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

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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 $J \times 5$ ?

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Important Questions on Algebra

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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 $Q$ ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 35

Find the remainder when

x^{2008} + 2008 x + 2008

is divided by

x + 1

.

EASY

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 36

Given that

x

and

y

are positive real numbers such that

{(x + y)}^{2} = 2500

and

x y = 500

, find the exact value of

\frac{x^{3} + y^{3}}{10^{3}}

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 37

Find the remainder when

{(x - 1)}^{100} + {(x - 2)}^{200}

is divided by

x^{2} - 3 x + 2

.

HARD

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IMPORTANT

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

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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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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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

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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

.