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 ) , ∞ )

Which of the following is divisible by x 2 - y 2 , ∀ x ≠ y ?

x n - y n x 2 n + 1 + y 2 n + 1 ∀ n ∈ N

x n - y n x m + y m ∀ m , n ∈ N

MEDIUM

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For how many integral values of $k$ , the equation $x^{2} - 4 x + k = 0$ , where $k$ is an integer has real roots and both of them lie in the interval $(0, 5)$ ?

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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

Which of the following is divisible by

x^{2} - y^{2}, \forall x \neq y ?

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

The number of real roots of the equation,

e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0

is:

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

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

Let $f (x) = x^{6} - 2 x^{5} + x^{3} + x^{2} - x - 1$ and $g (x) = x^{4} - x^{3} - x^{2} - 1$ be two polynomials. Let $a, b, c$ and $d$ be the roots of $g (x) = 0$ . Then, the value of

$f (a) + f (b) + f (c) + f (d)$ is

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

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

The number of cubic polynomials

P (x)

satisfying

P (1) = 2, P (2) = 4, P (3) = 6, P (4) = 8

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 number of integers

x

satisfying

- 3 x^{4} + det [\begin{matrix} 1 & x & x^{2} \\ 1 & x^{2} & x^{4} \\ 1 & x^{3} & x^{6} \end{matrix}] = 0

is equal to

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

Let

p (x)

be a quadratic polynomial such that

p (0) = 1 .

p (x)

leaves remainder

4

when divided by

x - 1

and it leaves remainder

6

when divided by

x + 1

then:

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

Let

P

be a non-zero polynomial such that

P (1 + x) = P (1 - x)

for all real

x

and

P (1) = 0

. Let

m

be the largest integer such that

{(x - 1)}^{m}

divides

P (x)

for all such

P (x)

. Then,

m

equals

MEDIUM

Mathematics>Algebra>Theory of Equation>Location of Roots

The number of ordered pairs

(x, y)

of integers satisfying

x^{3} + y^{3} = 65

EASY

Mathematics>Algebra>Theory of Equation>Location of Roots

x + 2

is a factor of

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

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:

For how many integral values of k, the equation x2-4x+k=0, where k is an integer has real roots and both of them lie in the interval 0, 5?

Important Questions on Theory of Equation

Learn and Prepare for any exam you want !

For how many integral values of $k$ , the equation $x^{2} - 4 x + k = 0$ , where $k$ is an integer has real roots and both of them lie in the interval $(0, 5)$ ?