Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 93

MEDIUM

CAT

IMPORTANT

Earn 100

If one root of the equation $24 x^{3} - 14 x^{2} - 63 x + 45 = 0$ is double the other, then the third root of this equation is

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Important Questions on Theory of Equations

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 94

If sum of the two roots of the equation

4 x^{3} + 16 x^{2} - 9 x - 36 = 0

is the first whole number, then the third root of this equation is:

MEDIUM

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 95

Find a cubic equation in which each root is greater by unity than a root of the equation

x^{3} - 5 x^{2} + 6 x - 3 = 0 .

Which of the following could be the required equation?

MEDIUM

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 96

Find a cubic equation in which each root is the cube of the roots of

x^{3} + 3 x^{2} + 2 = 0 .

Which of the following could be the required equation?

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 97

Let

a_{1}, a_{2}, a_{3}

be the roots of the polynomial

x^{3} + x^{2} + 3 x + 1

Find the value of

a_{1} + a_{2} + a_{3} + a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{1} + a_{1} a_{2} a_{3}

HARD

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 98

For a cubic polynomial,

a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0},

the roots

r_{1}, r_{2}, r_{3}

come in three symmetric combinations :

C_{1} = r_{1} + r_{2} + r_{3}, C_{2} = r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1}, C_{3} = r_{1} r_{2} r_{3} .

The sum of the

k^{th}

powers of the roots is defined as

S_{k} = r_{1}^{k} + r_{2}^{k} + r_{3}^{k} .

Then
(i)

a_{3} S_{1} + a_{2} = 0

(ii)

a_{3} S_{2} + a_{2} S_{1} + 2 a_{1} = 0

MEDIUM

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 100

The equation

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

has four positive roots, then

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 101

If

f (x) = 4 x^{2} + px + (p - 3)

is negative for at least one

x,

find all possible values of

p .

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 103

Let

a, b, c \in R

and

a \neq 0

Let

α, β

be the roots of equation

{ax}^{2} + bx + c = 0,

where

α < - p

and

β > p .

Then, for every

n \in N,

which one of the following is correct?