Mathematics>Algebra>Theory of Equation>Nature of Roots

MEDIUM

Earn 100

If $x$ is real, then the maximum and minimum values of expression $\frac{x^{2} + 14 x + 9}{x^{2} + 2 x + 3}$ respectively will be-

50% studentsanswered this correctly

Important Questions on Theory of Equation

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

If

y = \frac{x^{2} + 14 x + 9}{x^{2} + 2 x + 3} \forall x \in ℝ,

then the interval of maximum length in which

y

lies is

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

The expression

a x^{2} + b x + c, (a, b

and

c

are real

)

has the same sign as that of

a

for all

x

if

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

f (x) = (1 + b^{2}) x^{2} + 2 b x + 1

and

m (b)

be the minimum value of

f (x)

. As

b

varies, the range of

m (b)

is

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

If $x \in R$ then determine the range of the expression $\frac{x^{2} + x + 1}{x^{2} - x + 1}$ .

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

Assertion $(A) : 3 x^{2} - 16 x + 4 > - 16$ is satisfied for some values of real $x$ in $(0, \frac{10}{3})$ .

Reason $(R) : a x^{2} + b x + c$ and $a$ will have the same sign for some values of $x \in R$ when $b^{2} - 4 a c > 0$ .

The correct option among the following is

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

Find the maximum of the expression

2 x + 5 - 3 x^{2}

as

x

varies over

R

.

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

The maximum value of

z

in the following equation

z = 6 x y + y^{2},

where

3 x + 4 y \leq 100

and

4 x + 3 y \leq 75

for

x \geq 0

and

y \geq 0

is

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

Consider the quadratic equation

n x^{2} + 7 \sqrt{n} x + n = 0

, where

n

is a positive integer. Which of the following statements are necessarily correct?

I. For any

n

, the roots are distinct.

II. There are infinitely many values of

n

for which both roots are real.

III. The product of the roots is necessarily an integer.

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

f : [2, \infty) \to R

be the function defined by

f (x) = x^{2} - 4 x + 5

, then the range of

f

is

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

f (x) = (x - a) (x - b) - (\frac{a + b}{2}) .

If

f (x) = 0

has both non-negative roots, then the minimum value of

f (x)

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

If

l, m (l < m)

are roots of

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

, then

\lim_{x \to a} \frac{|a x^{2} + b x + c|}{a x^{2} + b x + c} =

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

Suppose a parabola

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

has two

x

intercepts, one positive and negative, and its vertex is

(2, - 2)

. Then which of the following is true?

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

If

λ \in R

is such that the sum of the cubes of the roots of the equation

x^{2} + (2 - λ) x + (10 - λ) = 0

is minimum, then the magnitude of the difference of the roots of this equation is :

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

f (x)

be a quadratic polynomial with

f (2) = 10

and

f (- 2) = - 2

. Then the coefficient of

x

in

f (x)

is

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

The number of all possible positive integral value of

α

for which the roots of the quadratic equation

6 x^{2} - 11 x + α = 0

are rational numbers is:

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

The quadratic equation

P (x) = 0

with real coefficients has purely imaginary roots. Then the equation

P (P (x)) = 0

has

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

If equations

a x^{2} + b x + c = 0, (a, b, c \in R, a \neq 0)

and

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

have a common root, then

a : b : c

equals :

EASY

Mathematics>Algebra>Theory of Equation>Nature of Roots

Let

a, b, c

be real numbers such that

a + b + c < 0

and the quadratic equation

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

has imaginary roots. Then

HARD

Mathematics>Algebra>Theory of Equation>Nature of Roots

If

x, y, z \in R, x + y + z = 5, x^{2} + y^{2} + z^{2} = 9

, then length of interval in which

x

lies is

MEDIUM

Mathematics>Algebra>Theory of Equation>Nature of Roots

The value of

λ

such that sum of the squares of the roots of the quadratic equation,

x^{2} + (3 - λ) x + 2 = λ

has the least value is: