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Find the maximum and minimum values of $\frac{x^{2} - x + 1}{x^{2} + x + 1}$ for real values of $x$ .

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Important Questions on Complex Numbers and Quadratic Equations

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

\frac{8 x}{2 x^{2} + 7 x - 2} = 1, x > 0,

then what is the value of

x^{3} + \frac{1}{x^{3}}

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

Let

α

and

β

be the roots of equation

x^{2} - 6 x - 2 = 0

. If

a_{n} = α^{n} - β^{n}, \forall n \geq 1,

then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

is equal to

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

Let

α

and

β

be the roots of

x^{2} - 6 x - 2 = 0

. If

a_{n} = α^{n} - β^{n}

for

n ⩾ 1

, then the value of

\frac{a_{10} - 2 a_{8}}{3 a_{9}}

is:

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

x - y = 4

and

x^{3} - y^{3} = 316

, then the value of

x^{4} + y^{4}

is:

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

α

and

β

be the roots of the equation

x^{2} - 2 x + 2 = 0,

then the least value of

n

for which

{(\frac{α}{β})}^{n} = 1

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

a + b = 24

and

a^{2} + b^{2} = 306

, where

a > b

, then the value of

4 a - 5 b

is:

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

The number of real roots of the polynomial equation

x^{4} - x^{2} + 2 x - 1 = 0

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

Difference between the corresponding roots of

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

is same and

a \neq b

,⁡ then

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

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>Complex Numbers and Quadratic Equations>Quadratic Equations

The factors of equation $x^{2} + 7 x + 12 = 0$ are-

(A) -3 and -4

(B) 3 and -4

(D) 3 and 4

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

9 x^{2} - 6 x + 1 = 0

, then the value of

27 x^{3} + {(27 x^{3})}^{- 1}

will be:

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

FInd the factors of $(x^{2} - x - 132) :$

1. $(x - 11) (x - 12)$
2. $(x + 12) (x - 11)$
3. $(x + 11) (x + 12)$
4. $(x - 12) (x + 11)$

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

x + \frac{81}{x} = 18

where

x > 0

, then the value of

x^{2} + \frac{162}{x^{2}}

is:

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

x^{2} + 1 - 2 x = 0, x > 0

, then

x^{2} (x^{2} - 2)

______.

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

The sum of all the real roots of the equation

{|x - 2|}^{2} + |x - 2| - 2 = 0

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

Let

r

be a root of the equation

x^{2} + 2 x + 6 = 0

. The value of

(r + 2) (r + 3) (r + 4) (r + 5)

is equal to -

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

For how many different values of $a$ does the following system have at least two distinct solutions?

$a x + y = 0$

$x + (a + 10) y = 0$

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

Which of the following is not a root of

f (x) = x^{3} - 11 x^{2} + 36 x - 36 ?

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

If roots of binomial equation

6 x^{2} - 5 x + 1 = 0

are a and β find the sum of their squares.

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Quadratic Equations

Let

α and β

be the roots of equation

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

p \neq 0

. If

p, q, r

are in A.P. and

\frac{1}{α} + \frac{1}{β} = 4

, then the value of

|α - β|

Find the maximum and minimum values of x 2 - x + 1 x 2 + x + 1 for real values of x.

Important Questions on Complex Numbers and Quadratic Equations

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Find the maximum and minimum values of $\frac{x^{2} - x + 1}{x^{2} + x + 1}$ for real values of $x$ .