Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 19

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Let $m \neq n$ be two real numbers such that $m^{2} = n + 2$ and $n^{2} = m + 2$ . Find the value of $4 m n - m^{3} - n^{3}$ .

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Important Questions on Algebra

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 20

There are a few integer values of

a

such that

\frac{a^{2} - 3 a - 3}{a - 2}

is an integer. Find the sum of all these integer values of

a

.

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 21

Suppose that

a, b, x

and

y

are real numbers such that

a x + b y = 3

,

a x^{2} + b y^{2} = 7

,

a x^{3} + b y^{3} = 16

and

a x^{4} + b y^{4} = 42

. Find the value of

a x^{5} + b y^{5}

.

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 22

Let

a

and

b

be positive real numbers such that

\frac{1}{a} - \frac{1}{b} - \frac{1}{a + b} = 0

. Find the value of

{(\frac{b}{a} + \frac{a}{b})}^{2}

.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 23

Let the sum of the coefficients of the polynomial ${(4 x^{2} - 4 x + 3)}^{4} {(4 + 3 x - 3 x^{2})}^{2}$ is $S$ . Find $\frac{S}{16}$ .

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 24

If

f (x) = \frac{1}{\sqrt[3]{x^{2} + 2 x + 1} + \sqrt[3]{x^{2} - 1} + \sqrt[3]{x^{2} - 2 x + 1}}

for all positive integers

x

, find the value of

f (1) + f (3) + f (5) + \dots + f (997) + f (999)

.

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 25

Let

a

and

b

be two integers. Suppose

x^{2} - x - 1

is a factor of the polynomial

a x^{5} + b x^{4} + 1

. Find the value of

a

.

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IOQM - PRMO and RMO

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 26

Suppose

f

is a function satisfying

f (x + x^{- 1}) = x^{6} + x^{- 6}

, for all

x \neq 0 .

Determine

400 - f (3)

.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 27

Suppose

x_{1}, x_{2}

and

x_{3}

are roots of

{(11 - x)}^{3} + {(13 - x)}^{3} = {(24 - 2 x)}^{3}

. What is the sum of

x_{1} + x_{2} + x_{3} ?