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

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JEE Main

IMPORTANT

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If $f (x)$ is a polynomial of degree four with the leading coefficient one satisfying $f (1) = 1$ , $f (2) = 2$ and $f (3) = 3$ , then $[\frac{f (- 1) + f (5)}{f (0) + f (4)}]$ (where $[\cdot]$ represents the greatest integer function) is equal to

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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 6

Let

P (x) = x^{6} - x^{5} - x^{3} - x^{2} - x

&

Q (x) = x^{4} - x^{3} - x^{2} - 1

. If

α

,

β

,

γ

and

δ

are the roots of

Q (x) = 0

then the value of

P (α) + P (β) + P (γ) + P (δ)

is equal to

MEDIUM

JEE Main

IMPORTANT

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

The line

y = m x + 1

touches the curve

y = - x^{4} + 2 x^{2}

+ x

at two points

P (x_{1}, y_{1})

and

Q (x_{2}, y_{2})

. The value of

x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2}

is

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JEE Main

IMPORTANT

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

If

a + b + c = 24

,

a^{2} + b^{2} + c^{2} = 210

and

a b c = 440

, then the least value of

a - b - c

is

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JEE Main

IMPORTANT

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

If the roots of

x^{4} + q x^{2} + k x + 225 = 0

are in arithmetic progression, then the value of

q

is

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JEE Main

IMPORTANT

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

If

α

,

β

and

γ

are the roots of the equation

9 x^{3} - 7 x + 6 = 0

, then the equation

x^{3} + A x^{2} + B x + C = 0

has the roots

3 α + 2

,

3 β +

2

and

3 γ + 2

, where

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 11

Let

m

be a real number and assume that two of the three solutions of the cubic equation

x^{3} + 3 x^{2} - 34 x = m

differ by

1

. Then, the possible value(s) of

m

is/are

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 2 - Theory of Equations>DPP>Q 12

Let

f (x) = x^{3} + x + 1

. Let

p (x)

be a cubic polynomial such that the roots of

p (x) = 0

are the squares of the roots of

f (x) = 0

. Then