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${(x - \frac{1}{x})}^{2} + p |x + \frac{1}{x}| + 29 = 0, p \in R$ exactly four distinct real solutions, then the true set of values of $p$ is

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Important Questions on Quadratic Equations

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

The number of real roots of the equation

5 + |2^{x} - 1| = 2^{x} (2^{x} - 2)

is :

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Directions: Each of the following sentences in this section has a blank space and four words or group of words given after the sentence. Select the word or group of words you consider most appropriate for the blank space and indicate your response on the Answer Sheet accordingly.

Every rash driver becomes a ____ killer.

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Let

S = \{x \in R : x \geq 0 & 2 |\sqrt{x} - 3| + \sqrt{x} (\sqrt{x} - 6) + 6 = 0\}

. Then

S

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

The set of all positive real values of

k,

for which the equation

x^{3} - 9 x^{2} + 24 x - k = 0

has three distinct real roots is the interval

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Let

a, b, c, d

be real numbers between

- 5

and

5

such that

|a| = \sqrt{4 - \sqrt{5 - a}}, |b| = \sqrt{4 + \sqrt{5 - b}}, |c| = \sqrt{4 - \sqrt{5 + c}}, |d| = \sqrt{4 + \sqrt{5 + d}}

, Then the product

a b c d

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

The number of real roots of the equation

{(x + \frac{1}{x})}^{3} + x + \frac{1}{x} = 0

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Let

n \geq 3

be an integer. For a permutation

σ = (a_{1}, a_{2}, \dots ., a_{n})

(1, 2, \dots ., n)

we let

f_{σ} (x) = a_{n} x^{n - 1} + a_{n - 1} x^{n - 2} + \dots + a_{2} x + a_{1}

. Let

S_{σ}

be the sum of the roots of

f_{σ} (x) = 0

and let

S

denotes the sum over all permutations

σ

(1, 2, \dots \dots, n)

of the number

S_{σ} .

Then-

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Which of the following is not a root of

f (x) = x^{3} - 11 x^{2} + 36 x - 36 ?

EASY

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Two sentences are given below and you are required to find the correct sentence which combines both the sentences.

Which is the correct combination of the given two sentences?

He is too tired. He could not stand.

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

For a real number

x

, let

[x]

denote the largest integer less than or equal to

x

, and let

\{x\} = x - [x]

. The number of solutions

x

to be equation

[x] \{x\} = 5

with

0 \leq x \leq 2015

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

In the following sentence, a blank space with four options is given. Select whichever preposition or determiner you consider the most appropriate for the blank space.

These are the good rules to live _____.

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Suppose

a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}

are integers such that

\frac{5}{7} = \frac{a_{2}}{2!} + \frac{a_{3}}{3!} + \frac{a_{4}}{4!} + \frac{a_{5}}{5!} + \frac{a_{6}}{6!} + \frac{a_{7}}{7!}

, where

0 \leq a_{j} < j

for

j = 2, 3, 4, 5, 6 7

. The sum

a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

In the real number system, the equation

\sqrt{x + 3 - 4 \sqrt{x - 1}} + \sqrt{x + 8 - 6 \sqrt{x - 1}} = 1

has

EASY

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

Let

S

be the set of all real roots of the equation

3^{x} (3^{x} - 1) + 2 = |3^{x} - 1| + |3^{x} - 2|

, then

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

If the bi-quadratic equation

f (x) = x^{4} + 2 x^{3} - 16 x^{2} - 22 x + 7 = 0

has

2 + \sqrt{3}

as one of its roots, then which of the following is not a root of

f (x) ?

MEDIUM

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

α, β, γ

are the roots of the equation

x^{3} + a x^{2} - b x + c = 0

, then

\sum β^{2} (γ + α) =

EASY

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

α, β, γ

are the roots of

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

, then find

\sum α β (α + β)

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

The number of integers

n

for which

3 x^{3} - 25 x + n = 0

has three real roots is-

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

The number of integers

a

in the interval

[1, 2014]

for which the system of equations

x + y = a, \frac{x^{2}}{x - 1} + \frac{y^{2}}{y - 1} = 4

has finitely many solutions is -

HARD

Mathematics>Algebra>Quadratic Equations>Algebraic Equations of Higher Degree

The polynomial equation

x^{3} - 3 a x^{2} + (27 a^{2} + 9) x + 2016 = 0

has

x-1x2+px+1x+29=0, p∈R exactly four distinct real solutions, then the true set of values of p is

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

${(x - \frac{1}{x})}^{2} + p |x + \frac{1}{x}| + 29 = 0, p \in R$ exactly four distinct real solutions, then the true set of values of $p$ is