EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 1 - 2014>Q 8

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JEE Advanced

IMPORTANT

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Let $a \in R$ and let $f : R \to R$ be given by $f (x) = x^{5} - 5 x + a,$ then

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Important Questions on Quadratic Equations

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 2 - 2015>Q 2

Let

S

be the set of all non-zero real numbers

α

such that the quadratic equation

α x^{2} - x + α = 0

has two distinct real roots

x_{1} & x_{2}

satisfying the inequality

|x_{1} - x_{2}| < 1 .

Which of the following intervals is(are) a subset(s) of

S

?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 2 - 2014>Q 3

The quadratic equation

P (x) = 0

with real coefficients has purely imaginary roots. Then the equation

P (P (x)) = 0

has

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JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 1 - 2021>Q 4

For

x \in R

, the number of real roots of the equation

3 x^{2} - 4 |x^{2} - 1| + x - 1 = 0

is

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 1 - 2020>Q 5

Suppose

a, b

denote the distinct real roots of the quadratic polynomial

x^{2} + 20 x - 2020

and suppose

c, d

denote the distinct complex roots of the quadratic polynomial

x^{2} - 20 x + 2020

. Then the value of

a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)

is equal to

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 1 - 2019>Q 6

Let

α

and

β

be the roots of

x^{2} - x - 1 = 0,

with

α > β .

For all positive integers

n,

define

a_{n} = \frac{α^{n} - β^{n}}{α - β}, n \geq 1

b_{1} = 1

and

b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2 .

Then which of the following options is/are correct?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 1 - 2016>Q 7

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 .

If

α_{1} > β_{1}

and

α_{2} > β_{2}

, then

α_{1} + β_{2}

equals

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 2 - Quadratic Equations>JEE Advanced Paper 1 - 2014>Q 8

Let

a \in R

and let

f : R \to R

be given by

f (x) = x^{5} - 5 x + a,

then