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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If $a > b > c > d$ and the roots of equation $(x - a) (x - c) + k (x - b) (x - d) = 0$ are real, then k can be equal to

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Important Questions on Theory of Equation

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

The number of ordered pairs

(x, y)

of real numbers that satisfy the simultaneous equations

x + y^{2} = x^{2} + y = 12

EASY

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

The harmonic mean of the roots of the equation

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

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

The sum of all the real values of

x

Satisfying the equation

2^{(x - 1) (x^{2} + 5 x - 50)} = 1

is:

HARD

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

Let

S

be the set of all non-zero real numbers

α

such that the quadratic equation

α x^{2} - x + α = 0

has two distinct real roots

x_{1} & x_{2}

satisfying the inequality

|x_{1} - x_{2}| < 1 .

Which of the following intervals is(are) a subset(s) of

S

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^{2} + (a + b) x + b = 0$ is necessarily true ?

$(I)$ It has at least one negative root.

$(II)$ It has at least one positive root.

$(III)$ Both its roots are real.

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

The equation

\sqrt{3 x^{2} + x + 5} = x - 3

, where

x

is real, has

HARD

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

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>Theory of Equation>Quadratic Equation and its Solutions

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>Theory of Equation>Quadratic Equation and its Solutions

\sin α

and

\cos α

are the roots of the equation

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

,

then

HARD

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

Let

f (x) = p x^{2} + q x + r

,

where

p, q, r

are constants and

p \neq 0 .

f (5) = - 3 f (2)

and

f (- 4) = 0

,

then the other root of

f

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

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:

EASY

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

If the difference between the roots of the equation

x^{2} + k x + 1 = 0

is strictly less than

\sqrt{5},

where

|k| \geq 2,

then

k

can be any element of the interval

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

The sum of the solutions of the equation

|\sqrt{x} - 2| + \sqrt{x} (\sqrt{x} - 4) + 2 = 0, (x > 0)

is equal to

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

\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>Theory of Equation>Quadratic Equation and its Solutions

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 :

MEDIUM

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

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>Theory of Equation>Quadratic Equation and its Solutions

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>Theory of Equation>Quadratic Equation and its Solutions

α \neq β, α^{2} = 5 α - 3, β^{2} = 5 β - 3

,

then the equation having

\frac{α}{β}

and

\frac{β}{α}

as its roots is

EASY

Mathematics>Algebra>Theory of Equation>Quadratic Equation and its Solutions

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>Theory of Equation>Quadratic Equation and its Solutions

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 :

If a > b > c > d and the roots of equation x - a x - c + k x - b x - d = 0 are real, then k can be equal to

Important Questions on Theory of Equation

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If $a > b > c > d$ and the roots of equation $(x - a) (x - c) + k (x - b) (x - d) = 0$ are real, then k can be equal to