Let, f ( x ) = a x 3 + b x 2 + c x + 1 have extrema at x = α , β ( β > α ) , such that α β < 0 and f ( α ) · f ( β ) < 0 , then equation f ( x ) = 0 , has

One negative root if f ( α ) > 0 and f ( β ) < 0

Let, f ( x ) = a x 3 + b x 2 + c x + 1 have extrema at x = α , β ( β > α ) , such that α β < 0 and f ( α ) · f ( β ) < 0 , then equation f ( x ) = 0 , has

One negative root if f ( α ) > 0 and f ( β ) < 0

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Let, $f (x) = a x^{3} + b x^{2} + c x + 1$ have extrema at $x = α, β (β > α),$ such that $α β < 0$ and $f (α) \cdot f (β) < 0,$ then equation $f (x) = 0,$ has

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Important Questions on Theory of Equation

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Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

In the real number system, the equation

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

has

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of distinct solutions of the equation,

\log_{\frac{1}{2}} |\sin x| = 2 - \log_{\frac{1}{2}} |\cos x|

in the interval

[0, 2 π],

is ________

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

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

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The curves

x = \log (y + e)

and

y = \log (\frac{1}{x})

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

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 Equation>Algebraic Equations of Higher Degree

The number of pairs

(a, b)

of positive real numbers satisfying

a^{4} + b^{4} < 1

and

a^{2} + b^{2} > 1

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of real roots of the equation

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

is :

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of ordered pairs

(x, y)

of positive integers satisfying

2^{x} + 3^{y} = 5^{x y}

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of integers

n

for which

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

has three real roots is-

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Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Let

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

. Then

S

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of solutions of

\sqrt{4 - x} + \sqrt{x + 9} = 5

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

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.

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

e^{x} = y + \sqrt{1 + y^{2}}

then the value of

y

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

One of the real roots of the equation

x^{3} - 6 x^{2} + 6 x - 2 = 0

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Value of

a

for which equations

x^{3} + a x + 1 = 0

and

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

have a common root

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of roots of the equation,

{(81)}^{\sin^{2} x} + {(81)}^{\cos^{2} x} = 30

in the interval

[0, π]

is equal to :

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of elements in the set

\{x \in R : (| x | - 3) |x + 4| = 6\}

is equal to

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

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 Equation>Algebraic Equations of Higher Degree

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 Equation>Algebraic Equations of Higher Degree

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}

Let, f(x)=ax3+bx2+cx+1 have extrema at x=α, β (β>α), such that αβ<0 and f(α)·f(β)<0, then equation f(x)=0, has

Important Questions on Theory of Equation

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Let, $f (x) = a x^{3} + b x^{2} + c x + 1$ have extrema at $x = α, β (β > α),$ such that $α β < 0$ and $f (α) \cdot f (β) < 0,$ then equation $f (x) = 0,$ has