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QUESTION: 1

Which of the following is not a quadratic equation?

Solution:

2x^{2 }+ 3 + 2√6x + x^{2} = 3x^{2} - 5x

3 + 2√6x + 5x= 0

Which is not a quadratic equation.

QUESTION: 2

Which of the following equations has no real roots ?

Solution:

(a) The given equation is x^{2} - 4x + 3√2 = 0.

On comparing with ax^{2} + bx + c = 0, we get

a = 1, b = -4 and c = 3√2

The discriminant of x^{2} - 4x + 3√2 = 0 is

D = b^{2} - 4ac

= (-4)^{2} - 4(1)(3√2) = 16 - 12√2 = 16 - 12 x (1.41)

= 16 - 16.92 = -0.92

⇒ b^{2} - 4ac < 0

(b) The given equation is x^{2} + 4x - 3√2 = 0

On comparing the equation with ax^{2} + bx + c = 0, we get

a = 1, b = 4 and c = -3√2

Then, D = b^{2} - 4ac = (-4)^{2} - 4(1)(-3√2)

= 16 + 12√2 > 0

Hence, the equation has real roots.

(c) Given equation is x^{2} - 4x - 3√2 = 0

On comparing the equation with ax^{2} + bx + c = 0, we get

a = 1, b = -4 and c = -3√2

Then, D = b^{2} - 4ac = (-4)^{2} - 4(1) (-3√2)

= 16 + 12√2 > 0

Hence, the equation has real roots.

(d) Given equation is 3x^{2} + 4√3x + 4 = 0.

On comparing the equation with ax^{2} + bx + c = 0, we get

a = 3, b = 4√3 and c = 4

Then, D = b^{2} - 4ac = (4√3)^{2} - 4(3)(4) = 48 - 48 = 0

Hence, the equation has real roots.

Hence, x^{2} - 4x + 3√2 = 0 has no real roots.

QUESTION: 3

Which of the following equations has the sum of its roots as 3?

Solution:

(a) Given that, 2x^{2} - 3x + 6 = 0

On comparing with ax^{2} + bx + c = 0, we get

a = 2, b = -3 and c = 6

∴ Sum of the roots =

So, sum of the roots of the quadratic equation 2x^{2} - 3x + = 0 is not 3, so it is not the answer

(b) Given that, -x^{2} + 3x - 3 =0

On compare with ax^{2} + bx + c = 0, we get

a = -1, b = 3 and c = -3

∴ Sum of the roots =

So, sum of the roots of the quadratic equation -x^{2} + 3x - 3 = 0 is 3, so it is the answer.

(c) Given that,

⇒ 2x^{2} - 3x + √2 = 0

On comparing with ax^{2} + bx + c = 0, we get

a = 2, b = -3 and c = √2

∴ Sum of thee roots =

So, sum of the roots of the quadratic equation is not 3, so it is not the answer.

(d) Given that, 3x^{2} - 3x + 3 = 0

⇒ x^{2} - x + 1 = 0

On comparing with ax^{2} + bx + c = 0, we get

a = 1, b = -1 and c = 1

∴ Sum of the roots =

So, sum of the roots of the quadratic equation 3x^{2} - 3x + 3 = 0 is not 3, so it is not the answer.

QUESTION: 4

(x^{2} + l)^{2} - x^{2} = 0 has

Solution:

Given equation is (x^{2} + 1)^{2} - x^{2} = 0

⇒ x^{4} + 1 + 2x^{2} - x^{2} = 0 [∵ (a + b)^{2} = a^{2} + b^{2} + 2ab]

⇒ x^{4} + x^{2} + 1 = 0

Let x^{2} = y

∴ (x^{2})^{2} + x^{2} + 1 = 0

y^{2} + y + 1 = 0

On comparing with ay^{2} + by + c = 0, we get

a = 1, b = 1 and c = 1

Discriminant, D = b^{2} - 4ac

= (1)^{2 }- 4(1)(1)

= 1 - 4 = -3

Since, D < 0

∴ y^{2} + y + 1 = 0 i.e., x^{4} + x^{2} + 1 = 0 or (x^{2} + 1)^{2} - x^{2} = 0 has no real roots.

QUESTION: 5

The quadratic equation has

Solution:

**We have a quadratic equation: **

If we have standard equation ax^{2} + bx + c then D = b^{2} - 4ac

a= 2, b= -√5, c= 1

D = (-√5)^{2} - (4x2x1)

D= 5 - 8

D = -3

As the value of D<0 so there is no real root

QUESTION: 6

Root of the equation x^{2 }- 0.09 = 0 is

Solution:

x^{2} - 0.09 = 0

x^{2} = 0.09

x = √0.09

x = 0.3

QUESTION: 7

If 1/2 is a root of the equation then the value of k is

Solution:

As 1/2 is a root then it will satisfy the given equation.

Put x = 1/2

1/4 +(k×1/2) - 5/4 =0

k×1/2 =5/4 -1/4

k =2

QUESTION: 8

If the equation x^{2} - kx +9 = 0 does not possess real roots, then

Solution:

QUESTION: 9

Which of the following equations has - 1 as a root?

Solution:

3x² - 2x - 5 = 0

3x² + 3x - 5x - 5 = 0

3x(x+1) - 5(x+1) = 0

(x+1)(3x-5) = 0

x = -1 and 5/3.

Hence roots of given polynomial are -1 and 5/3.

QUESTION: 10

The quadratic equation has

Solution:

As, discriminant (b^{2} - 4ac) of the equation is negative. Therefore, no real roots.

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