Let f x = x 3 + x , then the equation 2 y - f ( 2 ) + 3 y - f ( 3 ) + 4 y - f ( 4 ) = 0 , has

exactly one root lying in ( f ( 3 ) , f ( 4 ) )

both roots lying in ( f ( 2 ) , f ( 3 ) )

exactly one root lying in ( - ∞ , f ( 2 ) )

exactly one root lying in ( f ( 4 ) , ∞ )

Consider the function f ( x ) = x 3 4 - sin π x + 3

f ( x ) takes on the value 3 1 4 in the interval [ - 2 , 2 ]

f ( x ) does not attain value within the interval [ - 2 , 2 ]

f ( x ) takes no value p , 1 < p < 5 in the interval [ - 2 , 2 ]

MEDIUM

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The smallest value of $k,$ for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least $4$ , is ........

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Important Questions on Theory of Equation

EASY

Mathematics>Algebra>Theory of Equation>Location of Roots

The number of integral values of

k

, for which one root of the equation

2 x^{2} - 8 x + k = 0

lies in the interval

(1, 2)

and its other root lies in the interval

(2, 3)

, is :

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

Let

f (x) = x^{3} + x,

then the equation

\frac{2}{y - f (2)} + \frac{3}{y - f (3)} + \frac{4}{y - f (4)} = 0,

has

EASY

Mathematics>Algebra>Theory of Equation>Location of Roots

In which interval does the root of equation

x^{3} - x - 2 = 0

lie?

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

Let

a, b, c, d

be distinct real numbers such that

a, b

are roots of

x^{2} - 5 c x - 6 d = 0,

and

c, d

are roots of

x^{2} - 5 a x - 6 b = 0 .

Then

b + d

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

Consider the quadratic equation

(c - 5) x^{2} - 2 c x + (c - 4) = 0, c \neq 5 .

Let

S

be the set of all integral values of

c

for which one root of the equation lies in the interval

(0, 2)

and its other root lies in the interval

(2, 3) .

Then the number of elements in

S

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

Consider the function

f (x) = \frac{x^{3}}{4} - \sin π x + 3

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

If both roots of the equation

x^{2} - 2 (a - 1) x + (2 a + 1) = 0

aré positive, where

a

is a real number, then

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

Let $p (x) = x^{2} + a x + b$ have two distinct real roots, where $a, b$ are real number. Define $g (x) = p (x^{3})$ for all real number $x$

Then, which of the following statements are true?
I. $g$ has exactly two distinct real roots.
II. $g$ can have more than two distinct real roots.
III. There exists a real number $α$ such that $g (x) \geq α$ for all real $x$

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

Suppose a parabola

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

has two

x

intercepts, one positive and one negative, and its vertex is

(2, - 2)

, then which of the following is true?

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

If the roots of the equation

x^{2} + x + a = 0

exceed

a

, then

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

Let

f

and

g

be differentiable on the interval

I

and let

a, b \in I, a < b .

Then

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

Suppose

a

is a positive real number such that

a^{5} - a^{3} + a = 2

. Then

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

The set of all real values of

λ

for which the quadratic equation

(λ^{2} + 1) x^{2} - 4 λ x + 2 = 0

always have exactly one root in the interval

(0, 1)

is :

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

If both the roots of the quadratic equation

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

are real and distinct and they lie in the interval

(1, 5),

then

m

lies in the interval:

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

If exactly one root of the cubic equation 4x³ - 3x + r = 0 will lie in the interval (-1, 0), then the value of of r cannot be equal to

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

2 a + 3 b + 6 c = 0

, then at least one root of the equation

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

lies in the interval

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

If both the roots of the quadratic equation

x^{2} - 2 k x + k^{2} + k - 5 = 0

are less than

5

, then

k

lies in the interval

EASY

Mathematics>Algebra>Theory of Equation>Location of Roots

The values of

a

for which the roots of the equation

x^{2} + x + a = 0

are real and exceed

a

are -

HARD

Mathematics>Algebra>Theory of Equation>Location of Roots

All the values of

m

for which both roots of the equation

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

are greater than

- 2

but less than

4

lie in the interval

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

e_{1}

and

e_{2}

are the roots of the equation

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

, where

e_{1}, e_{2}

are the eccentricities of an ellipse and a hyperbola, respectively, then the value of

a

belongs to

The smallest value of k, for which both the roots of the equation x2-8kx+16k2-k+1=0 are real, distinct and have values at least 4, is ........

Important Questions on Theory of Equation

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The smallest value of $k,$ for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least $4$ , is ........