Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 2

HARD

JEE Main

IMPORTANT

Earn 100

Let $α$ and $β (α < β)$ be the roots of the equation $a x^{2} + b x + c = 0$ . If $\lim_{x \to m} \frac{|a x^{2} + b x + c|}{a x^{2} + b x + c} = 1$ , then which of the following is/are correct,

50% studentsanswered this correctly

Important Questions on Theory of Equations

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 3

If the quadratic polynomials defined on the real coefficients

P (x) = a_{1} x^{2} + 2 b_{1} x + c_{1}

and

Q (x) = a_{2} x^{2} + 2 b_{2} x + c_{2}

take positive values

\forall x \in R

, then what can we say for the trinomial

g (x) = a_{1} a_{2} x^{2} + b_{1} b_{2} x + c_{1} c_{2}

?

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 4

For which of the following graphs of the quadratic expression

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

, the product

a b c

is negative?

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 5

The difference of the maximum and minimum values of

\frac{x^{2} + 4 x + 9}{x^{2} + 9}

is

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 6

If

a > 1

, then the roots of the equation

(1 - a) x^{2} + 3 a x - 1 = 0

are

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 7

The values of

a

, for which the quadratic expression

a x^{2} + (a - 2) x - 2

is negative for exactly two integral values of

x

, belongs to

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 8

The values of

a

for which the roots of the equation

(a + 1) x^{2} - 3 a x + 4 a = 0

,

(a \neq - 1)

are real and greater than

1

are

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 9

The equation

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

will have real and unequal roots if

EASY

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 10

If

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

,

a \neq 0

,

a

,

b

,

c \in R

has distinct real roots in

(1, 2)

, then

a

and

5 a + 2 b + c

have