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Let $[x]$ be the greatest integer less than or equal to $x$ , for a real number $x$ . Then the equation $[x^{2}] = x + 1$ has

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Important Questions on Quadratic Expressions

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let

r

be a root of the equation

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

. The value of

(r + 2) (r + 3) (r + 4) (r + 5)

is equal to -

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let

p, q

and

r

be real numbers

(p \neq q, r \neq 0)

, such that the roots of the equation

\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}

are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^{2} + (a + b) x + b = 0$ is necessarily true ?

$(I)$ It has at least one negative root.

$(II)$ It has at least one positive root.

$(III)$ Both its roots are real.

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let

f (x) = p x^{2} + q x + r

,

where

p, q, r

are constants and

p \neq 0 .

f (5) = - 3 f (2)

and

f (- 4) = 0

,

then the other root of

f

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

λ

be the ratio of the roots of the quadratic equation in

x, 3 m^{2} x^{2} + m (m - 4) x + 2 = 0

, then the least value of

m

for which

λ + \frac{1}{λ} = 1

, is :

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

\sin α

and

\cos α

are the roots of the equation

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

,

then

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Ten ants are on the real line. At time

t = 0,

the

k

^$th$ ant starts at the point

k^{2}

and travelling at uniform speed, reaches the point

{(11 - k)}^{2}

at time

t = 1 .

The number of distinct times at which at least two ants are at the same location is

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The number of ordered pairs

(x, y)

of real numbers that satisfy the simultaneous equations

x + y^{2} = x^{2} + y = 12

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

If the two roots of the equation,

(a - 1) (x^{4} + x^{2} + 1) + (a + 1) {(x^{2} + x + 1)}^{2} = 0

are real and distinct, then the set of all values of

a

is equal to

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let

α and β

be the roots of equation

p x^{2} + q x + r = 0

p \neq 0

. If

p, q, r

are in A.P. and

\frac{1}{α} + \frac{1}{β} = 4

, then the value of

|α - β|

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

α \neq β, α^{2} = 5 α - 3, β^{2} = 5 β - 3

,

then the equation having

\frac{α}{β}

and

\frac{β}{α}

as its roots is

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Suppose the quadratic polynomial

P (x) = a x^{2} + b x + c

has positive coefficients

a, b, c

in arithmetic progression in that order. If

P (x) = 0

has integer roots

α and β,

then

α + β + α β

equals

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The solutions of

x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} - 4 = 0

are

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The sum of all the real values of

x

Satisfying the equation

2^{(x - 1) (x^{2} + 5 x - 50)} = 1

is:

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The equation

\sqrt{3 x^{2} + x + 5} = x - 3

, where

x

is real, has

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Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

\frac{1}{\sqrt{α}}, \frac{1}{\sqrt{β}}

are the roots of the equation

a x^{2} + b x + 1 = 0, (a \neq 0, a, b \in R)

, then the equation

x (x + b^{3}) + (a^{3} - 3 a b x) = 0

has roots:

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

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>Quadratic Expressions>Algebraic Inequalities

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>Quadratic Expressions>Algebraic Inequalities

The harmonic mean of the roots of the equation

(5 + \sqrt{2}) x^{2} - (4 + \sqrt{5}) x + (8 + 2 \sqrt{5} = 0)

Let x be the greatest integer less than or equal to x, for a real number x. Then the equation x2=x+1 has

Important Questions on Quadratic Expressions

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Let $[x]$ be the greatest integer less than or equal to $x$ , for a real number $x$ . Then the equation $[x^{2}] = x + 1$ has