Kindergarten

□ Click to Reveal Kindergarten Answer Key

• Counting Objects: 6
• Number Recognition: 7
• Comparing Sizes: The tree
• Identifying Shapes: Circle
• Spatial Position: Table
• Sorting Patterns: Blue
• Number Sequencing: 4
• Basic Coin Looks: Penny
• Beginning Addition: 3

Ordering Numbers

Rewrite these numbers in order from least to greatest:

12, 20, 11

Fractions

Color half of each shape.

Graphing Data

Draw a bar graph representing the favorite colors of 10 random people. Remember to label.

Data: Red (2), Blue (1), Yellow (3)

Geometry

Shape: Cube

• How many faces: ____
• How many edges: ____
• How many vertices: ____

Positions

Draw a red apple to the RIGHT of the cow. Then, draw a yellow star ABOVE the cow.

Time

What time is it?

[Clock face showing 11:00]

Place Value / Number Sense

How much?
[Counting sets exercise placeholder]

Money

Match the values to the coin names:

• Dime, Penny, Nickel, Quarter
• $.25, $.10, $.01, $.05

Operations

Solve the math strings:

• 4 + 6 = ___
• 10 - 3 = ___
• The sum of 5 and 5 = ___

Grade 1 

□ Click to Reveal Grade 1 Answer Key

• Ordering Numbers: 19, 32, 45
• Fractions: One-half (or half / 1/2)
• Graphing Data: 1 more person (6 - 5 = 1)
• Geometry: 3 straight sides | 3 corners (vertices)
• Positions: Ball is underneath the table; box is directly next to it.
• Time: Long hand = minutes | Short hand = hours
• Place Value: 2 tens and 4 ones
• Money: 13¢
• Operations: 8 + 5 = 13 | 12 - 4 = 8 | 10 + 7 = 17

Ordering Numbers

Rewrite these numbers in order from least to greatest:

45, 19, 32

_____, _____, _____

Fractions

If you cut a sweet pizza pie directly down the middle into 2 equal halves, what do we call one of those parts?

A _______________

Graphing Data

Look at our tally chart totals:

• Cats: ||||
• Dogs: |||| |

How many more people picked dogs than cats?
_______

Geometry

Shape: Triangle

• How many straight sides: ____
• How many corners (vertices): ____

Positions

Draw a small ball UNDER the table. Then, draw a box NEXT TO the table.

Time

Write down the missing time sequence numbers:

The long hand on a clock tells the _______________. The short hand tells the _______________.

Place Value / Number Sense

Break down the mystery number:

24 = ____ tens and ____ ones

Money

Count up the shiny coins:

How many total cents do you have if you hold 1 dime and 3 pennies?

_______¢

Operations

Solve the equation strings:

• 8 + 5 = ___
• 12 - 4 = ___
• 10 + 7 = ___

Grade 2

□ Click to Reveal Grade 2 Answer Key

• Ordering Numbers: 111, 112, 120
• Fractions: 3/4 (Three-quarters)
• Graphing Data: Red (5 blocks), Blue (3 blocks), Yellow (4 blocks)
• Geometry: A. triangle | B. quadrilateral
• Positions: Tree on left side of house; sun floating directly above roof
• Time: 1:45 (Quarter to two)
• Place Value: 50 (The digit 5 is in the tens place)
• Money: 63¢ (50¢ + 10¢ + 3¢)
• Operations: 14 + 16 = 30 | 50 - 25 = 25 | Sum of 12 and 12 = 24

Ordering Numbers

Rewrite these numbers in order from least to greatest:

112, 120, 111

Fractions

Divide the circle into 4 equal quadrants. Shade 3 of them. What fraction is shaded?

_______

Graphing Data

Draw a bar graph representing the favorite colors of 12 random people. Remember to label.

Data: Red (5), Blue (3), Yellow (4)

Geometry

Identify if each shape is a triangle or a quadrilateral:

A. _____

B. _____

Positions

Draw a green tree to the LEFT of the house. Then, draw a yellow sun ABOVE the house.

Time

What time is shown on the clock face below?

[ Clock time reads: 1:45 ]

_______

Place Value / Number Sense

Write the value of the underlined digit:

152 = _______

Money

How much money do you have in total if you hold:

2 quarters, 1 dime, and 3 pennies?

$_______

Operations

Solve the math strings:

• 14 + 16 = ___
• 50 - 25 = ___
• The sum of 12 and 12 = ___

Grade 3

□ Click to Reveal Grade 3 Answer Key

• Adding & Subtracting: 4572 + 963 = 5535 | 1349 - 234 = 1115 | 23 + 45 + 47 + 52 = 167
• Fractional Parts: Circle shaded with 3 out of 4 segments; Fraction = 3/4
• Graphing Data: Heights must hit Red (10), Blue (4), Green (8), Yellow (8)
• Geometry: Faces = 6 | Edges = 12 | Vertices = 8
• Equal Groups: Small rectangles = 12 | Challenge combinations = 30
• Time: 1:45
• Place Value: Standard = 4,347 | Words = Four thousand, three hundred forty-seven
• Measurement Word Problem: 68 meters (45 + 23)
• Money Word Problems: 70¢ or $0.70 (85¢ - 15¢)

Adding & Subtracting

Solve the expressions:

• 4572 + 963 = _______
• 1349 – 234 = _______
• 23 + 45 + 47 + 52 = _______

Fractional Parts

Cut the circle into 4 equal parts. Shade in 3 of them and write a fraction that represents the shaded part.

Fraction: _______

Graphing Data

Draw a bar graph representing the Favorite Colors of a Random 30 people. Remember to label.

Color	Red	Blue	Green	Yellow
# of people	10	4	8	8

Geometry

Shape: 3D Object

• How many faces: _______
• How many edges: _______
• How many vertices: _______

Equal Groups

How many small rectangles?
_______

Challenge: How many rectangles in all?
_______

Time

What time is it?

_______

Place Value

Given: 4000 + 300 + 40 + 7

Standard Form: ________________

Words: ____________________________________

Measurement Word Problem

Joey ran 45 meters. Sarah ran 23 more meters than Joey. How many meters did Sarah run?

_______ meters

Money Word Problems

Mary found two quarters and three dimes and one nickel, then gave her little sister 15¢. How much money does Mary have now?

$_______

Grade 4

□ Click to Reveal Grade 4 Answer Key

• Value and Place Value: Value of 5 = 500 | Position of 3 = Thousands place
• Fractions: Equal to (2/3 = 4/6)
• Long Multiplication: 540 (45 × 12)
• Geometry: Parallel lines drawn equidistant apart; they will never intersect.
• Operations Mixed: 345 + 963 = 1308 | 1349 - 234 = 1115 | 144 ÷ 12 = 12
• Time: 6:50 PM (4:15 + 2 hours 35 mins)
• Number Patterns: Missing terms = 32, 64 | Rule = Multiply by 2
• Measurement Conversions: 4 meters = 400 cm | 3 gallons = 12 quarts
• Money & Word Problems: $8.00 change ($20 - $12)

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Value and Place Value

Given the number: 43,572

• What is the value of the 5? _______
• What place value position is the 3 in? ______________

Fractions

Compare using less than, greater than, or equal to words:

2/3 is ____________________ 4/6

Long Multiplication

Find the product:

45 × 12 = _______

Geometry

Draw a set of parallel lines. Then, write a brief sentence explaining what makes them parallel.

Explanation: _____________________________________

Operations Mixed

Solve the expressions:

• 345 + 963 = _______
• 1349 – 234 = _______
• 144 ÷ 12 = _______

Time

If a movie starts at 4:15 PM and lasts for 2 hours and 35 minutes, what time does the movie end?

_______ PM

Number Patterns

Find the missing numbers in the sequence and determine the pattern rule:

4, 8, 16, ____, ____, 128

Rule: ____________________

Measurement Conversions

Convert the units:

• 4 meters = _______ centimeters
• 3 gallons = _______ quarts

Money & Multi-Step Word Problems

Sam bought 3 books for $4 each. He paid with a $20 bill. How much change should he receive back?

$_______

Grade 5

□ Click to Reveal Grade 5 Answer Key

• Place Value & Decimals: Value of 2 = 0.02 (two-hundredths) | Position of 9 = Thousandths place
• Fractions (Unlike Denominators): 11/12 (8/12 + 3/12)
• Long Division with Remainders: 37 3/4 (or 37 R 9, or 37.75)
• Geometry (Volume): 72 cm³ (6 × 4 × 3)
• Algebraic Expressions: 55 (20 × 3 - 5)
• Coordinate Graphing: (4, 7)
• Number Patterns & Relationships: 3, 7, 15 (Multiply by 2, then add 1)
• Measurement (Metric Conversions): 4.5 kilometers = 4500 meters | 3500 milliliters = 3.5 liters
• Multi-Step Word Problems (Decimals): $1.75 change ($20.00 - $18.25)

Place Value & Decimals

Given the number: 435.729

• What is the value of the 2? _______
• What place value position is the 9 in? ______________

Fractions (Unlike Denominators)

Find the sum. Reduce your answer to its simplest form:

2/3 + 1/4 = _______

Long Division with Remainders

Find the quotient. Express your remainder as a fraction:

453 ÷ 12 = _______

Geometry (Volume)

A rectangular prism has a length of 6 cm, a width of 4 cm, and a height of 3 cm. Find its total volume.

Volume = _______ cm³

Algebraic Expressions

Evaluate the expression following the proper order of operations:

(14 + 6) × 3 – 15 ÷ 3 = _______

Coordinate Graphing

Start at the origin (0,0). Move 4 units right along the x-axis and 7 units up along the y-axis. What ordered pair balances this point?

( ____ , ____ )

Number Patterns & Relationships

Given the rule "Multiply by 2, then add 1" starting at 1:

Write out the first 4 terms of the sequence:
1, ____ , ____ , ____

Measurement (Metric Conversions)

Convert the structural metrics:

• 4.5 kilometers = _______ meters
• 3500 milliliters = _______ liters

Multi-Step Word Problems (Decimals)

Sam bought 3 notebooks for $4.25 each and a pack of pens for $5.50. If he paid with a $20 bill, how much change does he get back?

$_______

Grade 6

□ Click to Reveal Grade 6 Answer Key

• Ratios and Rates: Ratio = 3:2 | Sugar needed = 6 cups
• Division of Fractions: 1 1/2 (or 3/2)
• Rational Numbers & Integers: -3, -2.5, -1.2, 1/2, 0.75
• Expressions & Exponents: 66 (Substitute x=4: 48 + 20 - 2)
• Equations & Inequalities: m = 8 | Graph features open circle at -2 pointing right.
• Geometry (Area of Triangles): 20 square inches (1/2 × 8 × 5)
• Data & Statistics: Median = 18 | Mean = 19
• Dependent & Independent Variables: Equation: C = 1.50m + 3
• Percent Word Problems: $45.00 ($60 - $15 sale discount)

Ratios and Rates

A recipe calls for 3 cups of flour for every 2 cups of sugar.

• Write the ratio of flour to sugar: _______
• If you use 9 cups of flour, how many cups of sugar do you need? _______

Division of Fractions

Find the quotient. Write your answer in simplest form:

(3/4) ÷ (1/2) = _______

Rational Numbers & Integers

Order these rational numbers from least to greatest:

-2.5, 0.75, -3, 1/2, -1.2

Answer: _____________________________________

Expressions & Exponents

Evaluate the expression when x = 4:

3x² + 5x – 2 = _______

Equations & Inequalities

Solve for variables or graph the parameters:

• Solve: 2.5m = 20 → m = _______
• Graph inequality bounds: x > -2

Geometry (Area of Triangles)

Find the area of a triangle with a base of 8 inches and a height of 5 inches.

Area = _______ square inches

Data & Statistics

Given the data set: 12, 15, 18, 12, 22, 25, 30

• Find the Median: _______
• Find the Mean: _______

Dependent & Independent Variables

An Uber ride costs a flat fee of $3.00 plus $1.50 per mile (m). Write an equation to find the total cost (C).

Equation: ____________________

Percent Word Problems

A jacket originally costs $60.00. It is on sale for 25% off. What is the sale price of the jacket before tax?

$_______

Grade 7

□ Click to Reveal Grade 7 Answer Key

• Integers & Rational Numbers: -15 + (-23) = -38 | 14 - (-8) = 22 | (-5) × (-9) = 45 | -48 ÷ 6 = -8
• Proportional Relationships: Unit rate = 50 mph | Equation: d = 50t
• Percents & Interest: Interest = $60.00 ($500 × 0.04 × 3)
• Expressions & Linear Equations: x = -7
• Multi-Step Inequalities: x ≥ -3 (Dividing by negative flips the operator)
• Geometry (Circles & Area): Circumference = 44 cm | Area = 154 cm²
• Sampling & Statistics: Predicted Students = 120 (0.3 × 400)
• Probability Modules: Probability = 1/6 (1/2 chance for Heads × 2/6 chance for 5 or 6)
• Angles & Geometric Conditions: Measure of Angle B = 55° (90° - 35°)

Integers & Rational Numbers

Simplify the expressions:

• -15 + (-23) = ____
• 14 - (-8) = ____

• (-5) × (-9) = ____
• -48 ÷ 6 = ____

Proportional Relationships

An object travels 150 miles in 3 hours at a constant speed.

• Find the unit rate: _______ mph
• Write an equation representing distance (d) over time (t): _________________

Percents & Interest

Calculate the total value:

Find the total simple interest earned on an investment of $500 at an annual rate of 4% over a span of 3 years.

Interest = $_______

Expressions & Linear Equations

Solve for the unknown variable:

3x + 7 = -14

x = _______

Multi-Step Inequalities

Solve the inequality and define its solution bounds:

-2x + 5 ≤ 11

Solution: ____________________

Geometry (Circles & Area)

A circle has a radius of 7 cm. Find its total metrics (Use π ≈ 22/7):

• Circumference = _______ cm
• Area = _______ cm²

Sampling & Statistics

A survey of 50 randomly selected students found that 15 preferred online learning. Out of 400 total students, predict how many prefer online learning.

Predicted Students = _______

Probability Modules

You flip a fair coin and roll a standard six-sided die simultaneously.

What is the exact theoretical probability of flipping heads and rolling a number greater than 4?

Probability = _______

Angles & Geometric Conditions

Angle A and Angle B are complementary. If the measure of Angle A is 35°, find the measure of Angle B.

Measure of Angle B = _______°

Grade 8

□ Click to Reveal Grade 8 Answer Key

• Exponents & Scientific Notation: Simplified = 3⁴ (or 81) | Standard Decimal = 0.00045
• Rational & Irrational Numbers: √25 = Rational (5) | π = Irrational | 0.333... = Rational (1/3)
• Linear Equations: x = -9
• Systems of Linear Equations: Intersection Point = (2, 5)
• Functions & Graphing: Slope (m) = 2 | Equation: y = 2x + 1
• Pythagorean Theorem: c = 10 cm (√(36 + 64))
• Geometry (Transformations): A' = (2, -7)
• Volume of 3D Shapes: Volume = 90π in³
• Bivariate Data: Temp vs Sales = Negative association | Study hours vs Scores = Positive association

Exponents & Scientific Notation

Simplify or convert the values:

• Simplify: (3² × 3⁵) ÷ 3³ = _______
• Write 4.5 × 10⁻⁴ in standard decimal form: _________________

Rational & Irrational Numbers

Identify each number as either Rational or Irrational:

• √25 : _________________
• π : _________________
• 0.333... : _________________

Linear Equations

Solve for the unknown variable x:

4x + 12 = 2x - 6

x = _______

Systems of Linear Equations

Solve the system of linear equations to find the intersection point (x, y):

y = 2x + 1
x + y = 7

Intersection Point = ( ____ , ____ )

Functions & Graphing

A linear equation passes through the coordinates (1, 3) and (3, 7). Find its characteristics:

• Find the slope (m): _______
• Write the equation in y = mx + b form: _________________

Pythagorean Theorem

A right triangle has leg lengths of a = 6 cm and b = 8 cm. Find the exact length of the hypotenuse (c).

c = _______ cm

Geometry (Transformations)

The point A(2, 4) is reflected across the x-axis, and then translated 3 units down. State the final coordinates of the transformed point A'.

A' = ( ____ , ____ )

Volume of 3D Shapes

Find the volume of a cylinder with a radius of 3 inches and a height of 10 inches. (Leave your answer in terms of π).

Volume = _______________ in³

Bivariate Data & Scatter Plots

State whether the given real-world relationships show a positive, negative, or no linear association:

• Outside temperature vs. hot chocolate sales: ______________
• Hours spent studying vs. test scores: ______________

Algebra 1

□ Click to Reveal Algebra I Answer Key

• Expressions & Equations: x = 3
• Linear Inequalities: Interval Notation = (-4, ∞)
• Absolute Value Equations: x = 7, -2
• Systems of Equations: Solution Pair = (3, -1)
• Exponent Rules & Radicals: Answer = 32x⁵
• Polynomial Operations: Answer = 2x³ + 5x² - 22x + 15
• Factoring Polynomials: Answer = (3x + 1)(x - 5)
• Quadratic Equations: x = 3, 1/2
• Rational Expressions: Simplified = (x - 3) / (x + 2) | Restrictions = x ≠ -2, -3

Expressions & Equations

Solve for the unknown variable x:

3(2x - 4) + 5 = 4x - 1

x = _______

Linear Inequalities

Solve the inequality and express the solution in interval notation:

-2(x - 3) < 14

Interval Notation: _________________

Absolute Value Equations

Find all possible real solutions for x:

|2x - 5| + 3 = 12

x = _________________

Systems of Equations

Solve the linear system using substitution or elimination:

3x - 2y = 11
y = 2x - 7

Solution Pair (x, y) = ( ____ , ____ )

Exponent Rules & Radicals

Simplify the expression completely leaving no negative exponents:

(4x³y⁻²)² × (2x⁻¹y⁴) = _________________

Polynomial Operations

Expand and simplify the product of the polynomials:

(2x - 3)(x² + 4x - 5)

Answer: _________________________________

Factoring Polynomials

Factor the quadratic expression completely:

3x² - 14x - 5

Answer: _________________________________

Quadratic Equations

Solve the equation using the quadratic formula or factoring:

2x² - 7x + 3 = 0

x = _______ , _______

Rational Expressions

Simplify the expression and state any restrictions on the variable x:

(x² - 9) / (x² + 5x + 6)

Simplified: _________________ (x ≠ _____, _____)

Geometry

□ Click to Reveal Geometry Answer Key

• Points, Lines, & Planes: x = 4 | BC = 15
• Angle Relationships: x = 20 | m∠1 = 75°
• Parallel Lines & Transversals: x = 30 (Supplementary consecutive interior angles)
• Triangle Congruence Proofs: Required statement = AC ≅ CE
• Similarity & Proportions: Height = 27 feet
• Right Triangle Trigonometry: cos(θ) = 12/13 | sin(θ) = 5/13
• Properties of Quadrilaterals: AC = 26
• Circle Theorems (Segments): Length = 6 cm
• Area & Perimeter Coordinates: Perimeter = 12 units

Points, Lines, & Planes

Points A, B, and C are collinear, and B is between A and C. If AB = 2x + 3, BC = 4x - 1, and AC = 26, solve for x and find the length of BC.

x = _______  |  BC = _______

Angle Relationships

∠1 and ∠2 are vertical angles. If m∠1 = (3x + 15)° and m∠2 = (5x - 25)°, find the value of x and the measure of both angles.

x = _______  |  m∠1 = _______°

Parallel Lines & Transversals

Two parallel lines are cut by a transversal. If a pair of consecutive interior angles are represented by (2x + 10)° and (3x + 20)°, solve for x.

x = _______

Triangle Congruence Proofs

Given: ΔABC with AB ≅ RC and ∠A ≅ ∠R. What additional piece of information is required to prove ΔABC ≅ ΔRCE by the SAS (Side-Angle-Side) postulate?

Required statement: _________________

Similarity & Proportions

A 6-foot tall person casts a 4-foot shadow. At the exact same time, a nearby flagpole casts an 18-foot shadow. Find the total height of the flagpole.

Height = _______ feet

Right Triangle Trigonometry

In a right triangle, the side adjacent to angle θ is 12 cm and the hypotenuse is 13 cm. Find the exact values for:

• cos(θ) = _______
• sin(θ) = _______

Properties of Quadrilaterals

In parallelogram ABCD, the diagonals intersect at point E. If AE = x + 5 and EC = 2x - 3, find the absolute length of the entire diagonal AC.

AC = _______

Circle Theorems (Segments)

Two chords intersect inside a circle. The segments of the first chord measure 4 cm and 9 cm. If one segment of the second chord measures 6 cm, find the length of its remaining segment.

Length = _______ cm

Area & Perimeter Coordinates

Find the perimeter of a triangle with coordinate vertices mapped at A(1, 1), B(1, 5), and C(4, 1) on a Cartesian plane.

Perimeter = _______ units

Algebra II

□ Click to Reveal Algebra II Answer Key

• Solving Linear Equations & Absolute Value: x = 5, -1
• Exponent Rules & Radical Expressions: Answer = 6x²y³√(2x)
• Complex Number Operations: Answer = 22 + 14i
• Quadratic Equations & Systems: x = 3, -1/2
• Polynomial Factoring & Division: Answer = (x - 2)(x + 2)(x + 3)
• Rational Expressions & Equations: Answer = [2(x - 3)] / x
• Logarithmic & Exponential Functions: x = 14
• Matrix Systems & Determinants: Determinant = 23
• Sequences & Series Patterns: a₁₀ = 68

Solving Linear Equations & Absolute Value

Solve for all real values of x:

|3x - 6| + 4 = 13

x = _________________

Exponent Rules & Radical Expressions

Simplify the radical expression completely:

√(72x&sup5;y&sup6;)

Answer: _________________

Complex Number Operations

Perform the multiplication and simplify your answer into standard a + bi form:

(4 - 2i)(3 + 5i)

Answer: _________________

Quadratic Equations & Systems

Solve the quadratic equation using any method (factoring, completing the square, or quadratic formula):

2x² - 5x - 3 = 0

x = _______ , _______

Polynomial Factoring & Division

Factor the cubic expression completely by grouping terms:

x³ + 3x² - 4x - 12

Answer: _________________

Rational Expressions & Equations

Simplify the product of the rational expressions:

[ (x² - 9) / (2x) ] × [ 4 / (x + 3) ]

Answer: _________________

Logarithmic & Exponential Functions

Solve for x by converting the equation to exponential form:

log³(2x - 1) = 3

x = _______

Matrix Systems & Determinants

Find the determinant of the 2×2 matrix:

| 4 -1 |
| 3 5 |

Determinant = _______

Sequences & Series Patterns

Find the 10th term of the arithmetic sequence given its parameters:

5, 12, 19, 26, ...

a₁₀ = _______

PreCalculus

□ Click to Reveal PreCalculus Answer Key

• Rational Functions Analysis: VA: x = 4, x = -1 | HA: y = 2
• Complex & Polynomial Roots: Roots = 2i, -2i, 3
• Trigonometric Functions & Identities: cos(θ) = -4/5 | tan(2θ) = 24/7
• Trigonometric Equations: x = π/6, 5π/6, 3π/2 over [0, 2π)
• Vectors & Polar Coordinates: (r, θ) = (6, 2π/3)
• Conic Sections: Center: (2, -1) | Foci: (6, -1) and (-2, -1)
• Matrix Systems: (x, y, z) = (1, 3, 1)
• Sequences & De Moivre's Theorem: Answer = -512 - 512i√3
• Limits Introduction: Limit Value = 3/2

Rational Functions Analysis

Given the rational function:
f(x) = (2x² - 8) / (x² - 3x - 4)

Find the equations of all vertical and horizontal asymptotes.

VA: _________________  |  HA: _________________

Complex & Polynomial Roots

Find all complex roots of the polynomial function given that x = 2i is a known zero:

P(x) = x³ - 3x² + 4x - 12

Roots: _________________

Trigonometric Functions & Identities

If sin(θ) = -3/5 and θ resides in Quadrant III, evaluate the exact values of:

• cos(θ) = _______
• tan(2θ) = _______

Trigonometric Equations

Solve the trigonometric equation over the real interval [0, 2π):

2 cos²(x) + sin(x) - 1 = 0

x = _________________

Vectors & Polar Coordinates

Convert the rectangular coordinates (-3, 3√3) into polar coordinates (r, θ), where r > 0 and 0 ≤ θ < 2π.

(r, θ) = ( ____ , ____ )

Conic Sections

An ellipse is defined by the following equation:
[ (x - 2)² / 25 ] + [ (y + 1)² / 9 ] = 1

Identify the coordinates of its center and its two foci.

Center: ( ____ , ____ )  |  Foci: _________________

Matrix Systems & Cramer's Rule

Solve the system of equations using matrix inverses or Gaussian elimination:

2x + y - z = 4
x - 2y + 3z = -6
3x + 3y + 2z = 14

(x, y, z) = ( ____ , ____ , ____ )

Sequences & De Moivre's Theorem

Use De Moivre's Theorem to find the exact simplified value of the complex expression:

( 1 + i √3 )¹°

Answer: _________________

Limits Introduction

Evaluate the algebraic limit analytically:

lim_x→3 [ (x² - 9) / (x² - 2x - 3) ]

Limit Value = _______

Calculus

□ Click to Reveal Calculus Answer Key

• Limits & Continuity: 5
• The Derivative Definition: 6x - 5
• Basic Derivative Rules: (2x⁴ + 6x² + 10x) / (x² + 1)²
• The Chain Rule: 12 sin²(4x) cos(4x)
• Implicit Differentiation: (2y - x²) / (y² - 2x)
• Applications of Derivatives: Absolute Max Value = 4
• Antiderivatives & Integration: x⁴ - 2x³ + x² - x + C
• Definite Integrals & FTC: 30
• Integration by U-Substitution: e^(x²) + C

Limits & Continuity

Evaluate the limit algebraically using factoring or rationalization:

lim_x→2 [ (x² + x - 6) / (x - 2) ]

Limit = _______

The Derivative Definition

Use the limit definition of the derivative to find f'(x) for the function:

f(x) = 3x² - 5x

f'(x) = _________________

Basic Derivative Rules

Find the first derivative dy/dx using Power, Product, or Quotient rules:

y = (2x³ - 5) / (x² + 1)

dy/dx = _________________

The Chain Rule

Find the derivative of the composite function with respect to x:

f(x) = sin³(4x)

f'(x) = _________________

Implicit Differentiation

Find dy/dx for the implicitly defined curve:

x³ + y³ = 6xy

dy/dx = _________________

Applications of Derivatives

Find the absolute maximum value of the function on the closed interval [-1, 2]:

f(x) = x³ - 3x + 2

Max Value = _______

Antiderivatives & Integration

Evaluate the indefinite integral. Don't forget the constant of integration:

∫ (4x³ - 6x² + 2x - 1) dx

Answer: _________________

Definite Integrals & FTC

Evaluate the definite integral using the Fundamental Theorem of Calculus:

∫₁³ (3x² + 2) dx

Value = _______

Integration by U-Substitution

Evaluate the integral using a change of variables (u-substitution):

∫ 2x · e^(x²) dx

Answer: _________________