Let α and β be the roots of the equation x 2 - x - 1 = 0 . If P k = α k + β k , k ≥ 1 , then which one of the following statements is not true?

P 1 + P 2 + P 3 + P 4 + P 5 = 26

HARD

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Let $α, β$ be the roots of the equation $x^{2} - x + 2 = 0$ with $Im (α) > Im (β)$ . Then $α^{6} + α^{4} + β^{4} - 5 α^{2}$ is equal to

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Important Questions on Quadratic Equations

EASY

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α

and

β

are roots of the equation

x^{2} + x + 1 = 0

then

α^{2} + β^{2}

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

Let

α

and

β

be two real roots of the equation

(k + 1) {t a n}^{2} x - \sqrt{2} \cdot λ \tan x = (1 - k),

where

k (\neq - 1)

and

λ

are real numbers. If

{t a n}^{2} (α + β) = 50,

then a value of

λ

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

Let

α

and

β

be the roots of the equation

x^{2} - x - 1 = 0

. If

P_{k} = {(α)}^{k} + {(β)}^{k}, k \geq 1,

then which one of the following statements is not true?

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α

and

β

are the roots of the equation

2 x (2 x + 1) = 1

, then

β

is equal to :

HARD

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

Suppose the quadratic polynomial

P (x) = a x^{2} + b x + c

has positive coefficients

a, b, c

in arithmetic progression in that order. If

P (x) = 0

has integer roots

α and β,

then

α + β + α β

equals

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α

and

β

are the roots of the equation,

7 x^{2} - 3 x - 2 = 0,

then the value of

\frac{α}{1 - α^{2}} + \frac{β}{1 - β^{2}}

is equal to:

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

Let

a, b \in R, a \neq 0

be such that the equation,

a x^{2} - 2 b x + 5 = 0

has a repeated root

α,

which is also a root of the equation,

x^{2} - 2 b x - 10 = 0 .

β

is the other root of this equation, then

α^{2} + β^{2}

is equal to:

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

For the equation

3 x^{2} + p x + 3 = 0, p > 0

, if one of the roots is square of the other, then

p

is equal to

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

\frac{1}{\sqrt{α}}, \frac{1}{\sqrt{β}}

are the roots of the equation

a x^{2} + b x + 1 = 0, (a \neq 0, a, b \in R)

, then the equation

x (x + b^{3}) + (a^{3} - 3 a b x) = 0

has roots:

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

m

is chosen in the quadratic equation

(m^{2} + 1) x^{2} - 3 x + {(m^{2} + 1)}^{2} = 0

such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α

and

β

be two roots of the equation

x^{2} - 64 x + 256 = 0 .

Then the value of

{(\frac{α^{3}}{β^{5}})}^{\frac{1}{8}} + {(\frac{β^{3}}{α^{5}})}^{\frac{1}{8}}

is :

EASY

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

λ

be the ratio of the roots of the quadratic equation in

x, 3 m^{2} x^{2} + m (m - 4) x + 2 = 0

, then the least value of

m

for which

λ + \frac{1}{λ} = 1

, is :

HARD

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α

and

β

are the roots of the equation

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

and

\frac{1}{α}

and

\frac{1}{β}

are the roots of the equation

2 x^{2} + 2 q x + 1 = 0,

then

(α - \frac{1}{α}) (β - \frac{1}{β}) (α + \frac{1}{β}) (β + \frac{1}{α})

is equal to :

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

Let

f (x)

be a quadratic polynomial such that

f (- 1) + f (2) = 0

. If one of the roots of

f (x) = 0

3

, then its other root lies in

HARD

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

Suppose

a, b

denote the distinct real roots of the quadratic polynomial

x^{2} + 20 x - 2020

and suppose

c, d

denote the distinct complex roots of the quadratic polynomial

x^{2} - 20 x + 2020

. Then the value of

a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)

is equal to

HARD

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α, β

are the real 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

has always

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α

and

β

are the roots of the quadratic equation

3 x^{2} - 16 x + 5 = 0,

then

\tan^{- 1} α + \tan^{- 1} β - \tan^{- 1} (\frac{α + β}{1 - α β}) =

MEDIUM

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

α

and

β

are the roots of the quadratic equation

x^{2} + x s i n θ - 2 s i n θ = 0, θ \in (0, \frac{π}{2})

, then

\frac{α^{12} + β^{12}}{(α^{- 12} + β^{- 12}) {. (α - β)}^{24}}

is equal to :

EASY

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

Let

p, q

and

r

be real numbers

(p \neq q, r \neq 0)

, such that the roots of the equation

\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}

are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to

EASY

Mathematics>Algebra>Quadratic Equations>Relation between Roots and Coefficients

The harmonic mean of the roots of the equation

(5 + \sqrt{2}) x^{2} - (4 + \sqrt{5}) x + (8 + 2 \sqrt{5} = 0)

Let α,β be the roots of the equation x2-x+2=0 with Im(α)>Im(β). Then α6+α4+β4-5α2 is equal to

Important Questions on Quadratic Equations

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Let $α, β$ be the roots of the equation $x^{2} - x + 2 = 0$ with $Im (α) > Im (β)$ . Then $α^{6} + α^{4} + β^{4} - 5 α^{2}$ is equal to