UPCAT Mathematics — Algebra — Sets, Exponents, Radicals, Polynomials & EquationsExam Answer Templates

Examiner tip

List elements in ascending order for clarity and avoid repetition

Model answer

A ∪ B = {1, 2, 3, 4, 5, 6}

Answer structure

• Direct statement of the union set [1 mark]

Scoring breakdown

Common mark deductions

• Writing repeated elements
• Missing elements
• Incorrect notation

Key phrases to include

• A ∪ B
• union
• all elements

Examiner tip

Always state which exponent rule you're using to demonstrate understanding

Model answer

Given: (3x²)³
To Find: Simplified form
Solution:
(3x²)³ = 3³ × (x²)³ [Using (ab)ⁿ = aⁿbⁿ]
= 27 × x⁶ [Using (xᵐ)ⁿ = xᵐⁿ]
= 27x⁶
Answer: 27x⁶

Answer structure

• Apply power rule (ab)ⁿ = aⁿbⁿ [1 mark]
• Apply power rule (xᵐ)ⁿ = xᵐⁿ and calculate final answer [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect exponent calculation
• Not showing intermediate steps
• Arithmetic errors

Key phrases to include

• power rule
• simplified form
• exponent laws

Examiner tip

Always verify radical equations as squaring can introduce extraneous solutions

Model answer

Given: √(2x + 3) = 5
To Find: Value of x
Solution:
√(2x + 3) = 5
Squaring both sides:
(√(2x + 3))² = 5²
2x + 3 = 25
2x = 25 - 3
2x = 22
x = 11
Verification: √(2(11) + 3) = √(22 + 3) = √25 = 5 ✓
Answer: x = 11

Answer structure

• Square both sides to eliminate the radical [1 mark]
• Solve the resulting linear equation [1 mark]
• Verify the solution by substitution [1 mark]

Scoring breakdown

Common mark deductions

• Forgetting to verify
• Arithmetic errors
• Not isolating the radical first

Key phrases to include

• square both sides
• verification
• substitute back

Examiner tip

Always write the formula first before substituting values

Model answer

Given: n(A) = 25, n(B) = 30, n(A ∩ B) = 10
To Find: n(A ∪ B)
Solution:
Using the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∪ B) = 25 + 30 - 10
n(A ∪ B) = 45
Answer: n(A ∪ B) = 45

Answer structure

• State the union formula [1 mark]
• Substitute values and calculate correctly [1 mark]

Scoring breakdown

Common mark deductions

• Using wrong formula
• Calculation errors
• Not showing the formula

Key phrases to include

• union formula
• n(A ∪ B)
• substitute

Examiner tip

Label your FOIL steps (First, Outer, Inner, Last) to avoid confusion

Model answer

Given: (2x - 3)(x + 4)
To Find: Expanded and simplified form
Solution:
Using FOIL method:
First: 2x × x = 2x²
Outer: 2x × 4 = 8x
Inner: (-3) × x = -3x
Last: (-3) × 4 = -12
Combining: 2x² + 8x - 3x - 12
Simplifying: 2x² + 5x - 12
Answer: 2x² + 5x - 12

Answer structure

• Apply FOIL method correctly [1 mark]
• Calculate each term correctly [1 mark]
• Combine like terms and simplify [1 mark]

Scoring breakdown

Common mark deductions

• Sign errors
• Incorrect multiplication
• Not combining like terms

Key phrases to include

• FOIL method
• expand
• combine like terms

Examiner tip

Show both methods (factoring and quadratic formula) if time permits for full understanding

Model answer

Given: x² - 5x + 6 = 0
To Find: Values of x
Solution:
Method 1: Factoring
x² - 5x + 6 = 0
We need two numbers that multiply to 6 and add to -5
Those numbers are -2 and -3
(x - 2)(x - 3) = 0
Using zero product property:
x - 2 = 0 or x - 3 = 0
x = 2 or x = 3

Verification:
For x = 2: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓
For x = 3: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓

Answer: x = 2 or x = 3

Answer structure

• Identify the quadratic equation in standard form [1 mark]
• Choose appropriate method (factoring/quadratic formula) [1 mark]
• Execute the method correctly [2 marks]
• Verify both solutions [1 mark]

Scoring breakdown

Common mark deductions

• Not finding both solutions
• Arithmetic errors
• Not verifying answers
• Incorrect factoring

Key phrases to include

• quadratic equation
• factoring
• zero product property
• verification

Examiner tip

Always look for the largest perfect square factor first

Model answer

Given: √48
To Find: Simplified radical form
Solution:
√48 = √(16 × 3)
= √16 × √3 [Using √(ab) = √a × √b]
= 4√3
Answer: 4√3

Answer structure

• Factor out perfect squares [1 mark]
• Simplify using radical properties [1 mark]

Scoring breakdown

Common mark deductions

• Not factoring completely
• Incorrect simplification
• Leaving answer unsimplified

Key phrases to include

• perfect square
• simplified radical form
• factor

Examiner tip

Always convert set-builder notation to roster form first to avoid errors

Model answer

Given: A = {x | x ≤ 5, x ∈ N} and B = {x | 2 ≤ x ≤ 7, x ∈ N}
To Find: A ∩ B
Solution:
First, let's write sets A and B in roster form:
A = {1, 2, 3, 4, 5} [Natural numbers ≤ 5]
B = {2, 3, 4, 5, 6, 7} [Natural numbers from 2 to 7]
A ∩ B = {2, 3, 4, 5} [Common elements]
Answer: A ∩ B = {2, 3, 4, 5}

Answer structure

• Convert set-builder notation to roster form for set A [1 mark]
• Convert set-builder notation to roster form for set B [1 mark]
• Find intersection (common elements) [1 mark]

Scoring breakdown

Common mark deductions

• Misunderstanding set notation
• Including wrong elements
• Missing elements

Key phrases to include

• roster form
• natural numbers
• intersection
• common elements

Examiner tip

Write out each step clearly and state which exponent rule you're using

Model answer

Given: (x²y³)⁴ ÷ (xy)²
To Find: Simplified form
Solution:
(x²y³)⁴ ÷ (xy)²
= x⁸y¹² ÷ x²y² [Using (aᵐ)ⁿ = aᵐⁿ]
= x⁸⁻²y¹²⁻² [Using aᵐ ÷ aⁿ = aᵐ⁻ⁿ]
= x⁶y¹⁰
Answer: x⁶y¹⁰

Answer structure

• Apply power rule to simplify (x²y³)⁴ and (xy)² [1 mark]
• Apply quotient rule for exponents [1 mark]
• Simplify to final form [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect exponent calculations
• Not applying rules properly
• Arithmetic errors

Key phrases to include

• power rule
• quotient rule
• exponent laws

Examiner tip

Read carefully - this is basic counting, not set operations

Model answer

Given: 15 red marbles, 10 blue marbles, 5 red-blue striped marbles
To Find: Total number of marbles
Solution:
Total marbles = Red marbles + Blue marbles + Striped marbles
Total marbles = 15 + 10 + 5
Total marbles = 30
Answer: 30 marbles

Answer structure

• Identify that this is a simple addition problem [1 mark]
• Add all categories correctly [1 mark]

Scoring breakdown

Common mark deductions

• Misunderstanding the problem as a set theory problem
• Arithmetic errors

Key phrases to include

• total marbles
• addition
• all categories

Examiner tip

Memorize special factoring patterns for quick recognition

Model answer

Given: x² - 9
To Find: Complete factorization
Solution:
x² - 9 = x² - 3² [Recognizing difference of squares]
= (x + 3)(x - 3) [Using a² - b² = (a + b)(a - b)]
Answer: (x + 3)(x - 3)

Answer structure

• Recognize as difference of squares pattern [1 mark]
• Apply the difference of squares formula correctly [1 mark]

Scoring breakdown

Common mark deductions

• Not recognizing the pattern
• Incorrect factorization
• Sign errors

Key phrases to include

• difference of squares
• factor
• a² - b² = (a + b)(a - b)

Examiner tip

The conjugate changes the sign between terms in binomial denominators

Model answer

Given: 3/(√5 - 2)
To Find: Rationalized form
Solution:
Multiply both numerator and denominator by the conjugate (√5 + 2):
= 3(√5 + 2)/[(√5 - 2)(√5 + 2)]
= 3(√5 + 2)/[(√5)² - (2)²] [Difference of squares]
= 3(√5 + 2)/(5 - 4)
= 3(√5 + 2)/1
= 3√5 + 6
Answer: 3√5 + 6

Answer structure

• Identify the conjugate of the denominator [1 mark]
• Multiply by conjugate and apply difference of squares [1 mark]
• Simplify to final rationalized form [1 mark]

Scoring breakdown

Common mark deductions

• Using wrong conjugate
• Arithmetic errors
• Not simplifying completely

Key phrases to include

• rationalize
• conjugate
• difference of squares

Examiner tip

Always double-check your discriminant calculation as it affects the final answer

Model answer

Given: 2x² + 3x - 1 = 0
To Find: Values of x
Solution:
Comparing with ax² + bx + c = 0:
a = 2, b = 3, c = -1

Using quadratic formula: x = [-b ± √(b² - 4ac)]/(2a)

Substituting values:
x = [-3 ± √(3² - 4(2)(-1))]/(2(2))
x = [-3 ± √(9 + 8)]/4
x = [-3 ± √17]/4

Therefore:
x = (-3 + √17)/4 or x = (-3 - √17)/4

Answer: x = (-3 + √17)/4 or x = (-3 - √17)/4

Answer structure

• Identify coefficients a, b, c [1 mark]
• Write the quadratic formula [1 mark]
• Substitute values correctly [1 mark]
• Simplify discriminant [1 mark]
• Write both solutions in simplest form [1 mark]

Scoring breakdown

Common mark deductions

• Wrong coefficient identification
• Formula errors
• Calculation mistakes
• Not giving both solutions

Key phrases to include

• quadratic formula
• discriminant
• coefficients
• both solutions

Examiner tip

Complement contains all universal set elements NOT in the original set

Model answer

A' = {1, 3, 5, 7}

Answer structure

• List all elements in U that are not in A [1 mark]

Scoring breakdown

Common mark deductions

• Missing elements
• Including elements from A
• Incorrect notation

Key phrases to include

• complement
• A'
• not in A

Examiner tip

Align like terms vertically to avoid mistakes when adding polynomials

Model answer

Given: (3x² - 2x + 5) + (x² + 4x - 3)
To Find: Sum of polynomials
Solution:
(3x² - 2x + 5) + (x² + 4x - 3)
= 3x² + x² - 2x + 4x + 5 - 3 [Group like terms]
= 4x² + 2x + 2 [Combine like terms]
Answer: 4x² + 2x + 2

Answer structure

• Group like terms together [1 mark]
• Combine like terms correctly [1 mark]

Scoring breakdown

Common mark deductions

• Not grouping like terms
• Arithmetic errors
• Missing terms

Key phrases to include

• like terms
• combine
• group

Do

• Write the answer clearly and legibly
• Use proper mathematical symbols
• Box or underline your final answer

Don't

• Don't show unnecessary working
• Don't write explanations unless asked
• Don't leave the answer unclear

Do

• State the formula or method you're using
• Show substitution of values
• Give clear final answer

Don't

• Don't skip the formula/method statement
• Don't make calculation errors
• Don't forget to label your answer

Do

• Use proper headings (Given, To Find, Solution)
• Show each step clearly
• Verify your answer by substitution

Don't

• Don't combine too many steps
• Don't make sign errors
• Don't forget verification for equations

Do

• Show complete systematic approach
• Verify all solutions thoroughly
• Present work neatly and logically
• State conclusions clearly

Don't

• Don't rush through steps
• Don't skip verification
• Don't present work messily
• Don't forget to answer what's asked