Let α and β be the roots of x 2 - x - 1 = 0 , with α > β . For all positive integers n , define a n = α n - β n α - β , n ≥ 1 b 1 = 1 and b n = a n - 1 + a n + 1 , n ≥ 2 . Then which of the following options is/are correct?

b n = α n + β n for all n ≥ 1

HARD

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The number of real roots of the equation $\sqrt{x^{2} - 4 x + 3} + \sqrt{x^{2} - 9} = \sqrt{4 x^{2} - 14 x + 6}$ , is:

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Important Questions on Complex Numbers

EASY

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

The number of real roots of the polynomial equation

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

EASY

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

Let

α

and

β

be the roots of the equation,

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

. If

S_{n} = α^{n} + β^{n}, n = 1, 2, 3, . . . .,

then

MEDIUM

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

α

and

β

are the roots of the equation

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

then

α^{12} + β^{12} =

MEDIUM

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

α

and

β

are the distinct roots of the equation

x^{2} + {(3)}^{\frac{1}{4}} x + 3^{\frac{1}{2}} = 0

, then the value of

α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)

is equal to:

HARD

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

The sum of all real values of

x

satisfying the equation

{(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = 1

HARD

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

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

then what is the value of

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

MEDIUM

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

Which of the following is not a root of

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

EASY

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

The value of discriminant of quadratic equation $x^{2} - 2 x + 1 = 0$ will be:

HARD

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

Let

α

and

β

be the roots of

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

with

α > β .

For all positive integers

n,

define

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

b_{1} = 1

and

b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2 .

Then which of the following options is/are correct?

MEDIUM

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

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

EASY

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

Let

α, β

be two roots of the equation

x^{2} + {(20)}^{1 / 4} x + {(5)}^{1 / 2} = 0

. Then

α^{8} + β^{8}

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

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>Quadratic Equation with Complex Roots

The sum of the roots of the equation, $x + 1 - 2 \log_{2} (3 + 2^{x}) + 2 \log_{4} (10 - 2^{- x}) = 0,$ is :

HARD

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

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:

EASY

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

Under what condition will

a x^{2} + 5 x + 7 = 0

be a quadratic equation?

EASY

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

From the following equations, which one is the quadratic equation?

EASY

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

The discriminant of the quadratic equation

x^{2} + 8 x + 16 = 0

MEDIUM

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

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

(A) -3 and -4

(B) 3 and -4

(D) 3 and 4

MEDIUM

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

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 -

HARD

Mathematics>Algebra>Complex Numbers>Quadratic Equation with Complex Roots

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

The number of real roots of the equation x2-4x+3+x2-9=4x2-14x+6, is:

Important Questions on Complex Numbers

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The number of real roots of the equation $\sqrt{x^{2} - 4 x + 3} + \sqrt{x^{2} - 9} = \sqrt{4 x^{2} - 14 x + 6}$ , is: