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How many roots of the equation $x^{3} - 3 x^{2} + 4 = 0$ are also roots of the equation $x^{2} + x - 6 = 0 ?$

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Important Questions on Theory of Equations

MEDIUM

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

The number of real roots of the equation

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

is :

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

Let

n \geq 3

be an integer. For a permutation

σ = (a_{1}, a_{2}, \dots ., a_{n})

(1, 2, \dots ., n)

we let

f_{σ} (x) = a_{n} x^{n - 1} + a_{n - 1} x^{n - 2} + \dots + a_{2} x + a_{1}

. Let

S_{σ}

be the sum of the roots of

f_{σ} (x) = 0

and let

S

denotes the sum over all permutations

σ

(1, 2, \dots \dots, n)

of the number

S_{σ} .

Then-

EASY

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

x

is a solution of the equation

\sqrt{2 x + 1} - \sqrt{2 x - 1} = 1, (x \geq \frac{1}{2})

, then

\sqrt{4 x^{2} - 1}

is equal to :

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

2 + 3 i

is one of the roots of the equation

2 x^{3} - 9 x^{2} + k x - 13 = 0, k \in R,

then the real root of this equation (where

i^{2} = - 1

) :

MEDIUM

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

Let

a, b, c, d

be real numbers between

- 5

and

5

such that

|a| = \sqrt{4 - \sqrt{5 - a}}, |b| = \sqrt{4 + \sqrt{5 - b}}, |c| = \sqrt{4 - \sqrt{5 + c}}, |d| = \sqrt{4 + \sqrt{5 + d}}

, Then the product

a b c d

MEDIUM

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

The number of real roots of the equation

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

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

Let

S = \{x \in R : x \geq 0 & 2 |\sqrt{x} - 3| + \sqrt{x} (\sqrt{x} - 6) + 6 = 0\}

. Then

S

MEDIUM

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

Which of the following is not a root of

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

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

The real number

k

for which the equation,

2 x^{3} + 3 x + k = 0

has two distinct real roots in

[0, 1]

belongs to

MEDIUM

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

For a real number

x

, let

[x]

denote the largest integer less than or equal to

x

, and let

\{x\} = x - [x]

. The number of solutions

x

to be equation

[x] \{x\} = 5

with

0 \leq x \leq 2015

MEDIUM

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

The set of all positive real values of

k,

for which the equation

x^{3} - 9 x^{2} + 24 x - k = 0

has three distinct real roots is the interval

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

Suppose

a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}

are integers such that

\frac{5}{7} = \frac{a_{2}}{2!} + \frac{a_{3}}{3!} + \frac{a_{4}}{4!} + \frac{a_{5}}{5!} + \frac{a_{6}}{6!} + \frac{a_{7}}{7!}

, where

0 \leq a_{j} < j

for

j = 2, 3, 4, 5, 6 7

. The sum

a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

In the real number system, the equation

\sqrt{x + 3 - 4 \sqrt{x - 1}} + \sqrt{x + 8 - 6 \sqrt{x - 1}} = 1

has

EASY

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

Let

S

be the set of all real roots of the equation

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

, then

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

If the bi-quadratic equation

f (x) = x^{4} + 2 x^{3} - 16 x^{2} - 22 x + 7 = 0

has

2 + \sqrt{3}

as one of its roots, then which of the following is not a root of

f (x) ?

MEDIUM

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

α, β, γ

are the roots of the equation

x^{3} + a x^{2} - b x + c = 0

, then

\sum β^{2} (γ + α) =

EASY

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

α, β, γ

are the roots of

x^{3} - 2 x^{2} + 3 x - 4 = 0

, then find

\sum α β (α + β)

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

The number of integers

n

for which

3 x^{3} - 25 x + n = 0

has three real roots is-

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

The number of integers

a

in the interval

[1, 2014]

for which the system of equations

x + y = a, \frac{x^{2}}{x - 1} + \frac{y^{2}}{y - 1} = 4

has finitely many solutions is -

HARD

Mathematics>Algebra>Theory of Equations>Polynomial Equation and Identity

The polynomial equation

x^{3} - 3 a x^{2} + (27 a^{2} + 9) x + 2016 = 0

has

How many roots of the equation x 3 - 3 x 2 + 4 = 0 are also roots of the equation x 2 + x - 6 = 0 ?

Important Questions on Theory of Equations

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How many roots of the equation $x^{3} - 3 x^{2} + 4 = 0$ are also roots of the equation $x^{2} + x - 6 = 0 ?$