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

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IMPORTANT

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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 $f (x)$ .

Important Questions on Algebra

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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?

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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

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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

.

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 15

Find an integer

x

that satisfies the equation

x^{5} - 101 x^{3} - 999 x^{2} + 100900 = 0

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 16

Let

x

and

y

be real numbers such that

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

. What is the largest possible value of

x^{2} + y^{2} - \sqrt{32}

?