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

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Suppose $f$ is a function satisfying $f (x + x^{- 1}) = x^{6} + x^{- 6}$ , for all $x \neq 0 .$ Determine $400 - f (3)$ .

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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 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

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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

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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}}

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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

?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 32

For certain real numbers

a, b

, and

c

, the polynomial

g (x) = x^{3} + a x^{2} + x + 10

has three distinct roots, and each root of

g (x)

is also a root of the polynomial

f (x) = x^{4} + x^{3} + b x^{2} + 100 x + c

. What is

\frac{- f (1)}{91}

?

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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

?

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 34

Let $a > 0$ , and let $P (x)$ be a polynomial with integer coefficients such that
$P (1) = P (3) = P (5) = P (7) = a$ , and
$P (2) = P (4) = P (6) = P (8) = - a$
The smallest possible value of $a = 5 \times Q$ . What is $Q$ ?