Civil Service Exam (Subprofessional) Numerical Ability — Algebra, Exponents & Number SeriesExam Answer Templates

Marks

2

Topic

Laws of Exponents - Product Rule

Difficulty

easy

Template Id

T1

Examiner Tip

Always write the exponent law you're using - this shows you understand the concept, not just memorized calculations

Model Answer

Given: 3^4 × 3^2 Using the Product Rule for exponents: a^m × a^n = a^(m+n) 3^4 × 3^2 = 3^(4+2) = 3^6 = 729 Therefore, 3^4 × 3^2 = 729

Question Type

short_answer

Answer Structure

• Line 1: State the given expression [0.5 marks]
• Line 2: Apply the correct exponent law [1 mark]
• Line 3: Calculate the final answer [0.5 marks]

Scoring Breakdown

Common Mark Deductions

• Not showing the rule used
• Arithmetic errors in final calculation
• Not stating the final answer clearly

Key Phrases To Include

• Product Rule
• a^m × a^n = a^(m+n)
• final answer

Marks

3

Topic

Number Series - Arithmetic Sequence

Difficulty

easy

Template Id

T2

Examiner Tip

Always show the pattern-finding process explicitly - don't just state the answer

Model Answer

Given series: 2, 5, 8, 11, 14, ? Step 1: Find the pattern 5 - 2 = 3 8 - 5 = 3 11 - 8 = 3 14 - 11 = 3 Step 2: Identify the sequence type This is an arithmetic sequence with common difference d = 3 Step 3: Find the next term Next term = 14 + 3 = 17 Therefore, the next term is 17.

Question Type

short_answer

Answer Structure

• Step 1: Calculate differences between consecutive terms [1 mark]
• Step 2: Identify the type of sequence [1 mark]
• Step 3: Apply pattern to find next term [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not showing difference calculations
• Incorrect identification of sequence type
• Arithmetic errors

Key Phrases To Include

• arithmetic sequence
• common difference
• pattern

Marks

3

Topic

Exponential Equations

Difficulty

medium

Template Id

T3

Examiner Tip

Always verify your answer by substituting back into the original equation

Model Answer

Given: 2^(x+1) = 16 Step 1: Express 16 as a power of 2 16 = 2^4 Step 2: Rewrite the equation 2^(x+1) = 2^4 Step 3: Since bases are equal, equate exponents x + 1 = 4 x = 4 - 1 x = 3 Verification: 2^(3+1) = 2^4 = 16 ✓ Therefore, x = 3

Question Type

short_answer

Answer Structure

• Step 1: Express the right side as power of same base [1 mark]
• Step 2: Apply the principle that equal bases mean equal exponents [1 mark]
• Step 3: Solve for the variable and verify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not converting to same base
• Algebraic errors
• Missing verification step

Key Phrases To Include

• express as power
• equal bases
• verification

Marks

2

Topic

Laws of Exponents - Quotient Rule

Difficulty

medium

Template Id

T4

Examiner Tip

Break down complex expressions into manageable parts - coefficients, then each variable separately

Model Answer

Given: (8x^3y^6)/(4x^2y^4) Step 1: Separate coefficients and variables = (8/4) × (x^3/x^2) × (y^6/y^4) Step 2: Apply quotient rule for exponents = 2 × x^(3-2) × y^(6-4) = 2 × x^1 × y^2 = 2xy^2 Therefore, (8x^3y^6)/(4x^2y^4) = 2xy^2

Question Type

short_answer

Answer Structure

• Step 1: Separate numerical and variable parts [1 mark]
• Step 2: Apply quotient rule and simplify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Incorrect application of quotient rule
• Arithmetic errors with coefficients
• Not simplifying completely

Key Phrases To Include

• quotient rule
• a^m/a^n = a^(m-n)
• simplify

Marks

1

Topic

Laws of Exponents - Zero Exponent Rule

Difficulty

easy

Template Id

T5

Examiner Tip

Remember: any non-zero number to the power 0 always equals 1

Model Answer

4^0 + 5^0 = 1 + 1 = 2 (Any non-zero number raised to power 0 equals 1)

Question Type

very_short_answer

Answer Structure

• Apply zero exponent rule and calculate [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not knowing that a^0 = 1
• Arithmetic errors

Key Phrases To Include

• zero exponent rule
• equals 1

Marks

3

Topic

Number Series - Geometric Sequence

Difficulty

medium

Template Id

T6

Examiner Tip

Always verify your answer fits the pattern by checking it works with surrounding terms

Model Answer

Given series: 3, 6, 12, ?, 48 Step 1: Find the common ratio r = 6/3 = 2 r = 12/6 = 2 Step 2: Verify it's a geometric series Common ratio r = 2 (constant) Step 3: Find the missing term Missing term = 12 × 2 = 24 Verification: 24 × 2 = 48 ✓ Therefore, the missing term is 24.

Question Type

short_answer

Answer Structure

• Step 1: Calculate common ratio from given terms [1 mark]
• Step 2: Confirm geometric sequence pattern [1 mark]
• Step 3: Apply ratio to find missing term [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not identifying the common ratio
• Calculation errors
• Missing verification

Key Phrases To Include

• geometric series
• common ratio
• verification

Marks

2

Topic

Laws of Exponents - Power Rule

Difficulty

easy

Template Id

T7

Examiner Tip

State the exponent law you're using - it shows conceptual understanding

Model Answer

Given: (3^2)^3 Using Power Rule: (a^m)^n = a^(mn) (3^2)^3 = 3^(2×3) = 3^6 Calculating: 3^6 = 729 Therefore, (3^2)^3 = 729

Question Type

short_answer

Answer Structure

• Apply power rule for exponents [1 mark]
• Calculate final numerical value [1 mark]

Scoring Breakdown

Common Mark Deductions

• Incorrect application of power rule
• Calculation errors
• Not showing the rule

Key Phrases To Include

• Power Rule
• (a^m)^n = a^(mn)

Marks

2

Topic

Laws of Exponents - Negative Exponent Rule

Difficulty

medium

Template Id

T8

Examiner Tip

Always express negative exponents as positive fractions unless specifically asked for decimal form

Model Answer

Given: 2^(-3) Using Negative Exponent Rule: a^(-n) = 1/a^n 2^(-3) = 1/2^3 = 1/8 Therefore, 2^(-3) = 1/8

Question Type

short_answer

Answer Structure

• Apply negative exponent rule [1 mark]
• Calculate final fraction form [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not knowing negative exponent rule
• Arithmetic errors
• Leaving answer as decimal instead of fraction

Key Phrases To Include

• Negative Exponent Rule
• a^(-n) = 1/a^n

Marks

3

Topic

Arithmetic Sequences

Difficulty

medium

Template Id

T9

Examiner Tip

Always state the formula you're using - it demonstrates method knowledge even if calculation errors occur

Model Answer

Given: Arithmetic sequence 5, 9, 13, 17, ... To find: 8th term (a₈) Step 1: Identify first term and common difference a₁ = 5 d = 9 - 5 = 4 Step 2: Use arithmetic sequence formula aₙ = a₁ + (n-1)d a₈ = 5 + (8-1)×4 a₈ = 5 + 7×4 a₈ = 5 + 28 a₈ = 33 Therefore, the 8th term is 33.

Question Type

short_answer

Answer Structure

• Step 1: Identify first term and common difference [1 mark]
• Step 2: Apply arithmetic sequence formula [1 mark]
• Step 3: Calculate final answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong common difference calculation
• Formula errors
• Arithmetic mistakes

Key Phrases To Include

• arithmetic sequence
• common difference
• formula aₙ = a₁ + (n-1)d

Marks

1

Topic

Basic Exponential Equations

Difficulty

easy

Template Id

T10

Examiner Tip

Memorize perfect cubes up to 10³ for quick recognition

Model Answer

x³ = 27 x = ∛27 = 3 Therefore, x = 3

Question Type

very_short_answer

Answer Structure

• Take cube root of both sides to find x = 3 [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not recognizing 27 as perfect cube
• Calculation errors

Key Phrases To Include

• cube root
• ∛27 = 3

Marks

3

Topic

Number Series - Perfect Squares

Difficulty

medium

Template Id

T11

Examiner Tip

Look for common mathematical sequences like squares, cubes, or factorial patterns

Model Answer

Given series: 1, 4, 9, 16, 25, ?, ? Step 1: Analyze the pattern 1 = 1² 4 = 2² 9 = 3² 16 = 4² 25 = 5² Step 2: Identify the sequence type This is a perfect squares sequence: n² Step 3: Find next two terms 6² = 36 7² = 49 Therefore, the next two terms are 36 and 49.

Question Type

short_answer

Answer Structure

• Step 1: Express each term as perfect square [1 mark]
• Step 2: Identify pattern as n² sequence [1 mark]
• Step 3: Calculate next two terms [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not recognizing perfect square pattern
• Calculation errors
• Finding only one term instead of two

Key Phrases To Include

• perfect squares
• n² sequence
• pattern

Marks

2

Topic

Laws of Exponents - Fractional Exponents

Difficulty

medium

Template Id

T12

Examiner Tip

Remember that fractional exponents represent roots - the denominator tells you which root to take

Model Answer

Given: 16^(1/4) Using Fractional Exponent Rule: a^(1/n) = ⁿ√a 16^(1/4) = ⁴√16 Since 16 = 2⁴, we have: ⁴√16 = ⁴√(2⁴) = 2 Therefore, 16^(1/4) = 2

Question Type

short_answer

Answer Structure

• Apply fractional exponent rule [1 mark]
• Evaluate the fourth root correctly [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not knowing fractional exponent rule
• Incorrect root calculation
• Not recognizing 16 as 2⁴

Key Phrases To Include

• Fractional Exponent Rule
• a^(1/n) = ⁿ√a
• fourth root

Marks

5

Topic

Complete Laws of Exponents Overview

Difficulty

hard

Template Id

T13

Examiner Tip

Structure your answer with clear headings and always include a worked example to demonstrate application

Model Answer

Steps to Simplify Exponential Expressions: Step 1: Apply Product Rule Multiply terms with same base by adding exponents: aᵐ × aⁿ = aᵐ⁺ⁿ Step 2: Apply Quotient Rule Divide terms with same base by subtracting exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ Step 3: Apply Power Rule Raise a power to another power by multiplying exponents: (aᵐ)ⁿ = aᵐⁿ Step 4: Apply Zero Exponent Rule Any non-zero number to power 0 equals 1: a⁰ = 1 Step 5: Apply Negative Exponent Rule Convert negative exponents to positive: a⁻ⁿ = 1/aⁿ Example: Simplify (2³ × 2⁵) ÷ 2⁴ Step 1: 2³ × 2⁵ = 2³⁺⁵ = 2⁸ (Product Rule) Step 2: 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ (Quotient Rule) Step 3: 2⁴ = 16 Therefore, the expression simplifies to 16.

Question Type

long_answer

Answer Structure

• Step 1: Explain Product Rule with formula [1 mark]
• Step 2: Explain Quotient Rule with formula [1 mark]
• Step 3: Explain Power Rule with formula [1 mark]
• Step 4: Explain Zero and Negative Exponent Rules [1 mark]
• Step 5: Provide complete worked example [1 mark]

Scoring Breakdown

Common Mark Deductions

• Missing any of the five rules
• Incorrect formulas
• Poor or missing example
• Not showing step-by-step solution

Key Phrases To Include

• Product Rule
• Quotient Rule
• Power Rule
• Zero Exponent Rule
• Negative Exponent Rule
• worked example

Marks

5

Topic

Geometric Series - Sum Formula

Difficulty

hard

Template Id

T14

Examiner Tip

Always verify your common ratio with at least two consecutive pairs before proceeding

Model Answer

Given: Geometric series 2, 6, 18, 54, ... To find: Sum of first 6 terms (S₆) Step 1: Identify first term and common ratio a₁ = 2 r = 6/2 = 3 Step 2: Verify common ratio 18/6 = 3 ✓ 54/18 = 3 ✓ Step 3: Find the 5th and 6th terms a₅ = 2 × 3⁴ = 2 × 81 = 162 a₆ = 2 × 3⁵ = 2 × 243 = 486 Step 4: Use geometric series sum formula Sₙ = a₁(rⁿ - 1)/(r - 1) S₆ = 2(3⁶ - 1)/(3 - 1) S₆ = 2(729 - 1)/2 S₆ = 2(728)/2 S₆ = 728 Therefore, the sum of first 6 terms is 728.

Question Type

long_answer

Answer Structure

• Step 1: Identify first term and common ratio [1 mark]
• Step 2: Verify the common ratio [1 mark]
• Step 3: Find remaining terms if needed [1 mark]
• Step 4: Apply geometric series sum formula [1 mark]
• Step 5: Calculate final answer correctly [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong common ratio
• Using wrong formula
• Calculation errors
• Not verifying the ratio

Key Phrases To Include

• geometric series
• common ratio
• sum formula Sₙ = a₁(rⁿ - 1)/(r - 1)
• verification

Marks

3

Topic

Sequence Identification and General Terms

Difficulty

medium

Template Id

T15

Examiner Tip

For geometric sequences, always express the general term in its simplest form

Model Answer

Given sequence: 1/2, 1/4, 1/8, 1/16, ... Step 1: Find the pattern (1/4)/(1/2) = 1/2 (1/8)/(1/4) = 1/2 (1/16)/(1/8) = 1/2 Step 2: Identify sequence type This is a geometric sequence with: a₁ = 1/2 r = 1/2 Step 3: Find general term For geometric sequence: aₙ = a₁ × rⁿ⁻¹ aₙ = (1/2) × (1/2)ⁿ⁻¹ aₙ = (1/2)ⁿ aₙ = 1/2ⁿ Therefore, this is a geometric sequence with general term aₙ = 1/2ⁿ

Question Type

short_answer

Answer Structure

• Step 1: Calculate ratios to find pattern [1 mark]
• Step 2: Identify as geometric sequence [1 mark]
• Step 3: Derive general term formula [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not identifying sequence type
• Wrong common ratio
• Incorrect general term formula

Key Phrases To Include

• geometric sequence
• common ratio
• general term
• aₙ = a₁ × rⁿ⁻¹