Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 59

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If $f (x)$ is a polynomial of degree three with leading coefficient $1$ such that $f (1) = 1, f (2) = 4, f (3) = 9,$ then

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Important Questions on Quadratic Equations

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 60

Let

P (x) = x^{32} - x^{25} + x^{18} - x^{11} + x^{4} - x^{3} + 1

. Which of the following are CORRECT?

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 61

If

α, β

are the real and distinct roots of

x^{2} + p x + q = 0

and

α^{4}, β^{4}

are the roots of

x^{2} - r x + s = 0

, then the equation

x^{2} - 4 q x + 2 q^{2} - r = 0

always has

(α \neq - β)

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 62

If

- 5 + i β, - 5 + i γ

(where

β^{2} \neq γ^{2}; β, γ \in R

and

i^{2} = - 1

) are roots of

x^{3} + 15 x^{2} + c x + 860 = 0, c \in R,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 63

Let

f (x) = a x^{2} + b x + c > 0, \forall x \in R

or

f (x) < 0, \forall x \in R

. Which of the following is/are CORRECT?

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 64

Let

f (x) = \frac{3}{x - 2} + \frac{4}{x - 3} + \frac{5}{x - 4}

, then

f (x) = 0

has

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 65

If the quadratic equations

a x^{2} + b x + c = 0 (a, b, c \in R, a \neq 0)

and

x^{2} + 4 x + 5 = 0

have a common root, then

a, b, c

must satisfy the relations

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 66

If the quadratic equations

x^{2} + a b x + c = 0

and

x^{2} + a c x + b = 0

have a common root, then the equation containing their other roots is/are

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 67

Consider the following statements

$S_{1} :$ The equation $2 x^{2} + 3 x + 1 = 0$ has irrational roots.

$S_{2} :$ If $a < b < c < d,$ then, the roots of the equation $(x - a) (x - c) + 2 (x - b) (x - d) = 0$ are real and distinct.

$S_{3} :$ If $x^{2} + 3 x + 5 = 0$ and $a x^{2} + b x + c = 0$ have a common root and $a, b, c \in N$ , then the minimum value of $(a + b + c)$ is $10$ .

$S_{4} :$ The value of the biquadratic expression $x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2,$ when $x = 2 + \sqrt{3}$ , is $1$ .

Which of the following are CORRECT?