t s = - b / a = \sqrt 3 / 4$

Q 9 :

If ½ is a root of the equation $x ² + k x - 5 / 4 = 0$ , then the value of k is

2
-2
1/4
1/2

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2

3

4

(1)

$G i v e n e q u a t i o n : x ² + k x - 5 / 4 = 0 . S i n c e ½ i s a r o o t, (½) ² + k ½ - 5 / 4 = 0 \Rightarrow 1 / 4 + k / 2 - 5 / 4 = 0 \Rightarrow k / 2 - 1 = 0 \Rightarrow k = 2$

Q 10 :

The product of roots of the equation $9 x ² + 3 / 4 x - \sqrt 2 = 0 i s$

$\sqrt 2$
$- \sqrt 2 / 9$
$9 / \sqrt 2$
1/9

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2

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4

(2)

Given quadratic equation: $9 x ² + 3 / 4 x - \sqrt 2 = 0$ .

Product of roots = $c / a = - \sqrt 2 / 9$

Topic Question Set

Q 1 :

Nature of roots of quadratic equation $2 x^{2} - 4 x + 3 = 0$ is

Q 2 :

Let p be a prime number. The quadratic equation having its roots as factors of p is

Q 3 :

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 4 :

If x = 5 is a solution of the quadratic equation $2 x^{2} + (k - 1) x + 10 = 0$ , then the value of k is :

Q 5 :

The quadratic equation $x^{2} + x + 1 = 0$ has ................. roots.

Q 6 :

The real roots of the equation $x^{2 / 3} + x^{1 / 3} - 2 = 0$ are:

(i) 1 (ii) -8 (iii) -1 (iv) 8

Choose the correct option from the following:

Q 7 :

If ‘p’ is a root of the quadratic equation $x^{2} - (p + q) x + k = 0,$ then the value of k is:

Q 8 :

The sum of the roots of $4 x ² - \sqrt 3 x - 5 = 0 i s :$

Q 9 :

If ½ is a root of the equation $x ² + k x - 5 / 4 = 0$ , then the value of k is

Q 10 :

The product of roots of the equation $9 x ² + 3 / 4 x - \sqrt 2 = 0 i s$

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Topic Question Set

Q 1 : Nature of roots of quadratic equation 2x2–4x+3=0 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 5 : The quadratic equation x2+x+1=0 has ................. roots. .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 8 : The sum of the roots of 4x² – √3x – 5 = 0 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 9 : If ½ is a root of the equation x² + kx – 5/4 = 0, then the value of k 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 10 : The product of roots of the equation 9x² + 3/4 x – √2 = 0 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; }

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Q 1 :

Nature of roots of quadratic equation $2 x^{2} - 4 x + 3 = 0$ is

Q 5 :

The quadratic equation $x^{2} + x + 1 = 0$ has ................. roots.

Q 8 :

The sum of the roots of $4 x ² - \sqrt 3 x - 5 = 0 i s :$

Q 9 :

If ½ is a root of the equation $x ² + k x - 5 / 4 = 0$ , then the value of k is

Q 10 :

The product of roots of the equation $9 x ² + 3 / 4 x - \sqrt 2 = 0 i s$