Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 24

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Find the set of all real values of $λ$ such that the root of the equation $x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ are always real for any choice of $a, b, c$ (Where $a, b, c$ represents sides of scalene triangle).

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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 25

If coefficients of biquadratic equation are all distinct and belong to the set

\{- 9, - 5, 3, 4, 7\},

then equation has

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 26

Let

p, q, r, s \in R, x^{2} + p x + q = 0, x^{2} + r x + s = 0

such that

2 (q + s) = p r

, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 27

The equation, $π^{x} = - 2 x^{2} + 6 x - 9$ has

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 28

If

(λ^{2} + λ - 2) x^{2} + (λ + 2) x < 1

(\forall x \in R)

then

λ

belongs to the interval

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 29

Let conditions

C_{1}

and

C_{2}

be defined as follows

: C_{1} : b^{2} - 4 a c \geq 0, C_{2} : a, - b, c

are of same sign. The roots of

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

are real and positive, if

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 30

If

x

is real, then

\frac{x^{2} - x + c}{x^{2} + x + 2 c}

can take all real values if

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 31

If both roots of the quadratic equation

(2 - x) (x + 1) = p

are distinct and positive, then

p

must lie in the interval

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-2>Q 32

If two roots of the equation

(a - 1) {(x^{2} + x + 1)}^{2} - (a + 1) (x^{4} + x^{2} + 1) = 0

are real and distinct, then

' a'

lies in the interval