Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 12

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Suppose $x - y = 1$ . Find the value of $x^{4} - x y^{3} - x^{3} y - 3 x^{2} y + 3 x y^{2} + y^{4}$ .

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Important Questions on Algebra

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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 13

If

x^{2} + x - 1 = 0

, find the value of

x^{4} - 3 x^{2} + 3

.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 14

If two positive integers

m

and

n

, both bigger than

1

, satisfy the equation

2005^{2} + m^{2} = 2004^{2} + n^{2}

, then find the value of

m + n - 200

.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 15

Find an integer

x

that satisfies the equation

x^{5} - 101 x^{3} - 999 x^{2} + 100900 = 0

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 16

Let

x

and

y

be real numbers such that

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

. What is the largest possible value of

x^{2} + y^{2} - \sqrt{32}

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 17

It is given that

x = \frac{1}{2 - \sqrt{3}}

. Find the value of

x^{6} - 2 \sqrt{3} x^{5} - x^{4} + x^{3} - 4 x^{2} + 2 x - \sqrt{3}

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 18

Let

f (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}

, where

a_{i}

are non-negative integers for

i = 0, 1, 2, \dots, n

. If

f (1) = 21

and

f (25) = 78357

, find the value of

(\frac{f (10) + 3}{100})

.

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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 19

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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IOQM - PRMO and RMO

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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

.