Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 6

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IMPORTANT

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Given that $f (x)$ is a polynomial of degree $3,$ and its remainders are $2 x - 5$ and $- 3 x + 4$ when divided by $x^{2} - 1$ and $x^{2} - 4$ , respectively. Find the $f (- 3)$

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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 7

Factorize

x^{4} + y^{4} + (x + y)^{4}

.

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 8

Given that

f (x) = x^{2} + a x + b

is a polynomial with integral coefficients. If

f

is a common factor of polynomials

g (x) = x^{4} - 3 x^{3} + 2 x^{2} - 3 x + 1

and

h (x) = 3 x^{4} - 9 x^{3} + 2 x^{2} + 3 x - 1

, find

f (x)

.

HARD

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 9

For any non-negative integers

m, n, p,

prove that the polynomial

x^{3 m} + x^{3 n + 1} + x^{3 p + 2}

has the factor

x^{2} + x + 1 .

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 10

When

f (x) = x^{3} + 2 x^{2} + 3 x + 2

is divided by

g (x)

which is a polynomial with integer coefficients, the quotient and remainder are both

h (x)

. Given that

h

is not

a

constant, find

g

and

h

.

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 11

The graph of the function $f$ is shown below. How many solutions does the equation $f (f (x)) = 6$ have?

Question Image

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 12

Suppose

x - y = 1

. Find the value of

x^{4} - x y^{3} - x^{3} y - 3 x^{2} y + 3 x y^{2} + y^{4}

.

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 13

If

x^{2} + x - 1 = 0

, find the value of

x^{4} - 3 x^{2} + 3

.

EASY

IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 14

If two positive integers

m

and

n

, both bigger than

1

, satisfy the equation

2005^{2} + m^{2} = 2004^{2} + n^{2}

, then find the value of

m + n - 200

.