Grade 12 math practice
Grade 12 courses (MAP4C, MCT4C, MCV4U, MDM4U, MEL4E, MHF4U): advanced functions, calculus and vectors, data management and college/workplace math. Exam-grade practice with step-by-step solutions.
Foundations for College Math (MAP4C)
Endless practice · MAP4C
Generated fresh every time — adaptive difficulty, never runs out
Use a line of best fit to predict
Use the equation of a line of best fit to predict a value, solve for the input, or interpret the slope.
Net area of a wall with cutouts
Find the paintable area of a wall after subtracting windows/doors, and (at higher difficulty) the number of paint cans needed.
Height from an angle of elevation
Use tan(angle) = opposite/adjacent to find the height of a structure from a measured distance and angle of elevation.
Solve exponential equations by isolating the power
Solve an exponential equation A0 * b^t = N for t by isolating the power and recognizing it as a power of b.
Build and use a linear cost model
Write a linear equation C = F + m*x for a flat-fee-plus-rate situation, then evaluate it or solve it for x.
Future value of a simple annuity
Use FV = R[(1+i)^n - 1]/i to find the future value of a series of equal annual deposits.
Renting vs. owning — compare monthly cost
Add up the components of the monthly cost of owning (mortgage + property tax + fees) and compare it to renting.
Designing a budget with recommended percentages
Apply a recommended budget percentage to a monthly income, and compare it to actual spending.
Math for College Technology (MCT4C)
Endless practice · MCT4C
Generated fresh every time — adaptive difficulty, never runs out
Solve exponential equations using a common base
Solve an exponential equation b^(px+q) = N algebraically by rewriting N as a power of b.
Solve growth/decay models for time using logarithms
Solve A0(1 +- i)^t = A0*M for t using logarithms, applied to a tech growth or decay context.
Volume and surface area of a composite tank
Find the volume or curved surface area of a composite cylinder-and-hemisphere tank.
Find the zeros of a polynomial in factored form
Read the zeros of a cubic function from its factored form, including a repeated (multiplicity-2) root at higher difficulty.
Solve a polynomial volume equation by testing values
Solve a cubic volume equation x(L-2x)(W-2x) = V for the whole-number side length x by systematic testing.
Resultant of two force vectors
Use the law of cosines (R^2 = F1^2 + F2^2 + 2F1F2*cos(theta)) to find the magnitude of the resultant of two force vectors.
Evaluate a sinusoidal model with a phase shift
Evaluate h(x) = a*sin(x - c) + d0 at a given angle to model a periodic reading from a rotating machine part.
Calculus & Vectors (MCV4U)
Curve Sketching Optimization · MCV4U
6 question sets · 10 questions each, easiest to hardest
Critical Points and the First Derivative Test
Curve Sketching Optimization · 30 questions, easiest to hardest
Concavity and the Second Derivative Test
Curve Sketching Optimization · 30 questions, easiest to hardest
Points of Inflection
Curve Sketching Optimization · 30 questions, easiest to hardest
Comprehensive Curve Sketching
Curve Sketching Optimization · 30 questions, easiest to hardest
Optimization Problems — Geometric
Curve Sketching Optimization · 30 questions, easiest to hardest
Optimization Problems — Real-World Applications
Curve Sketching Optimization · 30 questions, easiest to hardest
Derivatives Transcendental · MCV4U
6 question sets · 10 questions each, easiest to hardest
Derivatives of Trigonometric Functions
Derivatives Transcendental · 30 questions, easiest to hardest
Derivatives of Exponential Functions
Derivatives Transcendental · 30 questions, easiest to hardest
Derivatives of Logarithmic Functions
Derivatives Transcendental · 30 questions, easiest to hardest
Combining Rules with Trig/Exponential/Log Functions
Derivatives Transcendental · 30 questions, easiest to hardest
Implicit Differentiation
Derivatives Transcendental · 30 questions, easiest to hardest
Applications (Related Rates)
Derivatives Transcendental · 30 questions, easiest to hardest
Intro Calculus · MCV4U
6 question sets · 10 questions each, easiest to hardest
Limits and Continuity
Intro Calculus · 30 questions, easiest to hardest
The Derivative from First Principles
Intro Calculus · 30 questions, easiest to hardest
Power Rule and Basic Derivative Rules
Intro Calculus · 30 questions, easiest to hardest
Product Rule and Quotient Rule
Intro Calculus · 30 questions, easiest to hardest
Chain Rule
Intro Calculus · 30 questions, easiest to hardest
Applications of Derivatives (Tangent Lines, Rates of Change)
Intro Calculus · 30 questions, easiest to hardest
Lines Planes 3D · MCV4U
6 question sets · 10 questions each, easiest to hardest
Vector and Parametric Equations of Lines
Lines Planes 3D · 30 questions, easiest to hardest
Equations of Planes (Vector, Parametric, Scalar/Cartesian)
Lines Planes 3D · 30 questions, easiest to hardest
Intersections of Lines and Lines
Lines Planes 3D · 30 questions, easiest to hardest
Intersections of Lines and Planes, and Planes and Planes
Lines Planes 3D · 30 questions, easiest to hardest
Distance Problems (Point to Line, Point to Plane)
Lines Planes 3D · 30 questions, easiest to hardest
Comprehensive Applications
Lines Planes 3D · 30 questions, easiest to hardest
Vectors · MCV4U
6 question sets · 10 questions each, easiest to hardest
Introduction to Vectors (Magnitude, Direction, Representation)
Vectors · 30 questions, easiest to hardest
Vector Operations (Addition, Subtraction, Scalar Multiplication)
Vectors · 30 questions, easiest to hardest
Vectors in 2D and 3D (Components, Unit Vectors)
Vectors · 30 questions, easiest to hardest
The Dot Product
Vectors · 30 questions, easiest to hardest
The Cross Product
Vectors · 30 questions, easiest to hardest
Applications of Vectors (Force, Velocity, Work)
Vectors · 30 questions, easiest to hardest
Endless practice · MCV4U
Generated fresh every time — adaptive difficulty, never runs out
Power rule for derivatives
Differentiate a polynomial using the power, constant multiple, and constant rules.
Product rule
Differentiate a product of a linear and a quadratic factor using the product rule.
Quotient rule
Differentiate a ratio of two linear expressions using the quotient rule.
Chain rule
Differentiate a power of a linear expression using the chain rule.
Derivatives of exponential and trigonometric functions
Differentiate a*e^(kx), a*sin(kx), or a*cos(kx).
Equation of a tangent line
Find the equation of the tangent line to a polynomial curve at a given point.
Increasing/decreasing intervals and local extrema
Use the first derivative of a cubic to find and classify its local extrema.
Optimization word problems
Use derivatives to solve maximum-area and maximum-volume word problems with clean answers.
Velocity and acceleration from a position function
Differentiate a position function to find velocity, and differentiate again for acceleration.
Vector magnitude and direction
Find the magnitude of a vector (Pythagorean triples) or the direction angle of an axis-aligned/diagonal vector.
Vector addition, subtraction, and scalar multiplication
Compute a combination of vector addition, subtraction, and scalar multiplication in component form.
Dot product and the angle between two vectors
Compute the dot product of two vectors, and use it to identify perpendicular or parallel vectors.
Cross product of two vectors
Compute the cross product of two 3D vectors, and use it to find the area of a parallelogram.
Vector equation of a line and intersection of two lines
Write the parametric equation of a line through two points, evaluate a point on it, or find where two lines intersect.
Scalar equation of a plane
Find the value of D in the scalar equation ax + by + cz = D of a plane, given a normal vector or two direction vectors.
Average rate of change over an interval
Compute the average rate of change (secant slope) of a quadratic function over an interval.
Evaluating limits
Evaluate limits by direct substitution and by factoring to remove a 0/0 indeterminate form.
Data Management (MDM4U)
Counting Probability · MDM4U
6 question sets · 10 questions each, easiest to hardest
Fundamental Counting Principle
Counting Probability · 30 questions, easiest to hardest
Permutations
Counting Probability · 30 questions, easiest to hardest
Combinations
Counting Probability · 30 questions, easiest to hardest
Permutations with Repetition/Restrictions
Counting Probability · 30 questions, easiest to hardest
Probability Basics and Rules
Counting Probability · 30 questions, easiest to hardest
Conditional Probability and Independence
Counting Probability · 30 questions, easiest to hardest
Organizing Data · MDM4U
6 question sets · 10 questions each, easiest to hardest
Measures of Central Tendency
Organizing Data · 30 questions, easiest to hardest
Measures of Spread — Variance and Standard Deviation
Organizing Data · 30 questions, easiest to hardest
Percentiles and Quartiles
Organizing Data · 30 questions, easiest to hardest
Organizing Data — Frequency Tables and Grouped Data
Organizing Data · 30 questions, easiest to hardest
Standardized Scores and Comparing Data Sets
Organizing Data · 30 questions, easiest to hardest
Comprehensive Data Organization Applications
Organizing Data · 30 questions, easiest to hardest
Probability Distributions · MDM4U
6 question sets · 10 questions each, easiest to hardest
Discrete Random Variables and Probability Distributions
Probability Distributions · 30 questions, easiest to hardest
Expected Value
Probability Distributions · 30 questions, easiest to hardest
The Binomial Distribution
Probability Distributions · 30 questions, easiest to hardest
The Normal Distribution
Probability Distributions · 30 questions, easiest to hardest
Z-Scores and the Standard Normal Distribution
Probability Distributions · 30 questions, easiest to hardest
Applications of Probability Distributions
Probability Distributions · 30 questions, easiest to hardest
Statistical Analysis · MDM4U
6 question sets · 10 questions each, easiest to hardest
Correlation and the Correlation Coefficient
Statistical Analysis · 30 questions, easiest to hardest
Line of Best Fit and Linear Regression
Statistical Analysis · 30 questions, easiest to hardest
Sampling Distributions and Margin of Error
Statistical Analysis · 30 questions, easiest to hardest
Confidence Intervals
Statistical Analysis · 30 questions, easiest to hardest
Hypothesis Testing Basics
Statistical Analysis · 30 questions, easiest to hardest
Comprehensive Statistical Analysis Applications
Statistical Analysis · 30 questions, easiest to hardest
Endless practice · MDM4U
Generated fresh every time — adaptive difficulty, never runs out
Fundamental Counting Principle
Use the Fundamental Counting Principle to count outcomes across a sequence of independent choices.
Permutations of distinct objects
Count ordered arrangements of r objects chosen from n distinct objects using permutations, P(n, r).
Combinations
Count unordered selections of r objects chosen from n distinct objects using combinations, C(n, r).
Pascal's triangle applications
Apply Pascal's Rule and the row-sum and symmetry properties of Pascal's triangle.
Probability using counting techniques
Compute probabilities from a discrete sample space by counting favourable and total outcomes with combinations.
Data concepts, sampling, and bias
Classify data as categorical or quantitative, identify sampling methods, and recognize sources of bias in a study.
Conditional probability
Compute conditional probability P(A | B) from a two-way survey summary.
Binomial probability distribution
Compute P(X = k) for a binomial random variable with n independent trials and success probability p.
Hypergeometric probability distribution
Compute probabilities for sampling without replacement using the hypergeometric distribution.
Discrete uniform distributions
Compute probabilities and expected value for a discrete uniform random variable.
Mean, median, mode, and standard deviation
Compute the mean, median, mode, or standard deviation of a small, clean one-variable data set.
Z-scores
Convert between raw scores and z-scores using z = (x - mean) / sd.
Normal distribution: the 68-95-99.7 rule
Apply the 68-95-99.7 rule to estimate percentages and counts for a normally distributed variable.
Two-variable data: line of best fit and correlation
Use a line of best fit to predict a value, or classify a correlation coefficient by strength and direction.
Correlation vs. causation
Evaluate a described study or correlation and choose the most defensible conclusion about causation.
Math for Work & Everyday Life (MEL4E)
Endless practice · MEL4E
Generated fresh every time — adaptive difficulty, never runs out
Balance a monthly budget
Total a list of monthly expenses and find how much of a net income is left over (or overspent).
Budget guideline percentage
Compare actual spending in a category to a recommended percent-of-income guideline.
Refund or balance owing
Compute tax owed from a two-bracket rate table and compare it to tax already withheld to find a refund or balance owing.
How much paint to buy
Use a coverage rate and proportional reasoning to find how many whole units of material are needed for a job, rounding up.
Renting vs. owning — monthly cost
Compare the total monthly cost of owning (mortgage + property tax + maintenance) to renting.
Advanced Functions (MHF4U)
Characteristics Functions · MHF4U
6 question sets · 10 questions each, easiest to hardest
Combining Functions (Sum, Difference, Product, Quotient)
Characteristics Functions · 30 questions, easiest to hardest
Composition of Functions
Characteristics Functions · 30 questions, easiest to hardest
Average Rate of Change
Characteristics Functions · 30 questions, easiest to hardest
Instantaneous Rate of Change (Introduction to Derivatives)
Characteristics Functions · 30 questions, easiest to hardest
Increasing/Decreasing Intervals and Function Behavior
Characteristics Functions · 30 questions, easiest to hardest
Comprehensive Applications Combining Multiple Function Types
Characteristics Functions · 30 questions, easiest to hardest
Exponential Logarithmic · MHF4U
6 question sets · 10 questions each, easiest to hardest
Introduction to Logarithms
Exponential Logarithmic · 30 questions, easiest to hardest
Laws of Logarithms
Exponential Logarithmic · 30 questions, easiest to hardest
Solving Exponential Equations Using Logarithms
Exponential Logarithmic · 30 questions, easiest to hardest
Solving Logarithmic Equations
Exponential Logarithmic · 30 questions, easiest to hardest
Graphing Logarithmic Functions and Transformations
Exponential Logarithmic · 30 questions, easiest to hardest
Applications of Exponential and Logarithmic Functions
Exponential Logarithmic · 30 questions, easiest to hardest
Polynomial Functions · MHF4U
6 question sets · 10 questions each, easiest to hardest
Polynomial Functions — Characteristics and End Behavior
Polynomial Functions · 30 questions, easiest to hardest
Factoring Polynomials (Factor Theorem, Remainder Theorem)
Polynomial Functions · 30 questions, easiest to hardest
Solving Polynomial Equations and Inequalities
Polynomial Functions · 30 questions, easiest to hardest
Graphing Polynomial Functions
Polynomial Functions · 30 questions, easiest to hardest
Transformations of Polynomial Functions
Polynomial Functions · 30 questions, easiest to hardest
Applications of Polynomial Functions
Polynomial Functions · 30 questions, easiest to hardest
Rational Functions · MHF4U
6 question sets · 10 questions each, easiest to hardest
Simplifying Rational Expressions and Identifying Restrictions
Rational Functions · 30 questions, easiest to hardest
Vertical and Horizontal Asymptotes
Rational Functions · 30 questions, easiest to hardest
Graphing Rational Functions (Including Holes)
Rational Functions · 30 questions, easiest to hardest
Solving Rational Equations
Rational Functions · 30 questions, easiest to hardest
Solving Rational Inequalities
Rational Functions · 30 questions, easiest to hardest
Applications of Rational Functions
Rational Functions · 30 questions, easiest to hardest
Trigonometric Functions · MHF4U
6 question sets · 10 questions each, easiest to hardest
Reciprocal Trigonometric Functions (csc, sec, cot)
Trigonometric Functions · 30 questions, easiest to hardest
Pythagorean and Basic Trigonometric Identities
Trigonometric Functions · 30 questions, easiest to hardest
Sum and Difference Identities
Trigonometric Functions · 30 questions, easiest to hardest
Double Angle Identities
Trigonometric Functions · 30 questions, easiest to hardest
Proving Trigonometric Identities
Trigonometric Functions · 30 questions, easiest to hardest
Solving Advanced Trigonometric Equations
Trigonometric Functions · 30 questions, easiest to hardest
Endless practice · MHF4U
Generated fresh every time — adaptive difficulty, never runs out
Average and instantaneous rate of change
Find the average rate of change over an interval, and estimate the instantaneous rate of change at a point using a very small interval.
Function composition
Evaluate and build formulas for composite functions (f of g)(x), including a real-world discount-then-tax context.
Evaluate logarithmic expressions
Evaluate log_b(x) exactly, including negative and fractional exponent cases.
Laws of logarithms
Combine logarithms with the product, quotient, and power laws, then evaluate the result exactly.
Solving exponential equations with a common base
Solve exponential equations by rewriting both sides with the same base and matching exponents.
Solving exponential equations with logarithms
Solve exponential equations that don't share a common base by taking the logarithm of both sides.
Solving logarithmic equations
Solve logarithmic equations, including ones that combine into a quadratic and need an extraneous root check.
Remainder theorem
Use the remainder theorem to find the remainder when a polynomial is divided by (x - k), without doing long division.
Factor theorem and factoring a cubic completely
Use the factor theorem to test a possible root, then factor a cubic completely.
Solving polynomial equations by factoring
Solve a factorable polynomial equation for all real roots.
Solving polynomial inequalities
Solve a factored polynomial inequality using a sign chart / test-interval analysis.
Vertical and horizontal asymptotes of rational functions
Find the vertical and horizontal asymptotes of a rational function from its equation.
Radian measure and conversion
Convert between degrees and radians, and solve arc length problems using radian measure.
Exact trigonometric values at special angles (radians)
Find the exact value of sin, cos, or tan at special angles given in radians, in any quadrant.
Amplitude, period, and values of transformed trig functions
Identify the amplitude and period (in radians) of a transformed sine or cosine function, and evaluate it at a given input.
Compound angle formulas
Use the compound angle (sum/difference) formulas to find the exact value of sin or cos at a non-special angle.
Trigonometric identities and solving trigonometric equations
Simplify expressions using the Pythagorean/double-angle/quotient identities, and solve trig equations exactly on [0, 2pi).
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