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

EASY

IOQM - PRMO and RMO

IMPORTANT

Earn 100

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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Important Questions on Algebra

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

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

.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

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} ?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 28

Suppose that

a + x^{2} = 2006, b + x^{2} = 2007

and

c + x^{2} = 2008

and

a b c = 3

. Find the value of

\frac{a}{b c} + \frac{b}{c a} + \frac{c}{a b} - \frac{1}{a} - \frac{1}{b} - \frac{1}{c}

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 29

Find the value of

\frac{x^{4} - 6 x^{3} - 2 x^{2} + 18 x + 23}{x^{2} - 8 x + 15}

when

x = \sqrt{19 - 8 \sqrt{3}}

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

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

?