Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 19

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If $\frac{x^{2} + 5}{2} = x - 2 \cos (m + n x)$ has at least one real root, then

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Important Questions on Theory of Equations

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 20

Let three quadratic equations

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

,

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

and

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

, all have only positive roots. Then which of these are always true?

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 21

For the quadratic equation

x^{2} + 2 (a + 1) x + 9 a - 5 = 0

, which of the following is/are true?

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 22

If

a, b, c \in R

and

a b c < 0

, then the equation

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

has

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 23

The graph of the quadratic trinomial

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

has its vertex at

(4, - 5)

and two

x

-intercepts, one positive and one negative. Which of the following holds good?

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 24

Let

a, b, c \in Q^{+}

satisfying

a > b > c

. Which of the following statement(s) hold true for the quadratic polynomial

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

?

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 25

Let $f (x) = a x^{2} + b x + c$ . Consider the following diagram. Then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 26

Graph of $y = a x^{2} + b x + c$ is as shown in the figure. If $P Q = 9, O R = 5$ and $O B = 2.5$ , then which of the following is/are true?

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 2 - Theory of Equations>Exercises>Q 27

If the equation

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

has two real roots

α

and

β

, such that

α < - 2

and

β > 2

, then which of the following statements is/are true?