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

Marks

1

Topic

Sets

Difficulty

easy

Template Id

T1

Examiner Tip

List elements in ascending order for clarity and avoid repetition

Model Answer

A ∪ B = {1, 2, 3, 4, 5, 6}

Question Type

very_short_answer

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

Marks

2

Topic

Exponents

Difficulty

easy

Template Id

T2

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

Question Type

short_answer

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

Marks

3

Topic

Radicals

Difficulty

medium

Template Id

T3

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

Question Type

short_answer

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

Marks

2

Topic

Sets

Difficulty

medium

Template Id

T4

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

Question Type

short_answer

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

Marks

3

Topic

Polynomials

Difficulty

medium

Template Id

T5

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

Question Type

short_answer

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

Marks

5

Topic

Equations

Difficulty

medium

Template Id

T6

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

Question Type

long_answer

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

Marks

2

Topic

Radicals

Difficulty

easy

Template Id

T7

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

Question Type

short_answer

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

Marks

3

Topic

Sets

Difficulty

medium

Template Id

T8

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}

Question Type

short_answer

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

Marks

3

Topic

Exponents

Difficulty

medium

Template Id

T9

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

Question Type

short_answer

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

Marks

2

Topic

Sets

Difficulty

easy

Template Id

T10

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

Question Type

short_answer

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

Marks

2

Topic

Polynomials

Difficulty

easy

Template Id

T11

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)

Question Type

short_answer

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)

Marks

3

Topic

Radicals

Difficulty

hard

Template Id

T12

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

Question Type

short_answer

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

Marks

5

Topic

Equations

Difficulty

hard

Template Id

T13

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

Question Type

long_answer

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

Marks

1

Topic

Sets

Difficulty

easy

Template Id

T14

Examiner Tip

Complement contains all universal set elements NOT in the original set

Model Answer

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

Question Type

very_short_answer

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

Marks

2

Topic

Polynomials

Difficulty

easy

Template Id

T15

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

Question Type

short_answer

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