Math Problem-Solving Guide: 800+ Practice Questions with Step-by-Step Solutions – Master Algebra, Calculus, Statistics, and Test Prep for College, High School, and Standardized Exams

Math Problem-Solving Guide: 800+ Practice Questions with Step-
by-Step Solutions – Master Algebra, Calculus, Statistics, and Test
Prep for College, High School, and Standardized Exams


Simplify: (3a²b³)(2ab¶)
A. 6a³b¹
B. 6a²b¹
C. 5a³b¹
D. 5a²b¹²
Working: Multiply coefficients: 3×2 = 6. Add exponents of a: 2+1=3. Add exponents
of b: 3+4=7.
Answer: A

Find the x-intercept of y = 2x - 6
A. (0, 3)
B. (3, 0)
C. (-3, 0)
D. (0, -6)
Working: Set y = 0: 0 = 2x - 6 → 2x = 6 → x = 3.
Answer: B

What is the slope-intercept form of the line with slope -2 and y-intercept 5?
A. y = -2x + 5
B. y = 2x - 5
C. y = -5x + 2
D. y = 5x - 2
Working: y = mx + b, where m = slope, b = y-intercept. Substitute m = -2, b = 5.
Answer: A

Solve: |x - 3| = 7
A. x = 4 or x = 10
B. x = -4 or x = 10
C. x = -4 or x = 4
D. x = 10 only
Working: x - 3 = 7 → x = 10. Also x - 3 = -7 → x = -4.
Answer: B

What is the domain of f(x) = 1/(x - 5)?
A. All real numbers
B. x ≠ 5
C. x > 5

,D. x < 5
Working: The denominator cannot be zero. x - 5 = 0 → x = 5. Exclude 5.
Answer: B

Find the y-intercept of y = 3x² - 4x + 1
A. (0, 1)
B. (1, 0)
C. (0, -1)
D. (-1, 0)
Working: Set x = 0: y = 3(0)² - 4(0) + 1 = 1.
Answer: A

Solve: 2x² + 5x - 3 = 0
A. x = 1/2 or x = -3
B. x = -1/2 or x = 3
C. x = 2 or x = -3
D. x = -2 or x = 3
Working: Factor: (2x - 1)(x + 3) = 0. Set each to zero: 2x - 1 = 0 → x = 1/2. x + 3 = 0
→ x = -3.
Answer: A

What is the derivative of f(x) = sin(x)?
A. cos(x)
B. -cos(x)
C. sin(x)
D. -sin(x)
Working: Standard derivative: d/dx[sin(x)] = cos(x).
Answer: A

Evaluate: ∫ (1/x) dx
A. ln|x| + C
B. eˣ + C
C. x ln(x) + C
D. 1/x² + C
Working: Standard integral: ∫ 1/x dx = ln|x| + C.
Answer: A

What is the variance of: 2, 4, 6, 8, 10?
A. 6
B. 8
C. 10
D. 12
Working: Mean = 6. Deviations: -4, -2, 0, 2, 4. Squares: 16, 4, 0, 4, 16. Sum = 40.

,Variance = 40/5 = 8.
Answer: B

A coin is flipped 3 times. What is the probability of getting exactly 2 heads?
A. 1/8
B. 3/8
C. 1/2
D. 5/8
Working: Total outcomes = 8. Favorable: HHT, HTH, THH = 3. Probability = 3/8.
Answer: B

Simplify: (x³y²)/(x²y·)
A. x/y³
B. x·/y¹
C. y³/x
D. x/y¹
Working: Subtract exponents: x³½² = x´, y²½· = y½³ = 1/y³. Result: x/y³.
Answer: A

What is the equation of a line with slope 3 passing through (1, 4)?
A. y = 3x + 1
B. y = 3x + 2
C. y = 3x + 3
D. y = 3x + 4
Working: Use point-slope: y - 4 = 3(x - 1) → y - 4 = 3x - 3 → y = 3x + 1.
Answer: A

Evaluate: log₂(8)
A. 2
B. 3
C. 4
D. 8
Working: 2³ = 8, so log₂(8) = 3.
Answer: B

What is the derivative of f(x) = eˣ?
A. eˣ
B. x eˣ
C. eˣ + C
D. 1/eˣ
Working: Standard derivative: d/dx[eˣ] = eˣ.
Answer: A

Find the area of a circle with radius 5.
A. 25π

, B. 10π
C. 5π
D. 50π
Working: Area = πr² = π(5)² = 25π.
Answer: A

Solve: 2x - 5 = 3x + 1
A. x = -6
B. x = 6
C. x = -4
D. x = 4
Working: 2x - 5 = 3x + 1 → -5 - 1 = 3x - 2x → -6 = x.
Answer: A

What is the mode of: 1, 2, 2, 3, 4, 4, 4, 5?
A. 2
B. 3
C. 4
D. 5
Working: Mode is the most frequent value. 4 appears three times.
Answer: C

What is the derivative of f(x) = x³?
A. 3x²
B. x²
C. 3x¶
D. 2x³
Working: Power rule: d/dx[x³] = 3x².
Answer: A

Evaluate: ∫ 4x dx from 0 to 2
A. 4
B. 8
C. 12
D. 16
Working: ∫4x dx = 2x². Evaluate at 2: 2(4)=8. At 0: 0. Difference = 8.
Answer: B

What is the probability of drawing an ace from a standard deck of 52 cards?
A. 1/13
B. 1/52
C. 4/13
D. 1/4