Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 25

HARD

IOQM - PRMO and RMO

IMPORTANT

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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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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 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

?

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

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

?

EASY

IOQM - PRMO and RMO

IMPORTANT

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

.

MEDIUM

IOQM - PRMO and RMO

IMPORTANT

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

IOQM - PRMO and RMO

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)

.

MEDIUM

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}

.

MEDIUM

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