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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IOQM - PRMO and RMO

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

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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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 30

Let

x, y

and

z

be three real numbers such that

x y + y z + x z = 4

. Find the least possible value of

x^{2} + y^{2} + z^{2}

.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 31

Consider polynomials

P (x)

of degree at most

3

, each of whose coefficients is an element of

\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}

. If total

k

polynomials satisfy

P (- 1) = - 9

, then find

k - 200

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 33

Let

P

be a cubic polynomial with

P (0) = k, P (1) = 2 k

, and

P (- 1) = 3 k

. If

P (2) + P (- 2) = J K

. What is

J \times 5

?