If p > 1 , then prove that the expression x 2 - 2 x + p 2 x 2 + 2 x + p 2 lies between p - 1 p + 1 and p + 1 p - 1 for all real values of x .

Let y = x 2 - 2 x + p 2 x 2 + 2 x + p 2 ⇒ y x 2 + 2 y x + y p 2 - x 2 + 2 x - p 2 = 0 ⇒ ( y - 1 ) x 2 + 2 ( y + 1 ) x + ( y - 1 ) p 2 = 0 ∵ x ∈ R ∴ D ≥ 0 f o r a b o v e e q u a t i o n . ∴ 4 ( y + 1 ) 2 - 4 ( y - 1 ) 2 p 2 ≥ 0 ⇒ ( y + 1 ) 2 - ( y - 1 ) 2 p 2 ≥ 0 ⇒ [ y + 1 + ( y − 1 ) p ] [ y + 1 − ( y − 1 ) p ] ≥ 0 ⇒ f ( y ) = [ y ( p + 1 ) − ( p − 1 ) ] [ y ( 1 − p ) + p + 1 ] ≥ 0 ⇒ f ( y ) = [ y - ( p - 1 ) ( p + 1 ) ] [ y - ( p + 1 ) ( p - 1 ) ] ∵ p > 1 ∴ f ( y ) i s d e c r e a sin g . Therefore, the function is only positive between its zeroes. ∴ y ∈ ( p + 1 p - 1 , p - 1 p + 1 )

If p > 1 , then prove that the expression x 2 - 2 x + p 2 x 2 + 2 x + p 2 lies between p - 1 p + 1 and p + 1 p - 1 for all real values of x .

Let y = x 2 - 2 x + p 2 x 2 + 2 x + p 2 ⇒ y x 2 + 2 y x + y p 2 - x 2 + 2 x - p 2 = 0 ⇒ ( y - 1 ) x 2 + 2 ( y + 1 ) x + ( y - 1 ) p 2 = 0 ∵ x ∈ R ∴ D ≥ 0 f o r a b o v e e q u a t i o n . ∴ 4 ( y + 1 ) 2 - 4 ( y - 1 ) 2 p 2 ≥ 0 ⇒ ( y + 1 ) 2 - ( y - 1 ) 2 p 2 ≥ 0 ⇒ [ y + 1 + ( y − 1 ) p ] [ y + 1 − ( y − 1 ) p ] ≥ 0 ⇒ f ( y ) = [ y ( p + 1 ) − ( p − 1 ) ] [ y ( 1 − p ) + p + 1 ] ≥ 0 ⇒ f ( y ) = [ y - ( p - 1 ) ( p + 1 ) ] [ y - ( p + 1 ) ( p - 1 ) ] ∵ p > 1 ∴ f ( y ) i s d e c r e a sin g . Therefore, the function is only positive between its zeroes. ∴ y ∈ ( p + 1 p - 1 , p - 1 p + 1 )

Find the maximum of the expression 2 x + 5 - 3 x 2 as x varies over R .

Let f x = 2 x + 5 - 3 x 2 For maxima or minima f ' x = 0 ⇒ 2 - 6 x = 0 ⇒ x = 1 3 Here, f ' ' x = - 6 < 0 for x = 1 3 Therefore, maximum of the expression occurs at x = 1 3 So, f 1 3 = 2 3 + 5 - 3 1 9 = 16 3 Which is the maximum of the expression

If x ∈ R then determine the range of the expression x 2 + x + 1 x 2 - x + 1 .

Let y = x 2 + x + 1 x 2 - x + 1 ⇒ y x 2 - y x + y = x 2 + x + 1 ⇒ x 2 y - 1 + x ( - y - 1 ) + y - 1 = 0 On comparing with a x 2 + b x + c = 0 , we have: a = y - 1 ; b = - y - 1 ; c = y - 1 As x ∈ R , we have Discriminant, D ≥ 0 i.e., b 2 - 4 a c ≥ 0 ⇒ ( - y - 1 ) 2 - 4 ( y - 1 ) ( y - 1 ) ≥ 0 ⇒ y 2 + 2 y + 1 - 4 y 2 - 2 y + 1 ≥ 0 ⇒ - 3 y 2 + 10 y - 3 ≥ 0 ⇒ 3 y 2 - 10 y + 3 ≤ 0 ⇒ 3 y 2 - 9 y - y + 3 ≤ 0 ⇒ 3 y ( y - 3 ) - 1 ( y - 3 ) ≤ 0 ⇒ ( 3 y - 1 ) ( y - 3 ) ≤ 0 ⇒ 1 3 ≤ y ≤ 3 Thus, the range is y ∈ 1 3 , 3 .

EASY

Earn 100

If $p > 1$ , then prove that the expression $\frac{x^{2} - 2 x + p^{2}}{x^{2} + 2 x + p^{2}}$ lies between $\frac{p - 1}{p + 1}$ and $\frac{p + 1}{p - 1}$ for all real values of $x$ .

Important Questions on Theory of Equation

MEDIUM

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

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>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

EASY

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)

EASY

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

Find the maximum of the expression

2 x + 5 - 3 x^{2}

x

varies over

R

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

MEDIUM

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

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

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

HARD

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

Let

α

and

β

be the roots of equation

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

. If

a_{n} = α^{n} - β^{n}, \forall n \geq 1,

then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

is equal to

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

HARD

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

Let

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

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

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>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?

MEDIUM

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

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

f (x)

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

For how many different values of $a$ does the following system have at least two distinct solutions?

$a x + y = 0$

$x + (a + 10) y = 0$

HARD

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

Let

- \frac{π}{6} < θ < - \frac{π}{12}

. Suppose

α_{1}

and

β_{1}

are the roots of the equation

x^{2} - 2 x \sec θ + 1 = 0

and

α_{2}

and

β_{2}

are the roots of the equation

x^{2} + 2 x \tan θ - 1 = 0 .

α_{1} > β_{1}

and

α_{2} > β_{2}

, then

α_{1} + β_{2}

equals

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

HARD

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

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

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:

If p>1, then prove that the expression x2-2x+p2x2+2x+p2 lies between p-1p+1 and p+1p-1 for all real values of x.

Important Questions on Theory of Equation

Learn and Prepare for any exam you want !

If $p > 1$ , then prove that the expression $\frac{x^{2} - 2 x + p^{2}}{x^{2} + 2 x + p^{2}}$ lies between $\frac{p - 1}{p + 1}$ and $\frac{p + 1}{p - 1}$ for all real values of $x$ .