Question
Write 50 senior high school mathematics quizzing competition questions with their respective answers using step by step method
Answers
GPT-4o mini
Certainly! Below are 50 senior high school mathematics questions along with their answers that employ step-by-step methods. These questions cover various topics such as algebra, geometry, trigonometry, statistics, and calculus.
### Algebra
1. **Question:** Solve for \( x \): \( 2x + 3 = 11 \)
**Answer:**
Step 1: Subtract 3 from both sides:
\( 2x = 8 \)
Step 2: Divide by 2:
\( x = 4 \)
2. **Question:** Factor the quadratic \( x^2 - 5x + 6 \).
**Answer:**
Step 1: Look for two numbers that multiply to 6 and add to -5.
Step 2: The numbers are -2 and -3.
Step 3: Factor: \( (x - 2)(x - 3) \)
3. **Question:** Simplify \( \frac{3x^2 - 12}{3x} \).
**Answer:**
Step 1: Factor the numerator: \( 3(x^2 - 4) = 3(x - 2)(x + 2) \).
Step 2: Cancel \( 3x \):
\[ = \frac{(x - 2)(x + 2)}{x} = (x - 2)\frac{(x + 2)}{x} \]
4. **Question:** Solve for \( x \): \( 3(x - 1) + 4 = 10 \).
**Answer:**
Step 1: Distribute:
\( 3x - 3 + 4 = 10 \)
Step 2: Simplify:
\( 3x + 1 = 10 \)
Step 3: Subtract 1:
\( 3x = 9 \)
Step 4: Divide by 3:
\( x = 3 \)
5. **Question:** Find the roots of the equation \( x^2 + 6x + 8 = 0 \) using the quadratic formula.
**Answer:**
Step 1: Identify \( a = 1, b = 6, c = 8 \).
Step 2: Use quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-6 \pm \sqrt{36 - 32}}{2} = \frac{-6 \pm 2}{2} \)
Step 3: Solutions:
\( x = -2 \) or \( x = -4 \)
### Geometry
6. **Question:** Calculate the area of a circle with a radius of 5 cm.
**Answer:**
Step 1: Use the formula for the area \( A = \pi r^2 \):
\( A = \pi (5^2) = 25\pi\, \text{cm}^2 \)
7. **Question:** Find the volume of a cylinder with a radius of 3 cm and height of 10 cm.
**Answer:**
Step 1: Use the formula \( V = \pi r^2 h \):
\( V = \pi (3^2)(10) = 90\pi\, \text{cm}^3 \)
8. **Question:** What is the perimeter of a rectangle with length 8 cm and width 5 cm?
**Answer:**
Step 1: Use the perimeter formula \( P = 2(l + w) \):
\( P = 2(8 + 5) = 26\, \text{cm} \)
9. **Question:** Calculate the hypotenuse of a right triangle with legs 6 cm and 8 cm.
**Answer:**
Step 1: Use the Pythagorean theorem \( c = \sqrt{a^2 + b^2} \):
\( c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\, \text{cm} \)
10. **Question:** What is the area of a triangle with a base of 10 cm and height of 5 cm?
**Answer:**
Step 1: Use the area formula \( A = \frac{1}{2}bh \):
\( A = \frac{1}{2}(10)(5) = 25\, \text{cm}^2 \)
### Trigonometry
11. **Question:** What is \( \sin(30^\circ) \)?
**Answer:**
Step 1: From trigonometric values, \( \sin(30^\circ) = \frac{1}{2} \).
12. **Question:** Calculate \( \tan(45^\circ) \).
**Answer:**
Step 1: From trigonometric values, \( \tan(45^\circ) = 1 \).
13. **Question:** If \( \cos(\theta) = 0.6 \), find \( \sin(\theta) \).
**Answer:**
Step 1: Use \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\( \sin^2(\theta) + (0.6)^2 = 1 \)
Step 2: Solve for \( \sin^2(\theta) \):
\( \sin^2(\theta) + 0.36 = 1 \)
\( \sin^2(\theta) = 0.64 \)
Step 3: \( \sin(\theta) = \sqrt{0.64} = 0.8 \)
14. **Question:** Find \( \cot(60^\circ) \).
**Answer:**
Step 1: \( \cot(60^\circ) = \frac{1}{\tan(60^\circ)} = \frac{1}{\sqrt{3}} \).
15. **Question:** What is the value of \( \sec(0^\circ) \)?
**Answer:**
Step 1: \( \sec(0^\circ) = \frac{1}{\cos(0^\circ)} = 1 \).
### Statistics
16. **Question:** Calculate the mean of the following numbers: 4, 8, 6, 5.
**Answer:**
Step 1: Sum the numbers: \( 4 + 8 + 6 + 5 = 23 \).
Step 2: Divide by the number of values (4):
\( \text{Mean} = \frac{23}{4} = 5.75 \).
17. **Question:** What is the median of the data set: 5, 3, 9, 7, 2?
**Answer:**
Step 1: Arrange the data: 2, 3, 5, 7, 9.
Step 2: The median is the middle value:
\( \text{Median} = 5 \).
18. **Question:** Calculate the mode of the following set: 1, 2, 2, 3, 4.
**Answer:**
Step 1: Identify the most frequent number: 2.
\( \text{Mode} = 2 \).
19. **Question:** Find the standard deviation of the set: 2, 4, 4, 4, 5, 5, 7, 9.
**Answer:**
Step 1: Find the mean: \( (2+4+4+4+5+5+7+9)/8 = 5 \).
Step 2: Find the squared differences:
\( (2-5)^2, (4-5)^2, (4-5)^2, (4-5)^2, (5-5)^2, (5-5)^2, (7-5)^2, (9-5)^2 \)
Which gives: \( 9, 1, 1, 1, 0, 0, 4, 16 \).
Step 3: Mean of squared differences: \( (9 + 1 + 1 + 1 + 0 + 0 + 4 + 16) / 8 = 3 \).
Step 4: Standard deviation: \( \sqrt{3} \approx 1.73 \).
20. **Question:** What is the range of the set: 6, 2, 8, 4, 10?
**Answer:**
Step 1: Find maximum and minimum:
\( \text{Max} = 10, \text{Min} = 2 \).
Step 2: Calculate range:
\( \text{Range} = 10 - 2 = 8 \).
### Calculus
21. **Question:** Find the derivative of \( f(x) = 3x^2 + 5x - 4 \).
**Answer:**
Step 1: Use power rule:
\( f'(x) = 6x + 5 \).
22. **Question:** Evaluate the integral \( \int (4x^3) \, dx \).
**Answer:**
Step 1: Use power rule for integration:
\( \int (4x^3) \, dx = \frac{4}{4}x^4 + C = x^4 + C \).
23. **Question:** Find \( \lim_{x \to 0} \frac{\sin(x)}{x} \).
**Answer:**
Step 1: Apply the limit:
\( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \).
24. **Question:** Determine the second derivative of \( f(x) = x^3 - 3x + 2 \).
**Answer:**
Step 1: First derivative: \( f'(x) = 3x^2 - 3 \).
Step 2: Second derivative: \( f''(x) = 6x \).
25. **Question:** Find the area under the curve \( y = x^2 \) from \( x = 1 \) to \( x = 3 \).
**Answer:**
Step 1: Find the integral:
\( \int_1^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_1^3 \).
Step 2: Calculate:
\( \frac{3^3}{3} - \frac{1^3}{3} = \frac{27}{3} - \frac{1}{3} = 9 - \frac{1}{3} = \frac{26}{3} \).
### Functions
26. **Question:** What is the range of the quadratic function \( f(x) = x^2 - 4x + 3 \)?
**Answer:**
Step 1: Find the vertex:
\( x = -\frac{b}{2a} = -\frac{-4}{2 \cdot 1} = 2 \).
Step 2: Calculate \( f(2) = 2^2 - 4(2) + 3 = -1 \).
Step 3: Since it opens upwards, the range is \( [-1, \infty) \).
27. **Question:** Determine if \( f(x) = 2x + 3 \) is a one-to-one function.
**Answer:**
Step 1: Check if \( f(a) = f(b) \) implies \( a = b \):
\( 2a + 3 = 2b + 3 \Rightarrow 2a = 2b \Rightarrow a = b \).
Therefore, it is one-to-one.
28. **Question:** Find \( f(g(x)) \) if \( f(x) = x^2 \) and \( g(x) = x + 1 \).
**Answer:**
Step 1: Substitute:
\( f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1 \).
29. **Question:** What is the inverse of the function \( f(x) = 3x - 5 \)?
**Answer:**
Step 1: Set \( y = 3x - 5 \).
Step 2: Solve for \( x \):
\( y + 5 = 3x \Rightarrow x = \frac{y + 5}{3} \).
Step 3: Therefore, \( f^{-1}(y) = \frac{y + 5}{3} \).
30. **Question:** Is \( f(x) = |x| \) continuous everywhere?
**Answer:**
Step 1: Check for continuity at \( x = 0 \):
\( \lim_{x \to 0^-} f(x) = 0 \) and \( \lim_{x \to 0^+} f(x) = 0 \).
Since \( f(0) = 0 \), it is continuous.
### Sequences and Series
31. **Question:** What is the sum of the first 10 positive integers?
**Answer:**
Step 1: Use the formula for the sum \( S_n = \frac{n(n + 1)}{2} \):
\( S_{10} = \frac{10(11)}{2} = 55 \).
32. **Question:** Find the 5th term of the arithmetic sequence where \( a_1 = 3 \) and \( d = 2 \).
**Answer:**
Step 1: Use the formula \( a_n = a_1 + (n - 1)d \):
\( a_5 = 3 + (5 - 1)(2) = 3 + 8 = 11 \).
33. **Question:** What is the 4th term of the geometric sequence where \( a_1 = 2 \) and \( r = 3 \)?
**Answer:**
Step 1: Use the formula \( a_n = a_1 r^{n-1} \):
\( a_4 = 2 \cdot 3^{3} = 2 \cdot 27 = 54 \).
34. **Question:** Determine the common difference of the arithmetic sequence: 2, 5, 8, 11, ...
**Answer:**
Step 1: Subtract the first term from the second:
\( d = 5 - 2 = 3 \).
35. **Question:** Find the first 5 terms of the Fibonacci sequence.
**Answer:**
Step 1: Start with \( 0, 1 \):
\( 0, 1, 1(0+1), 2(1+1), 3(1+2) \)
Result: \( 0, 1, 1, 2, 3 \).
### Probability
36. **Question:** What is the probability of rolling a sum of 7 with two dice?
**Answer:**
Step 1: Identify favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
Step 2: Total outcomes = \( 6 \times 6 = 36 \).
Step 3: Probability = \( \frac{6}{36} = \frac{1}{6}\).
37. **Question:** If a card is drawn from a standard deck, what is the probability it is a heart?
**Answer:**
Step 1: Total hearts = 13.
Step 2: Probability = \( \frac{13}{52} = \frac{1}{4} \).
38. **Question:** Calculate the probability of flipping a coin and getting heads 3 times in a row.
**Answer:**
Step 1: Probability of heads = \( \frac{1}{2} \).
Step 2: Probability for 3 heads = \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \).
39. **Question:** What is the probability of choosing a day of the week that is either a weekend or Friday?
**Answer:**
Step 1: Total favorable outcomes = 2 (Saturday, Sunday) + 1 (Friday) = 3.
Step 2: Probability = \( \frac{3}{7} \).
40. **Question:** In a box of marbles, there are 5 red, 3 blue, and 2 green. What is the probability of drawing a blue marble?
**Answer:**
Step 1: Total marbles = \( 5 + 3 + 2 = 10 \).
Step 2: Probability = \( \frac{3}{10} \).
### Miscellaneous
41. **Question:** What is the 5th roots of 243?
**Answer:**
Step 1: Find \( 243 = 3^5 \).
Step 2: Therefore, \( \sqrt[5]{243} = 3 \).
42. **Question:** Solve for \( x \): \( 4^x = 64 \).
**Answer:**
Step 1: Rewrite 64 as \( 4^3 \).
Step 2: Set exponents equal: \( x = 3 \).
43. **Question:** Simplify \( (2 + 3i)(4 - i) \).
**Answer:**
Step 1: Use distributive property:
\( = 8 - 2i + 12i - 3i^2 \).
Step 2: Simplify:
\( = 8 + 10i + 3 = 11 + 10i \).
44. **Question:** What is the difference between the two square roots: \( \sqrt{25} - \sqrt{9} \)?
**Answer:**
Step 1: Calculate roots: \( \sqrt{25} = 5, \sqrt{9} = 3 \).
Step 2: Result: \( 5 - 3 = 2 \).
45. **Question:** Calculate the percentage increase from 50 to 80.
**Answer:**
Step 1: Calculate increase: \( 80 - 50 = 30 \).
Step 2: Percentage increase:
\( \frac{30}{50} \times 100 = 60\% \).
46. **Question:** If \( x^2 - 5x + 6 = 0 \), find the nature of the roots.
**Answer:**
Step 1: Calculate discriminant: \( D = b^2 - 4ac = 25 - 24 = 1 \).
Step 2: Since \( D > 0 \), roots are real and distinct.
47. **Question:** Solve for \( x \): \( \log(x) + \log(10) = 1 \).
**Answer:**
Step 1: Combine logs: \( \log(10x) = 1 \).
Step 2: Exponentiate: \( 10x = 10^1 \Rightarrow x = 1 \).
48. **Question:** Find the coordinates of the vertex of the parabola represented by \( y = x^2 - 4x + 1 \).
**Answer:**
Step 1: Vertex \( x = -\frac{b}{2a} = 2 \).
Step 2: Find \( y \):
\( y = 2^2 - 4(2) + 1 = 4 - 8 + 1 = -3 \).
Vertex is \( (2, -3) \).
49. **Question:** Evaluate \( \frac{1}{\frac{1}{3} + \frac{1}{6}} \).
**Answer:**
Step 1: Find a common denominator:
\( \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \).
Step 2: Invert:
Result: \( 2 \).
50. **Question:** What is the value of \( \sqrt{144} + \left| -8 \right| \)?
**Answer:**
Step 1: Calculate \( \sqrt{144} = 12 \).
Step 2: Calculate \( | -8 | = 8 \).
Step 3: Sum:
Result: \( 12 + 8 = 20 \).
Feel free to modify any of these questions or ask for more questions in specific areas!
### Algebra
1. **Question:** Solve for \( x \): \( 2x + 3 = 11 \)
**Answer:**
Step 1: Subtract 3 from both sides:
\( 2x = 8 \)
Step 2: Divide by 2:
\( x = 4 \)
2. **Question:** Factor the quadratic \( x^2 - 5x + 6 \).
**Answer:**
Step 1: Look for two numbers that multiply to 6 and add to -5.
Step 2: The numbers are -2 and -3.
Step 3: Factor: \( (x - 2)(x - 3) \)
3. **Question:** Simplify \( \frac{3x^2 - 12}{3x} \).
**Answer:**
Step 1: Factor the numerator: \( 3(x^2 - 4) = 3(x - 2)(x + 2) \).
Step 2: Cancel \( 3x \):
\[ = \frac{(x - 2)(x + 2)}{x} = (x - 2)\frac{(x + 2)}{x} \]
4. **Question:** Solve for \( x \): \( 3(x - 1) + 4 = 10 \).
**Answer:**
Step 1: Distribute:
\( 3x - 3 + 4 = 10 \)
Step 2: Simplify:
\( 3x + 1 = 10 \)
Step 3: Subtract 1:
\( 3x = 9 \)
Step 4: Divide by 3:
\( x = 3 \)
5. **Question:** Find the roots of the equation \( x^2 + 6x + 8 = 0 \) using the quadratic formula.
**Answer:**
Step 1: Identify \( a = 1, b = 6, c = 8 \).
Step 2: Use quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-6 \pm \sqrt{36 - 32}}{2} = \frac{-6 \pm 2}{2} \)
Step 3: Solutions:
\( x = -2 \) or \( x = -4 \)
### Geometry
6. **Question:** Calculate the area of a circle with a radius of 5 cm.
**Answer:**
Step 1: Use the formula for the area \( A = \pi r^2 \):
\( A = \pi (5^2) = 25\pi\, \text{cm}^2 \)
7. **Question:** Find the volume of a cylinder with a radius of 3 cm and height of 10 cm.
**Answer:**
Step 1: Use the formula \( V = \pi r^2 h \):
\( V = \pi (3^2)(10) = 90\pi\, \text{cm}^3 \)
8. **Question:** What is the perimeter of a rectangle with length 8 cm and width 5 cm?
**Answer:**
Step 1: Use the perimeter formula \( P = 2(l + w) \):
\( P = 2(8 + 5) = 26\, \text{cm} \)
9. **Question:** Calculate the hypotenuse of a right triangle with legs 6 cm and 8 cm.
**Answer:**
Step 1: Use the Pythagorean theorem \( c = \sqrt{a^2 + b^2} \):
\( c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\, \text{cm} \)
10. **Question:** What is the area of a triangle with a base of 10 cm and height of 5 cm?
**Answer:**
Step 1: Use the area formula \( A = \frac{1}{2}bh \):
\( A = \frac{1}{2}(10)(5) = 25\, \text{cm}^2 \)
### Trigonometry
11. **Question:** What is \( \sin(30^\circ) \)?
**Answer:**
Step 1: From trigonometric values, \( \sin(30^\circ) = \frac{1}{2} \).
12. **Question:** Calculate \( \tan(45^\circ) \).
**Answer:**
Step 1: From trigonometric values, \( \tan(45^\circ) = 1 \).
13. **Question:** If \( \cos(\theta) = 0.6 \), find \( \sin(\theta) \).
**Answer:**
Step 1: Use \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\( \sin^2(\theta) + (0.6)^2 = 1 \)
Step 2: Solve for \( \sin^2(\theta) \):
\( \sin^2(\theta) + 0.36 = 1 \)
\( \sin^2(\theta) = 0.64 \)
Step 3: \( \sin(\theta) = \sqrt{0.64} = 0.8 \)
14. **Question:** Find \( \cot(60^\circ) \).
**Answer:**
Step 1: \( \cot(60^\circ) = \frac{1}{\tan(60^\circ)} = \frac{1}{\sqrt{3}} \).
15. **Question:** What is the value of \( \sec(0^\circ) \)?
**Answer:**
Step 1: \( \sec(0^\circ) = \frac{1}{\cos(0^\circ)} = 1 \).
### Statistics
16. **Question:** Calculate the mean of the following numbers: 4, 8, 6, 5.
**Answer:**
Step 1: Sum the numbers: \( 4 + 8 + 6 + 5 = 23 \).
Step 2: Divide by the number of values (4):
\( \text{Mean} = \frac{23}{4} = 5.75 \).
17. **Question:** What is the median of the data set: 5, 3, 9, 7, 2?
**Answer:**
Step 1: Arrange the data: 2, 3, 5, 7, 9.
Step 2: The median is the middle value:
\( \text{Median} = 5 \).
18. **Question:** Calculate the mode of the following set: 1, 2, 2, 3, 4.
**Answer:**
Step 1: Identify the most frequent number: 2.
\( \text{Mode} = 2 \).
19. **Question:** Find the standard deviation of the set: 2, 4, 4, 4, 5, 5, 7, 9.
**Answer:**
Step 1: Find the mean: \( (2+4+4+4+5+5+7+9)/8 = 5 \).
Step 2: Find the squared differences:
\( (2-5)^2, (4-5)^2, (4-5)^2, (4-5)^2, (5-5)^2, (5-5)^2, (7-5)^2, (9-5)^2 \)
Which gives: \( 9, 1, 1, 1, 0, 0, 4, 16 \).
Step 3: Mean of squared differences: \( (9 + 1 + 1 + 1 + 0 + 0 + 4 + 16) / 8 = 3 \).
Step 4: Standard deviation: \( \sqrt{3} \approx 1.73 \).
20. **Question:** What is the range of the set: 6, 2, 8, 4, 10?
**Answer:**
Step 1: Find maximum and minimum:
\( \text{Max} = 10, \text{Min} = 2 \).
Step 2: Calculate range:
\( \text{Range} = 10 - 2 = 8 \).
### Calculus
21. **Question:** Find the derivative of \( f(x) = 3x^2 + 5x - 4 \).
**Answer:**
Step 1: Use power rule:
\( f'(x) = 6x + 5 \).
22. **Question:** Evaluate the integral \( \int (4x^3) \, dx \).
**Answer:**
Step 1: Use power rule for integration:
\( \int (4x^3) \, dx = \frac{4}{4}x^4 + C = x^4 + C \).
23. **Question:** Find \( \lim_{x \to 0} \frac{\sin(x)}{x} \).
**Answer:**
Step 1: Apply the limit:
\( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \).
24. **Question:** Determine the second derivative of \( f(x) = x^3 - 3x + 2 \).
**Answer:**
Step 1: First derivative: \( f'(x) = 3x^2 - 3 \).
Step 2: Second derivative: \( f''(x) = 6x \).
25. **Question:** Find the area under the curve \( y = x^2 \) from \( x = 1 \) to \( x = 3 \).
**Answer:**
Step 1: Find the integral:
\( \int_1^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_1^3 \).
Step 2: Calculate:
\( \frac{3^3}{3} - \frac{1^3}{3} = \frac{27}{3} - \frac{1}{3} = 9 - \frac{1}{3} = \frac{26}{3} \).
### Functions
26. **Question:** What is the range of the quadratic function \( f(x) = x^2 - 4x + 3 \)?
**Answer:**
Step 1: Find the vertex:
\( x = -\frac{b}{2a} = -\frac{-4}{2 \cdot 1} = 2 \).
Step 2: Calculate \( f(2) = 2^2 - 4(2) + 3 = -1 \).
Step 3: Since it opens upwards, the range is \( [-1, \infty) \).
27. **Question:** Determine if \( f(x) = 2x + 3 \) is a one-to-one function.
**Answer:**
Step 1: Check if \( f(a) = f(b) \) implies \( a = b \):
\( 2a + 3 = 2b + 3 \Rightarrow 2a = 2b \Rightarrow a = b \).
Therefore, it is one-to-one.
28. **Question:** Find \( f(g(x)) \) if \( f(x) = x^2 \) and \( g(x) = x + 1 \).
**Answer:**
Step 1: Substitute:
\( f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1 \).
29. **Question:** What is the inverse of the function \( f(x) = 3x - 5 \)?
**Answer:**
Step 1: Set \( y = 3x - 5 \).
Step 2: Solve for \( x \):
\( y + 5 = 3x \Rightarrow x = \frac{y + 5}{3} \).
Step 3: Therefore, \( f^{-1}(y) = \frac{y + 5}{3} \).
30. **Question:** Is \( f(x) = |x| \) continuous everywhere?
**Answer:**
Step 1: Check for continuity at \( x = 0 \):
\( \lim_{x \to 0^-} f(x) = 0 \) and \( \lim_{x \to 0^+} f(x) = 0 \).
Since \( f(0) = 0 \), it is continuous.
### Sequences and Series
31. **Question:** What is the sum of the first 10 positive integers?
**Answer:**
Step 1: Use the formula for the sum \( S_n = \frac{n(n + 1)}{2} \):
\( S_{10} = \frac{10(11)}{2} = 55 \).
32. **Question:** Find the 5th term of the arithmetic sequence where \( a_1 = 3 \) and \( d = 2 \).
**Answer:**
Step 1: Use the formula \( a_n = a_1 + (n - 1)d \):
\( a_5 = 3 + (5 - 1)(2) = 3 + 8 = 11 \).
33. **Question:** What is the 4th term of the geometric sequence where \( a_1 = 2 \) and \( r = 3 \)?
**Answer:**
Step 1: Use the formula \( a_n = a_1 r^{n-1} \):
\( a_4 = 2 \cdot 3^{3} = 2 \cdot 27 = 54 \).
34. **Question:** Determine the common difference of the arithmetic sequence: 2, 5, 8, 11, ...
**Answer:**
Step 1: Subtract the first term from the second:
\( d = 5 - 2 = 3 \).
35. **Question:** Find the first 5 terms of the Fibonacci sequence.
**Answer:**
Step 1: Start with \( 0, 1 \):
\( 0, 1, 1(0+1), 2(1+1), 3(1+2) \)
Result: \( 0, 1, 1, 2, 3 \).
### Probability
36. **Question:** What is the probability of rolling a sum of 7 with two dice?
**Answer:**
Step 1: Identify favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
Step 2: Total outcomes = \( 6 \times 6 = 36 \).
Step 3: Probability = \( \frac{6}{36} = \frac{1}{6}\).
37. **Question:** If a card is drawn from a standard deck, what is the probability it is a heart?
**Answer:**
Step 1: Total hearts = 13.
Step 2: Probability = \( \frac{13}{52} = \frac{1}{4} \).
38. **Question:** Calculate the probability of flipping a coin and getting heads 3 times in a row.
**Answer:**
Step 1: Probability of heads = \( \frac{1}{2} \).
Step 2: Probability for 3 heads = \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \).
39. **Question:** What is the probability of choosing a day of the week that is either a weekend or Friday?
**Answer:**
Step 1: Total favorable outcomes = 2 (Saturday, Sunday) + 1 (Friday) = 3.
Step 2: Probability = \( \frac{3}{7} \).
40. **Question:** In a box of marbles, there are 5 red, 3 blue, and 2 green. What is the probability of drawing a blue marble?
**Answer:**
Step 1: Total marbles = \( 5 + 3 + 2 = 10 \).
Step 2: Probability = \( \frac{3}{10} \).
### Miscellaneous
41. **Question:** What is the 5th roots of 243?
**Answer:**
Step 1: Find \( 243 = 3^5 \).
Step 2: Therefore, \( \sqrt[5]{243} = 3 \).
42. **Question:** Solve for \( x \): \( 4^x = 64 \).
**Answer:**
Step 1: Rewrite 64 as \( 4^3 \).
Step 2: Set exponents equal: \( x = 3 \).
43. **Question:** Simplify \( (2 + 3i)(4 - i) \).
**Answer:**
Step 1: Use distributive property:
\( = 8 - 2i + 12i - 3i^2 \).
Step 2: Simplify:
\( = 8 + 10i + 3 = 11 + 10i \).
44. **Question:** What is the difference between the two square roots: \( \sqrt{25} - \sqrt{9} \)?
**Answer:**
Step 1: Calculate roots: \( \sqrt{25} = 5, \sqrt{9} = 3 \).
Step 2: Result: \( 5 - 3 = 2 \).
45. **Question:** Calculate the percentage increase from 50 to 80.
**Answer:**
Step 1: Calculate increase: \( 80 - 50 = 30 \).
Step 2: Percentage increase:
\( \frac{30}{50} \times 100 = 60\% \).
46. **Question:** If \( x^2 - 5x + 6 = 0 \), find the nature of the roots.
**Answer:**
Step 1: Calculate discriminant: \( D = b^2 - 4ac = 25 - 24 = 1 \).
Step 2: Since \( D > 0 \), roots are real and distinct.
47. **Question:** Solve for \( x \): \( \log(x) + \log(10) = 1 \).
**Answer:**
Step 1: Combine logs: \( \log(10x) = 1 \).
Step 2: Exponentiate: \( 10x = 10^1 \Rightarrow x = 1 \).
48. **Question:** Find the coordinates of the vertex of the parabola represented by \( y = x^2 - 4x + 1 \).
**Answer:**
Step 1: Vertex \( x = -\frac{b}{2a} = 2 \).
Step 2: Find \( y \):
\( y = 2^2 - 4(2) + 1 = 4 - 8 + 1 = -3 \).
Vertex is \( (2, -3) \).
49. **Question:** Evaluate \( \frac{1}{\frac{1}{3} + \frac{1}{6}} \).
**Answer:**
Step 1: Find a common denominator:
\( \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \).
Step 2: Invert:
Result: \( 2 \).
50. **Question:** What is the value of \( \sqrt{144} + \left| -8 \right| \)?
**Answer:**
Step 1: Calculate \( \sqrt{144} = 12 \).
Step 2: Calculate \( | -8 | = 8 \).
Step 3: Sum:
Result: \( 12 + 8 = 20 \).
Feel free to modify any of these questions or ask for more questions in specific areas!