Let f ( x ) = x 4 - 4 x 3 + 4 x 2 + c , c ∈ R . Then,

f ( x ) has exactly one zero in ( 1 , 2 ) if - 1 < c < 0

f ( x ) has infinitely many zeros in ( 1 , 2 ) for all c

f ( x ) has double zeros in ( 1 , 2 ) if - 1 < c < 0

whatever be the value of c , f ( x ) has no zero in ( 1 , 2 )

Let a ∈ R and let f : R → R be given by f x = x 5 - 5 x + a , then

f x has three real roots if - 4 < a < 4

f x has three real roots if a > 4

f x has three real roots if a < - 4

HARD

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Consider the equation $125^{x} + 45^{x} = (2) 27^{x}$

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Important Questions on Theory of Equations

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

EASY

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

2 + \sqrt{5}, 1

are roots of the cubic equation given by

MEDIUM

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

Let

a, b \in R

be such that the equation

a x^{2} - 2 b x + 15 = 0

has repeated root

α

and if

α

and

β

are the roots of the equation

x^{2} - 2 b x + 21 = 0

, then

α^{2} + β^{2}

is equal to:

MEDIUM

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

Let $R$ be the set of all real numbers and let $f$ be a function $R$ to $R$ such that $f (x) + (x + \frac{1}{2}) f (1 - x) = 1$ , for all $x \in R$ . Then $2 f (0) + 3 f (1)$ is equal to.

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) ?

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

The sum of all non-integer roots of the equation

x^{5} - 6 x^{4} + 11 x^{3} - 5 x^{2} - 3 x + 2 = 0

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

EASY

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

Let

f (x) = x^{13} + x^{11} + x^{9} + x^{7} + x^{5} + x^{3} + x + 12

. Then

MEDIUM

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

Let

f (x) = x^{4} - 4 x^{3} + 4 x^{2} + c, c \in R .

Then,

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

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

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 -

MEDIUM

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

The sum of all real values of

x

for which

\frac{3 x^{2} - 9 x + 17}{x^{2} + 3 x + 10} = \frac{5 x^{2} - 7 x + 19}{3 x^{2} + 5 x + 12}

is equal to

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

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 figure shows a portion of the graph $y = 2 x - 4 x^{3} .$ The line $y = c$ is such that the areas of the regions marked $I$ and $II$ are equal. If $a, b$ are the $x$ -coordinates of $A, B$ respectively, then $a + b$ equals-

Question Image

MEDIUM

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

The equation

x^{{(\log_{3} x)}^{2} - \frac{9}{2} \log_{3} x + 5} = 3 \sqrt{3}

has

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

Let

a \in R

and let

f : R \to R

be given by

f (x) = x^{5} - 5 x + a,

then

Consider the equation 125x+45x=(2)27x

Important Questions on Theory of Equations

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Consider the equation $125^{x} + 45^{x} = (2) 27^{x}$