Logical Reasoning
15-20%of exam
ConjecturesCounterexamplesSet TheoryVenn DiagramsLogic
Probability
30-35%of exam
Relations + Functions
45-55%of exam
Quick Facts
- Exam
- Math 30-2
- Machine
- 24 MC + 8 NR
- Written
- 2 responses
- Machine mark
- 75%
- Written mark
- 25%
- Time
- 3 h
- Extra
- Up to 3 h
- Course blend
- Exam 30%
- Standards
- 50% / 80%
Sets
Union or; intersection and
Union: allIntersection: overlapComplement: outside
Union vs Intersection
Union
- A or B
- Combine regions
- Add then subtract
Intersection
- A and B
- Overlap only
- Shared region
Or vs and
Logic Picker
- Pattern observed→Conjecture
- One false case→Counterexample
- All cases needed→Venn diagram
- Either set→Union
- Both sets→Intersection
- Outside set→Complement
- Inside another→Subset
- If-then needed→Conditional
Logic Core
- Conjecture
- Pattern claim
- Counterexample
- Single disproof
- Inductive
- Pattern reasoning
- Deductive
- Rule reasoning
- Conditional
- If-then statement
- Converse
- Reverse condition
- Biconditional
- Both directions
- Valid
- Reasoning holds
- Invalid
- Logic fails
Example vs Proof
Example
- Supports pattern
- Not enough
- Can mislead
Proof
- All cases
- Logical guarantee
- General result
Suggest vs establish
Sets
- Set
- Object collection
- Element
- Set member
- Subset
- Contained set
- Complement
- Outside set
- Union
- A or B
- Intersection
- A and B
- Empty set
- No elements
- Universal set
- All possible
- Venn diagram
- Set picture
Counting
nPr orders; nCr chooses
P: positionsC: committeesFCP: slots
Permutation vs Combination
Permutation
- Order matters
- Arrangements
- nPr
Combination
- Order ignored
- Selections
- nCr
Arrange vs choose
Counting Picker
- Sequential choices→FCP(Multiply slots)
- Order matters→Permutation(nPr)
- Order ignored→Combination(nCr)
- Identical items→Divide repeats
- Restrictions given→Casework
- At least one→Complement
- Groups selected→Combination
- Positions filled→Slot method
Counting Methods
- FCP
- Multiply choices
- Factorial
- Descending product
- Permutation
- Ordered arrangement
- Combination
- Unordered selection
- Repetition
- Choices may repeat
- Restriction
- Limit choices
- Casework
- Split scenarios
- Slot method
- Fill positions
Independent vs Dependent
Independent
- No effect
- Replacement common
- Multiply probabilities
Dependent
- Probability changes
- No replacement
- Update denominator
Fixed vs changing
Probability Picker
- Not A→Complement
- A or B→Addition rule
- A and B→Multiplication rule
- Overlap included→Subtract overlap
- Given condition→Conditional
- Replacement occurs→Independent
- No replacement→Dependent
- Odds given→Convert first
Probability Rules
- Probability
- Favorable / total
- Odds
- Favorable:unfavorable
- Complement
- 1 - P(A)
- Disjoint
- No overlap
- Overlap
- Shared outcomes
- Conditional
- Given event
- Independent
- No effect
- Dependent
- Probability changes
- Expected value
- Long-run average
Probability Models
- Tree diagram
- Branch outcomes
- Two-way table
- Category counts
- Venn diagram
- Overlap counts
- Simulation
- Repeated model
- Sample space
- All outcomes
- Outcome
- Single result
- Event
- Outcome set
- Trial
- One repetition
Sine Model
Amplitude, midline, period, shift
A: heightD: midlineB: periodC: shift
Exponential vs Log
Exponential
- Variable exponent
- Growth/decay
- Horizontal asymptote
Logarithmic
- Inverse form
- Positive inputs
- Vertical asymptote
Power vs inverse
Function Picker
- U-shape graph→Quadratic
- End behavior→Polynomial
- Repeated percent→Exponential
- Undo exponential→Logarithm
- Repeating pattern→Sinusoidal
- Excluded values→Rational
- Best-fit data→Regression
- Money growth→Compound interest
Polynomials
- Degree
- Highest exponent
- Leading coefficient
- First coefficient
- End behavior
- Far-left/right trend
- Zero
- x-intercept
- Multiplicity
- Repeated zero
- y-intercept
- x = 0
- Turning point
- Direction changes
- Factor
- Zero source
- Quadratic
- Degree two
Rational NPV
Restrictions before canceling
Denominator: zeroCanceled: holesUncanceled: asymptotes
Polynomial vs Rational
Polynomial
- No denominators
- Zeros
- End behavior
Rational
- Variable denominator
- NPVs
- Asymptotes
Smooth vs restricted
Exponential Models
- Growth
- Base above one
- Decay
- Base between zero-one
- Base
- Repeated multiplier
- Initial value
- Starting amount
- Common base
- Solve by matching
- Half-life
- 50% decay time
- Doubling time
- 100% growth time
- Asymptote
- Approached line
- Compound interest
- Repeated percent growth
Amplitude vs Period
Amplitude
- Vertical size
- Half height
- Max-min/2
Period
- Horizontal cycle
- Repeat length
- x-distance
Height vs width
Logarithms
- Logarithm
- Exponent question
- Log form
- log_b a = c
- Exponential form
- b^c = a
- Common log
- Base ten
- Domain
- Input positive
- Asymptote
- Input boundary
- Inverse
- Undo exponential
- Solve log
- Rewrite exponential
Sinusoidal Functions
- Amplitude
- Half height
- Midline
- Center line
- Period
- Cycle length
- Frequency
- Cycles per unit
- Phase shift
- Horizontal move
- Vertical shift
- Midline move
- Maximum
- Highest value
- Minimum
- Lowest value
- Cycle
- One repeat
Rational Expressions
- NPV
- Excluded x-values
- Hole
- Canceled factor
- Vertical asymptote
- Uncanceled zero
- Horizontal asymptote
- End behavior
- Simplify
- Cancel factors
- Rational equation
- Clear denominators
- Extraneous
- Invalid solution
- Restriction
- Denominator not zero
Finance + Regression
- Principal
- Starting money
- Interest rate
- Percent growth
- Compounding
- Interest earns interest
- Future value
- Ending amount
- Regression
- Best-fit model
- Residual
- Actual minus predicted
- Interpolation
- Within data
- Extrapolation
- Beyond data
- Correlation
- Association strength
Written Response
- State
- Name result
- Show work
- Visible reasoning
- Justify
- Explain why
- Units
- Context labels
- Restrictions
- Mention NPVs
- Technology
- Graphing tool
- Conclusion
- Answer context
- Check
- Validate solution
Common Traps
Order vs grouping
nPr orders ≠ nCr groups
Replacement matters
Replacement independent ≠ Removal dependent
Overlap double-count
Union adds ≠ Overlap subtracts
NPV forgotten
Before canceling ≠ Check final answers
Log domain
Input positive ≠ Base positive
Amplitude vs period
Amplitude vertical ≠ Period horizontal
Examples vs proof
Examples suggest ≠ Counterexample disproves
Last Minute
- 1.Weights: 15-20/30-35/45-55
- 2.Diploma exam = 30%
- 3.Machine = 75%; written = 25%
- 4.nPr orders; nCr chooses
- 5.Union: add, subtract overlap
- 6.Conditional: reduce sample space
- 7.NPVs before simplifying
- 8.Log inputs stay positive
- 9.Amplitude vertical; period horizontal
- 10.Answer written-response context
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