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TopicAlgebra
Easy
Solve: 3x + 7 = 22
Find the value of x that makes this linear equation true.
Hint — Step 1
Subtract 7 from both sides to isolate the term with x. Then divide both sides by 3. Answer: x = 5.
Medium
Solve: x² - 5x + 6 = 0
Find all values of x that satisfy this quadratic equation.
Hint — Method
Factor into (x - 2)(x - 3) = 0. Set each factor to zero: x = 2 or x = 3.
Medium
Simplify: (2x³y²) × (3x²y)
Multiply these expressions and write in simplest form.
Hint
Multiply coefficients: 2 × 3 = 6. Add exponents: x³⁺² = x⁵, y²⁺¹ = y³. Answer: 6x⁵y³.
Hard
Solve: |2x - 3| = 7
Find all values of x that satisfy this absolute value equation.
Key Concept
Split into two cases: 2x - 3 = 7 (x = 5) and 2x - 3 = -7 (x = -2). Both are solutions.
TopicGeometry
Easy
Area of a circle: r = 5 cm
Find the area of a circle with radius 5 cm. Leave in terms of pi or as a decimal.
Formula
A = pi × r². Substitute r = 5: A = 25pi ≈ 78.54 cm².
Easy
Right triangle: sides 3 cm, 4 cm. Find hypotenuse.
Apply the Pythagorean theorem to find the missing side.
Theorem
c² = 3² + 4² = 9 + 16 = 25. So c = 5 cm. This is the famous 3-4-5 right triangle.
Medium
Volume of a cylinder: r = 4, h = 9
Calculate the volume of a cylinder with these dimensions.
Formula
V = pi × r² × h = pi × 16 × 9 = 144pi ≈ 452.4 cm³.
Hard
Interior angle sum of a heptagon (n = 7)
Find the sum of all interior angles in a 7-sided polygon.
Formula
Sum = (n - 2) × 180° = (7 - 2) × 180° = 5 × 180° = 900°.
TopicTrigonometry
Easy
Find the exact value of sin(30°)
Use the unit circle or special triangle to find this value.
Key Values
sin(30°) = 1/2. Learn the special angles 30°, 45°, 60° for all three trig ratios.
Medium
Solve: 2sin(x) - 1 = 0, for 0 ≤ x ≤ 360°
Find all values of x in the given interval.
Approach
sin(x) = 0.5. Reference angle = 30°. Sine is positive in Q1 and Q2: x = 30° and x = 150°.
Medium
Prove: sin²x + cos²x = 1
Show this fundamental identity using the unit circle definition.
Origin
On the unit circle, any point is (cos x, sin x). By Pythagoras: cos²x + sin²x = 1²= 1.
Hard
Solve: tan(2x) = 1, for 0 ≤ x < 180°
Find all solutions in the given range for this double-angle equation.
Method
Let u = 2x. tan(u) = 1 gives u = 45° + n×180°. So x = 22.5° or x = 112.5° (within range).
TopicStatistics
Easy
Data: 4, 7, 2, 9, 3, 7, 5 — find the mean
Calculate the arithmetic mean (average) of this data set.
Formula
Mean = sum ÷ count = (4+7+2+9+3+7+5) ÷ 7 = 37 ÷ 7 ≈ 5.29.
Medium
Standard deviation of: 2, 4, 4, 4, 5, 5, 7, 9
Calculate the population standard deviation for this data set.
Steps
Mean = 5. Squared deviations: 9,1,1,1,0,0,4,16. Average = 4. SD = sqrt(4) = 2.
Medium
P(A) = 0.4, P(B) = 0.3. Find P(A or B) if mutually exclusive.
Calculate the probability that A or B occurs.
Rule
For mutually exclusive events: P(A or B) = P(A) + P(B) = 0.4 + 0.3 = 0.7.
Hard
Z-score: x = 80, mean = 75, SD = 5
Calculate the z-score and interpret what it means.
Formula
z = (x - mean) / SD = (80 - 75) / 5 = 1.0. This score is exactly 1 standard deviation above the mean.
TopicCalculus
Easy
Differentiate: f(x) = 3x² + 5x - 2
Find f'(x) using the power rule for each term.
Power Rule
d/dx[xⁿ] = n·xⁿ⁻¹. So f'(x) = 6x + 5.
Easy
Find: integral of (2x + 3) dx
Evaluate this indefinite integral. Include the constant of integration.
Reverse Power Rule
Integrate each term: 2x → x², 3 → 3x. Answer: x² + 3x + C.
Medium
Differentiate: y = x · sin(x)
Use the product rule to differentiate this expression.
Product Rule
d/dx[u·v] = u'v + uv'. Here u=x (u'=1) and v=sin(x) (v'=cos(x)). Answer: sin(x) + x·cos(x).
Hard
Evaluate: definite integral from 0 to 2 of (x² + 1) dx
Compute this definite integral and find the exact value.
FTC
Antiderivative: x³/3 + x. Apply F(2) - F(0) = (8/3 + 2) - 0 = 14/3.
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