x ∈ ℝ : 6 + x - x 2 2 x + 5 ≥ 6 + x - x 2 x + 4 =

( - ∞ , - 4 ] ∪ - 5 2 , - 1

( - ∞ , - 4 ] ∪ [ - 2 , - 1 ]

For 0 ≤ p ≤ 1 and for any positive a , b ; let I ( p ) = ( a + b ) p , J ( p ) = a p + b p , then

I ( p ) < J ( p ) in 0 , p 2 and l ( p ) > J ( p ) in p 2 , ∞

I ( p ) < J ( p ) in p 2 , ∞ and J ( p ) < I ( p ) in 0 , p 2

Let N be the set of positive integers. For all n ∈ N , let f n = n + 1 1 / 3 - n 1 / 3 and A = n ∈ N : f n + 1 < 1 3 n + 1 2 / 3 < f n Then,

the complement of A in N is nonempty, but finite

A and its complement in N are both infinite

HARD

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Find the least integral value of $k$ for which the quadratic polynomial $(k - 2) x^{2} + 8 x + k + 4 > 0 \forall x \in R$ .

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Important Questions on Quadratic Equations

MEDIUM

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

The integer

k,

for which the inequality

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

is valid for every

x

R

is:

EASY

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

The solution of the inequality

|x^{2} - 4 x| < 5

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

β

satisfies the equation

x^{2} - x - 6 > 0

, then a value exists for

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

Find the number of ordered triples

(a, b, c)

of positive integers such that

30 a + 50 b + 70 c \leq 343 .

MEDIUM

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

5, 5 r, 5 r^{2}

are the lengths of the sides of a triangle, then

r

can not be equal to:

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

Determine the sum of all possible positive integers

n,

the product of whose digits equals

n^{2} - 15 n - 27 .

EASY

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

The solution of

\frac{6 x}{4 x - 1} < \frac{1}{2}

EASY

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

n \in N

then the statement

8 n + 16 \leq 2^{n}

is true for

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

\{x \in ℝ : \frac{\sqrt{6 + x - x^{2}}}{2 x + 5} \geq \frac{\sqrt{6 + x - x^{2}}}{x + 4}\} =

EASY

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

For

0 \leq p \leq 1

and for any positive

a, b;

let

I (p) = (a + b)^{p}, J (p) = a^{p} + b^{p},

then

MEDIUM

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

The values of

x

for which

4^{x} + 4^{1 - x} - 5 < 0,

is given by

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

The least positive integer

n

for which

\sqrt[3]{n + 1} - \sqrt[3]{n} < \frac{1}{12}

is-

EASY

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

The cost and revenue functions of a product are given by

c (x) = 20 x + 4000

and

R (x) = 60 x + 2000

respectively where

x

is the number of items produced and sold. The value of

x

to earn profit is

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

Let

a, b, c, d, e

be real numbers such that

a + b < c + d, b + c < d + e, c + d < e + a, d + e < a + b .

Then

EASY

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

The smallest negative integer satisfying both the quadratic inequalities

x^{2} < 4 x + 77

and

x^{2} > 4

EASY

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

x^{2} - 5 x - 14 > 0 \Rightarrow x

lie outside

[α, β],

then

\frac{α}{β} =

MEDIUM

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

f (x) = a x^{2} - b x - a

is a quadratic expression. If

K

is the least real number such that

f (x) \leq K, \forall x \in R,

then

MEDIUM

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

Let

N

be the set of positive integers. For all

n \in N,

let

f_{n} = {(n + 1)}^{1 / 3} - n^{1 / 3}

and

A = \{n \in N : f_{n + 1} < \frac{1}{3 {(n + 1)}^{2 / 3}} < f_{n}\}

Then,

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

\frac{6 x^{2} - 5 x - 3}{x^{2} - 2 x + 6} \leq 4,

then the least and the highest values of

4 x^{2}

are

HARD

Mathematics>Algebra>Quadratic Equations>Graph of Quadratic Expression

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

Find the least integral value of k for which the quadratic polynomial k-2x2+8x+k+4>0 ∀x∈R.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Find the least integral value of $k$ for which the quadratic polynomial $(k - 2) x^{2} + 8 x + k + 4 > 0 \forall x \in R$ .