Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 38

HARD

IOQM - PRMO and RMO

IMPORTANT

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Let $p (x)$ be a polynomial with real coefficients such that for all real $x$ , $2 (1 + p (x)) = p (x - 1) + p (x + 1)$ and $p (0) = 8, p (2) = 32$ . Determine the value of $p (4)$ .

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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 39

Let

a, b, c, d, e

be five numbers satisfying the following conditions:

a + b + c + d + e = 0

, and

a b c + a b d + a b e + a c d + a c e + a d e + b c d + b c e + b d e + c d e = 33

Find the value of

\frac{a^{3} + b^{3} + c^{3} + d^{3} + e^{3}}{502}

.

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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 40

Given that

x

and

y

are both negative integers satisfying the equation

y = \frac{10 x}{10 - x}

, find the maximum value of

y + 20

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 41

Given that

x

and

y

are real numbers satisfying the following equations:

x + x y + y = 2 + 3 \sqrt{2}

and

x^{2} + y^{2} = 6

, then the value of

| x + y + 1 | = a + \sqrt{b}

. Find the product of

a

and

b

.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 42

Given that

y = (x - 16) (x - 14) (x + 14) (x + 16)

, find the maximum value of

(- 0.1 y)

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 43

Given that

a + \frac{1}{a + 1} = b + \frac{1}{b - 1} - 2

and

a - b + 2 \neq 0

, find the value of

a b - a + b

.

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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 44

If

|x| + x + 5 y = 2

and

|y| - y + x = 7

, find the value of

{(x + y)}^{3}

.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 45

Let

x_{1}, x_{2}, x_{3}, x_{4}

denote the four roots of the equation

x^{4} + k x^{2} + 90 x - 2009 = 0

. If

x_{1} x_{2} = 49

, find the value of

k

.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 46

Let

x

be a real number such that

x^{2} - 15 x + 1 = 0

. Find the last two digits of

x^{4} + \frac{1}{x^{4}}

.