Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 90

EASY

CAT

IMPORTANT

Earn 100

Number of positive integral solutions of the following inequation $\frac{x^{3} (2 x - 3)^{2} (x - 4)^{6}}{(x - 3)^{3} (3 x - 8)^{4}} \leq 0$

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Important Questions on Theory of Equations

EASY

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 91

Find the most appropriate solution of the following rational inequation.

\frac{x^{2} + 6 x - 7}{| x + 3 |} < 0

MEDIUM

CAT

IMPORTANT

Quantum CAT Also Useful for XAT>Chapter 14 - SNAP>CMAT>Q 92

If

x

is real, then find the minimum and maximum value of the following rational function

f (x)

f (x) = \frac{x^{2} - 3 x + 4}{x^{2} + 3 x + 4}

MEDIUM

CAT

IMPORTANT

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

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

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