Math 30-2 Diploma Exam Flashcards

Q: How many cards are in this Math 30-2 set?

This set includes 50 original active-recall cards aligned to Math 30-2 topics: logical reasoning, set theory, probability, counting methods, and relations and functions.

Q: Which Math 30-2 topic gets the most review weight?

Relations and Functions is the largest review area at 45-55%, followed by Probability at 30-35% and Logical Reasoning at 15-20%.

Q: Do these cards copy diploma exam items?

No. The cards are original study prompts covering permutations, combinations, odds, conditional probability, polynomial features, logs, exponentials, sinusoidal models, and rational expressions.

Q: What is a common Math 30-2 probability mistake?

A common mistake is choosing a permutation when order does not matter, or treating dependent events as independent after one item has been removed.

Q: What is a common Math 30-2 functions mistake?

A common mistake is simplifying a rational expression without recording non-permissible values, or solving a rational equation without checking for excluded values.

Question 1

What is a conjecture?

Accepted Answer

A conjecture is a statement believed to be true based on observations or patterns. It still needs proof or a valid counterexample test.

Question 2

What does one counterexample do to a conjecture?

Accepted Answer

One valid counterexample disproves a conjecture that claims something is always true. More supporting examples do not repair the false claim.

Question 3

Inductive vs. deductive reasoning: what is the difference?

Accepted Answer

Inductive reasoning builds a general pattern from examples. Deductive reasoning starts from accepted facts or rules and reaches a logically necessary conclusion.

Question 4

What does A' mean in set notation?

Accepted Answer

A' is the complement of A: all elements in the universal set that are not in A. The universal set must be known before a complement is meaningful.

Question 5

Union vs. intersection: which word matches each?

Accepted Answer

Union means elements in A or B or both. Intersection means elements in both A and B at the same time.

Question 6

What does it mean for two sets to be disjoint?

Accepted Answer

Disjoint sets have no elements in common, so their intersection is empty. In a Venn diagram, their circles do not overlap.

Question 7

What does A subset B mean?

Accepted Answer

Every element of A is also an element of B. A can be equal to B unless the statement says A is a proper subset of B.

Question 8

How many subsets does a set with n distinct elements have?

Accepted Answer

It has 2^n subsets because each element has two choices: included or not included. This count includes the empty set and the whole set.

Question 9

How do you count elements in A union B from a two-set Venn diagram?

Accepted Answer

Add the number in A, add the number in B, then subtract the overlap once: n(A union B) = n(A) + n(B) - n(A intersect B).

Question 10

Why can a Venn diagram prevent double-counting?

Accepted Answer

It separates only A, only B, both, and neither. Counting each region once keeps overlap from being added twice.

Question 11

When do you use the fundamental counting principle?

Accepted Answer

Use it when a process has stages with independent choices. Multiply the number of choices for each stage to count total outcomes.

Question 12

Permutation vs. combination: what is the deciding question?

Accepted Answer

Ask whether order matters. If order creates a different outcome, use a permutation; if the same group counts once, use a combination.

Question 13

What does nPr count?

Accepted Answer

nPr counts ordered arrangements of r objects chosen from n distinct objects. It is used for rankings, positions, or roles where order matters.

Question 14

What does nCr count?

Accepted Answer

nCr counts unordered selections of r objects from n distinct objects. It is used for committees, groups, or choices where order does not matter.

Question 15

How do repeated objects change arrangement counts?

Accepted Answer

Divide by factorials for repeated items. For n total objects with repeats of a, b, and c, distinct arrangements are n!/(a!b!c!).

Question 16

Why are circular arrangements counted differently?

Accepted Answer

Rotations of the same seating are not new arrangements, so n distinct people around a circle can be arranged in (n - 1)! ways when only rotations match.

Question 17

What is the complement rule for probability?

Accepted Answer

P(not A) = 1 - P(A). It is especially useful for at least one questions, where counting no successes may be easier.

Question 18

Probability vs. odds in favour: what is the difference?

Accepted Answer

Probability compares favourable outcomes to total outcomes. Odds in favour compare favourable outcomes to unfavourable outcomes.

Question 19

How do you convert odds in favour a:b to probability?

Accepted Answer

Use a/(a + b), because a outcomes are favourable and b are unfavourable, making a + b total outcomes in the ratio model.

Question 20

What are mutually exclusive events?

Accepted Answer

They cannot occur on the same trial. If A and B are mutually exclusive, P(A and B) = 0 and P(A or B) = P(A) + P(B).

Question 21

What changes when events are not mutually exclusive?

Accepted Answer

The overlap must be subtracted once: P(A or B) = P(A) + P(B) - P(A and B). Otherwise the overlap is counted twice.

Question 22

What makes two events independent?

Accepted Answer

A and B are independent when one event occurring does not change the probability of the other. Algebraically, P(A and B) = P(A)P(B).

Question 23

What makes probability dependent when drawing without replacement?

Accepted Answer

The first draw changes the sample space for the second draw. Update both the favourable count and the total count before multiplying.

Question 24

What does conditional probability P(B|A) mean?

Accepted Answer

It means the probability of B given that A has already occurred. The sample space is restricted to outcomes where A is true.

Question 25

How can a two-way table help with conditional probability?

Accepted Answer

Use the given condition as the denominator. For P(B|A), look only within the A row, column, or group, then count how many of those also have B.

Question 26

How do you find the probability of at least one success in repeated independent trials?

Accepted Answer

Use the complement: 1 - P(no successes). This is usually faster than adding many separate success cases.

Question 27

What does the degree of a polynomial tell you?

Accepted Answer

The degree is the highest exponent with a nonzero coefficient. It helps predict end behaviour and the maximum possible number of x-intercepts.

Question 28

What does the leading coefficient tell about polynomial end behaviour?

Accepted Answer

Together with degree, it describes what happens as x becomes very large positive or negative. Even and odd degrees behave differently at the two ends.

Question 29

What are zeros of a polynomial function?

Accepted Answer

Zeros are x-values where f(x) = 0. On a graph they are x-intercepts when the graph crosses or touches the x-axis.

Question 30

What does vertex form reveal for a quadratic?

Accepted Answer

In y = a(x - h)^2 + k, the vertex is (h,k). If a is positive, the vertex is a minimum; if a is negative, it is a maximum.

Question 31

How do you find a y-intercept from a function rule?

Accepted Answer

Set x = 0 and evaluate the function. The y-intercept is the output where the graph crosses the y-axis.

Question 32

Why are exponential and logarithmic functions inverses?

Accepted Answer

An exponential function gives a power from an exponent, while a logarithm returns the exponent needed to produce a value with the same base.

Question 33

What restrictions apply to logarithms?

Accepted Answer

The base must be positive and not equal to 1, and the argument must be positive. These restrictions define the domain of logarithmic functions.

Question 34

What does exponential growth look like in a model?

Accepted Answer

A growth model has multiplier greater than 1, such as A = a(b)^t with b > 1. Each equal time step multiplies by the same factor.

Question 35

What does exponential decay look like in a model?

Accepted Answer

A decay model has multiplier between 0 and 1, such as A = a(b)^t with 0 < b < 1. The amount decreases by a constant percent each step.

Question 36

How does half-life fit an exponential model?

Accepted Answer

Each half-life multiplies the amount by 1/2. After n half-lives, the remaining amount is initial amount times (1/2)^n.

Question 37

What does the growth factor in compound interest represent?

Accepted Answer

In A = P(1 + r)^t for annual compounding, 1 + r is the yearly multiplier. It includes the original 100% plus the interest rate.

Question 38

How do you rewrite log_b(a) = c in exponential form?

Accepted Answer

Write b^c = a. This rewrite is often the fastest way to evaluate logarithms or solve simple logarithmic equations.

Question 39

What is the horizontal asymptote of a basic exponential function a*b^x with a > 0?

Accepted Answer

Without vertical shifts, the horizontal asymptote is y = 0. The graph approaches zero but does not cross it in the basic model.

Question 40

What are amplitude and midline in a sinusoidal model?

Accepted Answer

Amplitude is half the distance between maximum and minimum. The midline is the average of maximum and minimum and represents the vertical centre.

Question 41

What does period mean in a sinusoidal or periodic model?

Accepted Answer

The period is the input length required for one complete cycle before the pattern repeats.

Question 42

How are period and frequency related?

Accepted Answer

Frequency is cycles per unit time, so frequency = 1/period when period is measured in that time unit.

Question 43

What does a vertical shift do to a sine graph?

Accepted Answer

It moves the entire graph up or down and changes the midline, but it does not change amplitude or period.

Question 44

What is a non-permissible value in a rational expression?

Accepted Answer

It is any value that makes a denominator zero. These values must be stated even if a factor later cancels.

Question 45

Why must restrictions be kept after simplifying a rational expression?

Accepted Answer

Cancelling a factor changes the appearance but not the original domain. Values that made the original denominator zero remain excluded.

Question 46

What is the first step before adding rational expressions?

Accepted Answer

Find a common denominator. Then rewrite each fraction, add or subtract numerators, and keep the denominator restrictions.

Question 47

How do you divide rational expressions?

Accepted Answer

Multiply by the reciprocal of the second expression, factor when possible, cancel common factors, and record all non-permissible values from the original expression.

Question 48

Why can rational equations have extraneous solutions?

Accepted Answer

Multiplying by denominators can hide values that make an original denominator zero. Check proposed solutions against the original restrictions.

Question 49

What does a vertical asymptote of a rational function usually indicate?

Accepted Answer

It usually comes from a denominator zero that does not cancel with the numerator. The function is undefined there and the graph approaches without crossing in the local model.

Question 50

How should written-response function work show reasoning?

Accepted Answer

State restrictions, define variables, show formulas or transformations used, include units when applicable, and check that the result fits the context.

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