Master Rational Expressions: A Comprehensive Guide for Sixth Form Students

Question 1

Given real numbers x and y, where x ≠ 0 and y ≠ 0, simplify the expression (x² - y²)/(x - y). Which of the following is equivalent to the simplified expression?

Accepted Answer

x - y

Answer

x + y

Answer

x² + y²

Answer

(x + y)²

Question 2

Find the least common multiple (LCM) of the rational expressions: 1 / x and 1 / (x + 2)

Accepted Answer

x(x + 2)

Answer

x + 2

Answer

x

Answer

2x

Question 3

Identify the domain of the rational function: f(x) = (x - 1) / (x^2 - 4)

Accepted Answer

{x | x ≠ ±2}

Answer

{x | x ≠ 2}

Answer

{x | x ≠ -2}

Answer

{x | x ≠ 0}

Question 4

Which of the following is equivalent to the rational expression: (x + 2) / (x - 1)?

Accepted Answer

1 + 3 / (x - 1)

Answer

1 - 3 / (x - 1)

Answer

3 + 1 / (x - 1)

Answer

3 - 1 / (x - 1)

Question 5

Which of the following rational functions has a vertical asymptote at x = 0?

Accepted Answer

(x - 1) / (x^2 + 1)

Answer

(x - 1) / (x + 1)

Answer

(x + 1) / (x - 1)

Answer

(x^2 + 1) / (x - 1)

Question 6

Perform the indicated operation: (3x / (x-1)) + (2 / (x+1))

Accepted Answer

(5x + 2) / (x² - 1)

Answer

(5x + 2) / (x² + 1)

Answer

(5x - 2) / (x² - 1)

Answer

(5x - 2) / (x² + 1)

Question 7

Solve the rational equation: (x - 1) / (x + 2) = 1/2

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 0

Answer

x = 2

Question 8

Identify the vertical asymptote of the rational function: f(x) = (x - 2) / (x² - 4)

Accepted Answer

x = 2

Answer

x = 0

Answer

x = -2

Answer

x = 4

Question 9

Find the least common denominator (LCD) of the fractions: (2 / (x - 3)) and (5 / (x + 2))

Accepted Answer

(x - 3)(x + 2)

Answer

(2x - 6)(5x + 10)

Answer

(x² - x - 6)

Answer

(x² + x - 6)

Question 10

Which of the following is NOT a property of rational expressions?

Accepted Answer

The sum of two rational expressions is always a rational expression.

Answer

The product of two rational expressions is a rational expression.

Answer

The quotient of two rational expressions is a rational expression.

Answer

A rational expression can be undefined for certain values of the variable.

Question 11

Which of the following is the reciprocal of the rational expression: (2x + 1) / (x - 4)?

Accepted Answer

(x - 4) / (2x + 1)

Answer

(2x - 1) / (x + 4)

Answer

-(2x + 1) / (x - 4)

Answer

-(2x - 1) / (x + 4)

Question 12

Find the least common denominator (LCD) of the expressions: 1/x and 1/(x - 2)

Accepted Answer

x(x - 2)

Answer

x + 2

Answer

x - 2

Answer

x^2

Question 13

Which of the following expressions is equivalent to (x + 1)/(x^2 - 1)?

Accepted Answer

1/(x - 1)

Answer

(x + 1)/(x + 1)

Answer

1/(x - 1)^2

Answer

x - 1

Question 14

Rationalize the denominator of the expression: 1/√(x)

Accepted Answer

√(x)/x

Answer

√(x)

Answer

x/√(x)

Answer

1/x

Question 15

Which of the following is not a property of rational expressions?

Accepted Answer

Their domain can include all real numbers.

Answer

They can be added and subtracted if they have the same denominator.

Answer

They can be multiplied and divided like fractions.

Answer

They can have undefined values.

Question 16

Identify the excluded value of the expression: (x² - 9) / (x - 3)

Accepted Answer

3

Answer

0

Answer

-3

Answer

9

Question 17

Rationalize the denominator of the expression: 1 / √2

Accepted Answer

√2/2

Answer

2/√2

Answer

1/2

Answer

2

Question 18

Find the domain of the rational expression: (x+2) / (x² - 4)

Accepted Answer

All real numbers except 2 and -2

Answer

All real numbers except -2

Answer

All real numbers except 0

Answer

All real numbers

Question 19

Determine the domain of the rational expression: (2x - 1) / (x² - 4)

Accepted Answer

All real numbers except x = 2 and x = -2

Answer

All real numbers except x = 0 and x = 2

Answer

All real numbers

Answer

No real numbers

Question 20

Simplify the rational expression: (x^2 - 9)/(x - 3)

Accepted Answer

(x - 3)

Answer

(x + 3)

Answer

(x - 6)

Answer

(x + 6)

Question 21

Which of the following is equivalent to the expression: (x/(x - 1)) + (1/(x + 1))

Accepted Answer

x^2 + 1/(x - 1)(x + 1)

Answer

x^2 - 1/(x - 1)(x + 1)

Answer

2x/(x - 1)(x + 1)

Answer

x/(x - 1)^2 + 1/(x + 1)^2

Question 22

Factorise the denominator of the rational expression: (2x + 3)/(x^2 - 9)

Accepted Answer

(x - 3)(x + 3)

Answer

(x - 3)^2

Answer

(x + 3)(x - 3)

Answer

(x + 3)^2

Question 23

Simplify the expression: (x^2 - 4) / (x - 2)

Accepted Answer

x - 2

Answer

x + 2

Answer

x^2 + 2x + 4

Answer

x^2 - 2x + 4

Question 24

Perform the operation: (3 / x) + (2 / y)

Accepted Answer

(5 / xy)

Answer

(5 / (x + y))

Answer

(5xy / (x + y))

Answer

(5 / x - 5 / y)

Question 25

Solve the equation for x: (x - 1) / (x + 2) = 1 / 3

Accepted Answer

x = 5/2

Answer

x = -5/2

Answer

x = 1/2

Answer

x = -1/2

Question 26

Which of the following is equivalent to the expression: (x^2 - 9) / (x + 3)?

Accepted Answer

x - 3

Answer

x + 3

Answer

x^2 + 3x - 9

Answer

x^2 - 3x - 9

Question 27

Find the least common multiple (LCM) of the denominators of the following fractions: 1 / x, 2 / (x - 1), and 3 / (x + 1)

Accepted Answer

x(x - 1)(x + 1)

Answer

x(x - 1)

Answer

x(x + 1)

Answer

None of these

Question 28

Identify the restriction on the variable in the expression: (x - 2) / (x^2 - 4)?

Accepted Answer

x cannot be 2 or -2

Answer

x cannot be 0

Answer

x cannot be 4

Answer

There are no restrictions

Question 29

Simplify the rational expression: (x² - 4)/(x + 2)

Accepted Answer

x - 2

Answer

x + 2

Answer

(x - 2)(x + 2)

Answer

Cannot be simplified further

Question 30

Which of the following is equivalent to the rational expression: (x+3)/(x² - 4)

Accepted Answer

1/(x+2)

Answer

1/(x-2)

Answer

1/(x² - 4)

Answer

Cannot be simplified further

Question 31

Find the least common multiple (LCM) of the expressions: 1 / (x - 2) and 1 / (x + 2)

Accepted Answer

(x - 2)(x + 2)

Answer

x² - 4

Answer

x² + 4

Answer

Cannot be found

Question 32

Solve the rational equation: (x + 1) / (x - 1) = 2

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 1

Answer

x = -1

Question 33

Simplify the rational expression: (x² - 9) / (x + 3)

Accepted Answer

(x - 3)

Answer

(x + 3)

Answer

(x - 3)(x + 3)

Answer

(x + 3)²

Question 34

Find the least common multiple (LCM) of the expressions: 1/x and 1/(x + 2)?

Accepted Answer

(x)(x + 2)

Answer

x

Answer

x + 2

Answer

(x - 2)

Question 35

Which of the following rational expressions is undefined?

Accepted Answer

(x² - 1)/(x - 1)

Answer

(x - 2)/(x + 2)

Answer

(x + 1)/(x - 1)

Answer

(x² + 1)/(x + 1)

Question 36

Simplify the rational expression: [(x - 1)/(x + 1)] / [(x + 1)/(x - 1)]?

Accepted Answer

1

Answer

-1

Answer

x - 1

Answer

x + 1

Question 37

Simplify the expression (3x - 6) / (x^2 - 4) - 1 / (x + 2).

Accepted Answer

2x / (x - 2)(x + 2)

Answer

2 / (x - 2)(x + 2)

Answer

(3x - 5) / (x - 2)(x + 2)

Answer

-(3x - 5) / (x - 2)(x + 2)

Question 38

Solve the equation (x - 1) / (x + 1) = 2 / 3.

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 5

Answer

x = -5

Question 39

The least common denominator of the rational expressions (2x) / (x - 3) and (x + 1) / (x^2 + 3x - 4) is:

Accepted Answer

x(x^2 + 3x - 4)

Answer

x - 3

Answer

x^2 + 3x - 4

Answer

x(x - 3)

Question 40

The horizontal asymptote of the rational function f(x) = (2x^2 + 5x + 3) / (x^2 - 1) is:

Accepted Answer

y = 2

Answer

y = 5

Answer

y = 3

Answer

The function does not have a horizontal asymptote

Question 41

Perform the operation and simplify: (2x/(x - 1)) + (3/(x - 1))

Accepted Answer

(2x + 3)/(x - 1)

Answer

5x - 1/(x - 1)

Answer

5x/(x - 1)

Answer

(5x + 3)/x

Question 42

Multiply the rational expressions: ((x + 2)/(x - 1)) * ((x - 1)/(x + 3))

Accepted Answer

(x + 2)/(x + 3)

Answer

(x - 2)/(x + 3)

Answer

(x - 1)/(x + 2)

Answer

(x + 1)/(x - 2)

Question 43

Which of the following rational expressions is undefined?

Accepted Answer

(x - 1)/(x² - 1)

Answer

(x + 1)/(x - 2)

Answer

(x - 2)/(x + 1)

Answer

(x² - 1)/(x - 1)

Question 44

The graph of the rational function f(x) = (x^2 - 4) / (x + 2) has a vertical asymptote at:

Accepted Answer

x = 2

Answer

x = -2

Answer

y = 2

Answer

y = -2

Question 45

Simplify the rational expression: (x² - 4) / (x + 2)

Accepted Answer

x - 2

Answer

x + 2

Answer

x² - 2

Question 46

What is the least common multiple (LCM) of the denominators in the expressions: 1/(x - 2) and 3/(x + 1)?

Accepted Answer

(x - 2)(x + 1)

Answer

x - 1

Answer

x² - 4

Answer

x² - 1

Question 47

Multiply the rational expressions: (x² + 3x + 2) / (x - 1) * (x² - 1) / (x + 2)

Accepted Answer

(x + 1)(x + 2)

Answer

(x + 1)(x - 2)

Answer

x² + 1

Question 48

Divide the rational expressions: (x² - 9) / (x + 2) ÷ (x - 3) / (x + 1)

Accepted Answer

(x + 3)(x + 1) / (x + 2)

Answer

(x - 3)(x + 1) / (x + 2)

Answer

(x + 3)(x + 1) / (x - 2)

Answer

(x - 3)(x + 1) / (x - 2)

Question 49

What values of 'x' are excluded from the domain of the rational expression: 5 / (x² - 25)?

Accepted Answer

x = 5, x = -5

Answer

x = 0

Answer

x = 5

Answer

x = -5

Question 50

Solve for 'x' in the rational equation: 1 / (x - 1) = 2 / (x + 1)

Accepted Answer

x = 3

Answer

x = -1

Answer

x = 1

Answer

x = 0

Question 51

Simplify the complex fraction: (1/x + 1/y) / (1/x - 1/y)

Accepted Answer

(x + y) / (x - y)

Answer

y/x

Answer

(x - y) / (x + y)

Answer

x/y

Question 52

Which expression represents a rational expression?

Accepted Answer

(2x + 1) / (x² - 4)

Answer

3x - 2

Answer

x³ + 2x - 5

Answer

√(x² + 1)

Question 53

Which is NOT a valid step in simplifying a rational expression?

Accepted Answer

Adding a constant to both the numerator and denominator.

Answer

Factoring the numerator and denominator.

Answer

Canceling common factors in the numerator and denominator.