Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 31

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Let $a > 0$ , and let $P (x)$ be a polynomial with integer coefficients such that
$P (1) = P (3) = P (5) = P (7) = a$ , and
$P (2) = P (4) = P (6) = P (8) = - a$
The smallest possible value of $a = 5 \times Q$ . What is $Q$ ?

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Important Questions on Quadratic Equations

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IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 1

If

α

and

β

are the roots of the equation

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

and

\frac{1}{α}

and

\frac{1}{β}

are the roots of the equation

2 x^{2} + 2 q x + 1 = 0,

then

(α - \frac{1}{α}) (β - \frac{1}{β}) (α + \frac{1}{β}) (β + \frac{1}{α})

is equal to :

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Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 2

Find the set of all real values of

λ

such that the root of the equation

x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0

are always real for any choice of

a, b, c

(Where

a, b, c

represents sides of scalene triangle).

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Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 3

If the equation

|x^{2} - 5 x + 6| - λ x + 7 λ = 0

has exactly three solutions, then

λ

is equal to

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Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 4

Let

a, b \in N, a \neq b

and the two quadratic equations

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

and

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

have a common root. The value of

a b

is

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Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 5

If

x

is real, then

\frac{x^{2} - x + c}{x^{2} + x + 2 c}

can take all real values if

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Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 6

If the equation

2^{2 x} + a \cdot 2^{x + 1} + a + 1 = 0

has roots of the opposite sign, then the exhaustive set of real values of

a

is

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Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 7

If both roots of the quadratic equation

(2 - x) (x + 1) = p

are distinct and positive, then

p

must lie in the interval

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Mathematics Crash Course JEE Advanced>Chapter 2 - Quadratic Equations>Exercise 1>Q 8

If

α, β

are the real and distinct roots of

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

and

α^{4}, β^{4}

are the roots of

x^{2} - r x + s = 0

, then the equation

x^{2} - 4 q x + 2 q^{2} - r = 0

always has

(α \neq - β)