Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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If $a > 0$ and $b^{2} - 4 a c < 0,$ then the graph of $y = a x^{2} + b x + c$

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Important Questions on Theory of Equation

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Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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:

HARD

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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

MEDIUM

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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

MEDIUM

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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

EASY

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

Let

f : [2, \infty) \to R

be the function defined by

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

, then the range of

f

is

EASY

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

Find the maximum of the expression

2 x + 5 - 3 x^{2}

as

x

varies over

R

.

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Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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

EASY

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

If

f : R - \{2\} \to R

is a function defined by

f (x) = \frac{x^{2} - 4}{x - 2},

then range is

HARD

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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>Range and Graph of Quadratic Expressions

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

EASY

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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} =

HARD

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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>Range and Graph of Quadratic Expressions

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>Range and Graph of Quadratic Expressions

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>Range and Graph of Quadratic Expressions

Let

m

and

n

be the numbers of real roots of the quadratic equations

x^{2} - 12 x + [x] + 31 = 0

and

x^{2} - 5 | x + 2 | - 4 = 0

respectively, where

[x]

denotes the greatest integer

\leq x

. Then

m^{2} + m n + n^{2}

is equal to

MEDIUM

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

The integer

k,

for which the inequality

x^{2} - 2 (3 k - 1) x + 8 k^{2} - 7 > 0

is valid for every

x

in

R

is:

HARD

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

The number of integral values of

m

for which the quadratic expression

(1 + 2 m) x^{2} - 2 (1 + 3 m) x + 4 (1 + m), x \in R

is always positive, is

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Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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

MEDIUM

Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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

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Mathematics>Algebra>Theory of Equation>Range and Graph of Quadratic Expressions

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?