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

EASY

Earn 100

The condition that one of the roots of the equation $a x^{2} + b x + c = 0$ may be negative of the other is

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Important Questions on Quadratic Expressions

HARD

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

Let

α = \frac{- 1 + i \sqrt{3}}{2}

. If

a = (1 + α) \sum_{k = 0}^{100} α^{2 k}

and

b = \sum_{k = 0}^{100} α^{3 k}

, then

a

and

b

are the roots of the quadratic equation

HARD

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of 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?

EASY

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

α

and

β

are the roots of the equation

375 x^{2} - 25 x - 2 = 0,

then

\underset{n \to \infty}{l i m} \sum_{r = 1}^{n} α^{r} + \underset{n \to \infty}{l i m} \sum_{r = 1}^{n} β^{r}

is equal to:

HARD

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

Let

- \frac{π}{6} < θ < - \frac{π}{12}

. Suppose

α_{1}

and

β_{1}

are the roots of the equation

x^{2} - 2 x \sec θ + 1 = 0

and

α_{2}

and

β_{2}

are the roots of the equation

x^{2} + 2 x \tan θ - 1 = 0 .

α_{1} > β_{1}

and

α_{2} > β_{2}

, then

α_{1} + β_{2}

equals

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of 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 -

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

Let

p

and

q

be roots of the equation

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

and let

r

and

s

be the roots of the equation

x^{2} - 18 x + B = 0

. If

p < q < r < s

are in arithmetic progression, then

A

and

B

are

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of 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>Quadratic Expressions>Special Roots and Signs of Roots

The number of real roots of the polynomial equation

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

EASY

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

The solutions of

x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} - 4 = 0

are

HARD

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

The sum of all real values of

x

satisfying the equation

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

EASY

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

Solve:

i x^{2} - 3 x - 2 i = 0

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of 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>Quadratic Expressions>Special Roots and Signs of Roots

Which of the following is not a root of

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

EASY

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of 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>Quadratic Expressions>Special Roots and Signs of Roots

α

and

β

are the roots of the equation

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

then

α^{12} + β^{12} =

HARD

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of 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:

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

Let

p

and

q

be two positive numbers such that

p + q = 2

and

p^{4} + q^{4} = 272

. Then

p

and

q

are roots of the equation:

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

The least positive value of

a

for which the equation,

2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a

has real roots is ___________.

HARD

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of Roots

Ten ants are on the real line. At time

t = 0,

the

k

^$th$ ant starts at the point

k^{2}

and travelling at uniform speed, reaches the point

{(11 - k)}^{2}

at time

t = 1 .

The number of distinct times at which at least two ants are at the same location is

HARD

Mathematics>Algebra>Quadratic Expressions>Special Roots and Signs of 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 condition that one of the roots of the equation ax2+bx+c=0 may be negative of the other is

Important Questions on Quadratic Expressions

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The condition that one of the roots of the equation $a x^{2} + b x + c = 0$ may be negative of the other is