Statement 1: The probability that y = 16 x 2 + 8 ( a + 5 ) x - 7 a - 5 defined in - 26 , 0 is strictly above the x -axis is 1 2 . Statement 2: If the graph of y = a x 2 + b x + c is strictly above the x -axis, then discriminant is negative and a > 0 .

Statement 1 is true, statement 2 is true, statement 2 is the correct explanation for statement 1

Statement 1 is true, statement 2 is true, statement 2 is not the correct explanation for statement 1

Statement 1 is true, statement 2 is false.

Statement 1 is false, statement 2 is true.

Statement 1: The probability that y = 16 x 2 + 8 ( a + 5 ) x - 7 a - 5 defined in - 26 , 0 is strictly above the x -axis is 1 2 . Statement 2: If the graph of y = a x 2 + b x + c is strictly above the x -axis, then discriminant is negative and a > 0 .

Statement 1 is true, statement 2 is true, statement 2 is the correct explanation for statement 1

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

Earn 100

Statement 1: The probability that $y = 16 x^{2} + 8 (a + 5) x - 7 a - 5$ defined in $[- 26, 0]$ is strictly above the $x$ -axis is $\frac{1}{2}$ .

Statement 2: If the graph of $y = a x^{2} + b x + c$ is strictly above the $x$ -axis, then discriminant is negative and $a > 0$ .

50% studentsanswered this correctly

Important Questions on Algebraic Inequalities

HARD

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

β

satisfies the equation

x^{2} - x - 6 > 0

, then a value exists for

MEDIUM

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

5, 5 r, 5 r^{2}

are the lengths of the sides of a triangle, then

r

can not be equal to:

EASY

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

The solution of

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

HARD

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

Find the number of ordered triples

(a, b, c)

of positive integers such that

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

HARD

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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

HARD

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

Determine the sum of all possible positive integers

n,

the product of whose digits equals

n^{2} - 15 n - 27 .

EASY

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

The solution of the inequality

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

EASY

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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

lie outside

[α, β],

then

\frac{α}{β} =

HARD

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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

EASY

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

The values of

x

for which

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

is given by

HARD

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

The least positive integer

n

for which

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

is-

EASY

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

The smallest negative integer satisfying both the quadratic inequalities

x^{2} < 4 x + 77

and

x^{2} > 4

MEDIUM

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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:

MEDIUM

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

x^{2} + 6 x - 27 > 0

and

x^{2} - 3 x - 4 < 0

, then

EASY

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

x \in I

(set of all integers) such that

x^{2} - 3 x < 4

, then the number of possible values of

x

MEDIUM

Mathematics>Algebra>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

\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>Algebraic Inequalities>Idea of Intervals and Signs of Inequations

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

Statement 1: The probability that y=16x2+8(a+5)x-7a-5 defined in -26, 0 is strictly above the x-axis is 12. Statement 2: If the graph of y=ax2+bx+c is strictly above the x-axis, then discriminant is negative and a>0.

Important Questions on Algebraic Inequalities

Learn and Prepare for any exam you want !

Statement 1: The probability that $y = 16 x^{2} + 8 (a + 5) x - 7 a - 5$ defined in $[- 26, 0]$ is strictly above the $x$ -axis is $\frac{1}{2}$ .

Statement 2: If the graph of $y = a x^{2} + b x + c$ is strictly above the $x$ -axis, then discriminant is negative and $a > 0$ .