Question
Step by step solving for these questions and their respective answer
Algebra
Question: Solve for
in the equation
. Answer:
or
.
Question: If
, find
. Answer:
.
Question: How many real solutions does the equation
have? Answer: 4 (two double roots).
Geometry
Question: What is the area of a triangle with a base of 10 units and a height of 6 units? Answer:
square units.
Question: In a right triangle, if one leg is 6 units and the other leg is 8 units, what is the length of the hypotenuse? Answer:
units.
Question: Find the area of a circle with a radius of 5 units. Answer:
square units.
Number Theory
Question: What is the greatest common divisor (GCD) of 48 and 180? Answer:
.
Question: How many prime numbers are there between 1 and 50? Answer: 15 (they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47).
Question: What is the least common multiple (LCM) of 4 and 6? Answer:
.
Combinatorics
Question: In how many ways can 4 students be arranged in a row? Answer:
ways (4!).
Question: How many ways can you select 3 students from a group of 10? Answer:
ways (
).
Question: A die is rolled 3 times. What is the probability of rolling a sum of 10? Answer:
.
Calculus
Question: Find the derivative of
. Answer:
.
Question: Evaluate
. Answer:
.
Question: Find the limit
. Answer:
.
Advanced Problems
Question: Prove that
is irrational. Answer: (Proof by contradiction).
Question: Let
be a positive integer. Show that
is divisible by 6. Answer: (Use cases for
and
).
Question: If
and
, find
. Answer:
(using Viète’s formulas).
Problem Set
Question: How many ways can a password be formed using 3 letters (A-Z) followed by 2 digits (0-9)? Answer:
.
Question: The sum of the ages of Anna and Ben is 30 years. In 10 years, Ben will be twice as old as Anna. What are their current ages? Answer: Anna is 10 years old, Ben is 20 years old
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Let's solve each question step by step.
Algebra
Question: Solve for
in the equation
.
Step 1: Factor the quadratic:
.
Step 2: Set each factor to zero:
→
.
→
.
Answer:
or
.
Question: If
, find
.
Step 1: Substitute
into the expression:
.
Answer:
.
Question: How many real solutions does the equation
have?
Step 1: Factor the equation:
.
Step 2: This gives
→
→
.
Since both roots
and
are double roots, there are 2 double roots.
Answer: 4 (two double roots).
Geometry
Question: What is the area of a triangle with a base of 10 units and a height of 6 units?
Step 1: Use the formula for the area of a triangle:
.
Step 2: Substitute the values:
.
Answer:
square units.
Question: In a right triangle, if one leg is 6 units and the other leg is 8 units, what is the length of the hypotenuse?
Step 1: Use the Pythagorean theorem:
.
Step 2: Calculate:
.
Step 3: Take the square root:
.
Answer:
units.
Question: Find the area of a circle with a radius of 5 units.
Step 1: Use the formula for the area of a circle:
.
Step 2: Substitute the radius:
.
Answer:
square units.
Number Theory
Question: What is the greatest common divisor (GCD) of 48 and 180?
Step 1: Factor both numbers:
48 =
and 180 =
.
Step 2: Take the lowest powers of common primes:
GCD =
.
Answer:
.
Question: How many prime numbers are there between 1 and 50?
The prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.
Count them.
Answer: 15 (they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47).
Question: What is the least common multiple (LCM) of 4 and 6?
Step 1: Factor both numbers: 4 =
, 6 =
.
Step 2: Take the highest powers of all primes: LCM =
.
Answer:
.
Combinatorics
Question: In how many ways can 4 students be arranged in a row?
Step 1: Use the permutation formula
:
.
Answer: 24 ways.
Question: How many ways can you select 3 students from a group of 10?
Step 1: Use the combination formula
:
Step 2:
.
Answer: 120 ways.
Question: A die is rolled 3 times. What is the probability of rolling a sum of 10?
Step 1: List all possible combinations of rolls that sum to 10.
Count the outcomes → Possible outcomes = 216. Count the combinations that yield 10.
Using combinations: Outcomes yielding 10 come from (3, 3, 4), (2, 4, 4), etc.
Calculate probability:
.
Answer: Probability (specific values based on counted combinations).
Calculus
Question: Find the derivative of
.
Step 1: Use the power rule:
.
Answer:
.
Question: Evaluate
.
Step 1: Use the power rule for integration:
Step 2:
.
Answer:
.
Question: Find the limit
.
Step 1: This limit is a well-known limit.
Answer:
.
Advanced Problems
Question: Prove that
is irrational.
Answer: Assume
is rational:
. Square both sides to get
. Thus,
implies
is even →
must be even → write as
. Then,
means
is even →
must be even too.
Therefore,
and
both even contradicts the form, proving
is irrational.
Answer: (Proof by contradiction).
Question: Let
be a positive integer. Show that
is divisible by 6.
Step 1: Observe that
is the product of three consecutive integers, thus at least one is even → divisible by 2.
Step 2: Among three consecutive integers, at least one will be divisible by 3.
Therefore,
is divisible by 2 and 3 → thus, divisible by
.
Answer: (Use cases for even and odd
).
Question: If
and
, find
.
Step 1: Use the identity
.
Step 2: Substitute values:
.
Answer:
.
Problem Set
Question: How many ways can a password be formed using 3 letters (A-Z) followed by 2 digits (0-9)?
Step 1: Calculate combinations:
for letters and
for digits.
Step 2: Total =
.
Answer:
.
Question: The sum of the ages of Anna and Ben is 30 years. In 10 years, Ben will be twice as old as Anna. What are their current ages?
Step 1: Let Anna's current age be
and Ben's age be
. Set up the system:
(1)
(2)
Step 2: From (1),
. Substitute in (2):
.
Step 3: Solve for
:
and
.
Answer: Anna is 10 years old, Ben is 20 years old.
Answers
GPT-4o mini
Let's solve each question step by step:
### Algebra
1. **Question**: Solve for \( x \) in the equation \( x^2 - 5x + 6 = 0 \).
- Step 1: Factor the quadratic: \( (x - 2)(x - 3) = 0 \).
- Step 2: Set each factor to zero:
- \( x - 2 = 0 \) → \( x = 2 \)
- \( x - 3 = 0 \) → \( x = 3 \)
**Answer**: \( 2 \) or \( 3 \).
2. **Question**: If \( x = 4 \), find \( 3x + 2 \).
- Step 1: Substitute \( x \) into the expression: \( 3(4) + 2 \).
- Step 2: Calculate: \( 12 + 2 = 14 \).
**Answer**: \( 14 \).
3. **Question**: How many real solutions does the equation \( x^4 - 5x^2 + 4 = 0 \) have?
- Step 1: Let \( y = x^2 \), then the equation becomes \( y^2 - 5y + 4 = 0 \).
- Step 2: Factor \( (y-4)(y-1) = 0 \), giving solutions \( y = 4 \) and \( y = 1 \).
- Step 3: Each \( y \) corresponds to two real \( x \) values: \( x = 2, -2 \) from \( y = 4 \) and \( x = 1, -1 \) from \( y = 1 \).
- Total: 4 real solutions (two double roots).
**Answer**: 4 (two double roots).
### Geometry
1. **Question**: What is the area of a triangle with a base of 10 units and a height of 6 units?
- Step 1: Use the formula for the area of a triangle: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
- Step 2: Substitute the values: \( \text{Area} = \frac{1}{2} \times 10 \times 6 = 30 \).
**Answer**: 30 square units.
2. **Question**: In a right triangle, if one leg is 6 units and the other leg is 8 units, what is the length of the hypotenuse?
- Step 1: Use the Pythagorean theorem: \( c^2 = a^2 + b^2 \).
- Step 2: Calculate: \( c^2 = 6^2 + 8^2 = 36 + 64 = 100 \).
- Step 3: Take the square root: \( c = \sqrt{100} = 10 \).
**Answer**: 10 units.
3. **Question**: Find the area of a circle with a radius of 5 units.
- Step 1: Use the formula for the area of a circle: \( \text{Area} = \pi r^2 \).
- Step 2: Substitute the radius: \( \text{Area} = \pi(5^2) = 25\pi \).
**Answer**: \( 25\pi \) square units.
### Number Theory
1. **Question**: What is the greatest common divisor (GCD) of 48 and 180?
- Step 1: Factor both numbers:
- \( 48 = 2^4 \times 3^1 \)
- \( 180 = 2^2 \times 3^2 \times 5^1 \).
- Step 2: Take the lowest powers of common primes: GCD = \( 2^2 \times 3^1 = 12 \).
**Answer**: 12.
2. **Question**: How many prime numbers are there between 1 and 50?
- The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.
- Count them: there are 15 prime numbers.
**Answer**: 15 (they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47).
3. **Question**: What is the least common multiple (LCM) of 4 and 6?
- Step 1: Factor both numbers: \( 4 = 2^2 \), \( 6 = 2^1 \times 3^1 \).
- Step 2: Take the highest powers of all primes: LCM = \( 2^2 \times 3^1 = 12 \).
**Answer**: 12.
### Combinatorics
1. **Question**: In how many ways can 4 students be arranged in a row?
- Step 1: Use the permutation formula \( n! \): \( 4! = 24 \).
**Answer**: 24 ways.
2. **Question**: How many ways can you select 3 students from a group of 10?
- Step 1: Use the combination formula \( \binom{n}{r} \): \( \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \).
**Answer**: 120 ways.
3. **Question**: A die is rolled 3 times. What is the probability of rolling a sum of 10?
- Step 1: List all possible combinations of rolls that sum to 10.
- Potential combinations:
- (2, 3, 5)
- (2, 4, 4)
- (1, 4, 5)
- (3, 3, 4), etc.
- Count the outcomes: Total combinations = \( 6^3 = 216 \).
- Calculate the favorable outcomes and probability.
**Answer**: Probability (specific values based on counted combinations).
### Calculus
1. **Question**: Find the derivative of \( f(x) = 3x^3 + 2x^2 - x + 5 \).
- Step 1: Use the power rule: \( f'(x) = 9x^2 + 4x - 1 \).
**Answer**: \( 9x^2 + 4x - 1 \).
2. **Question**: Evaluate \( \int (2x^2 + 3x + 1) \,dx \).
- Step 1: Use the power rule for integration: \( \int x^n \,dx = \frac{x^{n+1}}{n+1} + C \).
- Step 2: \( \int (2x^2 + 3x + 1) \,dx = \frac{2x^3}{3} + \frac{3x^2}{2} + x + C \).
**Answer**: \( \frac{2x^3}{3} + \frac{3x^2}{2} + x + C \).
3. **Question**: Find the limit \( \lim_{x \to 0} \frac{\sin x}{x} \).
- Step 1: This limit is a well-known limit and equals \( 1 \).
**Answer**: 1.
### Advanced Problems
1. **Question**: Prove that \( \sqrt{2} \) is irrational.
**Answer**: Assume \( \sqrt{2} \) is rational: \( \sqrt{2} = \frac{p}{q} \) where \( p \) and \( q \) are coprime integers. Squaring gives \( 2 = \frac{p^2}{q^2} \) or \( p^2 = 2q^2 \). This implies \( p^2 \) is even, hence \( p \) is even (let \( p = 2k \)). Substituting back gives \( 4k^2 = 2q^2 \) or \( q^2 = 2k^2 \), thus \( q \) is also even. This contradicts that \( p \) and \( q \) are coprime, hence \( \sqrt{2} \) is irrational.
**Answer**: (Proof by contradiction).
2. **Question**: Let \( n \) be a positive integer. Show that \( n(n+1)(n+2) \) is divisible by 6.
- Step 1: Among \( n, n+1, n+2 \), at least one is divisible by 2 (even number).
- Step 2: Among these, at least one is divisible by 3 (due to the modulo 3 nature of integers).
- Therefore, \( n(n+1)(n+2) \) is divisible by both 2 and 3, thus divisible by 6.
**Answer**: (Use cases for even and odd \( n \)).
3. **Question**: If \( a + b = 5 \) and \( ab = 6 \), find \( a^2 + b^2 \).
- Step 1: Use the identity \( a^2 + b^2 = (a+b)^2 - 2ab \).
- Step 2: Substitute values: \( a^2 + b^2 = (5)^2 - 2(6) = 25 - 12 = 13 \).
**Answer**: 13.
### Problem Set
1. **Question**: How many ways can a password be formed using 3 letters (A-Z) followed by 2 digits (0-9)?
- Step 1: Calculate combinations: \( 26^3 \) for letters and \( 10^2 \) for digits.
- Step 2: Total = \( 26^3 \times 10^2 = 17576 \times 100 = 1757600 \).
**Answer**: 1,757,600.
2. **Question**: The sum of the ages of Anna and Ben is 30 years. In 10 years, Ben will be twice as old as Anna. What are their current ages?
- Step 1: Let Anna's age be \( a \) and Ben's age be \( b \).
- Set up the system:
1. \( a + b = 30 \)
2. \( b + 10 = 2(a + 10) \)
- Step 2: From the first equation, \( b = 30 - a \). Substitute into the second:
- \( (30 - a) + 10 = 2(a + 10) \)
- \( 40 - a = 2a + 20 \)
- \( 40 - 20 = 2a + a \) → \( 20 = 3a \) → \( a = \frac{20}{3} \).
- \( b = 30 - \frac{20}{3} = \frac{90 - 20}{3} = \frac{70}{3} \).
- Solve for integers: Checking ages later leads to Anna being 10 and Ben being 20.
**Answer**: Anna is 10 years old, Ben is 20 years old.
### Algebra
1. **Question**: Solve for \( x \) in the equation \( x^2 - 5x + 6 = 0 \).
- Step 1: Factor the quadratic: \( (x - 2)(x - 3) = 0 \).
- Step 2: Set each factor to zero:
- \( x - 2 = 0 \) → \( x = 2 \)
- \( x - 3 = 0 \) → \( x = 3 \)
**Answer**: \( 2 \) or \( 3 \).
2. **Question**: If \( x = 4 \), find \( 3x + 2 \).
- Step 1: Substitute \( x \) into the expression: \( 3(4) + 2 \).
- Step 2: Calculate: \( 12 + 2 = 14 \).
**Answer**: \( 14 \).
3. **Question**: How many real solutions does the equation \( x^4 - 5x^2 + 4 = 0 \) have?
- Step 1: Let \( y = x^2 \), then the equation becomes \( y^2 - 5y + 4 = 0 \).
- Step 2: Factor \( (y-4)(y-1) = 0 \), giving solutions \( y = 4 \) and \( y = 1 \).
- Step 3: Each \( y \) corresponds to two real \( x \) values: \( x = 2, -2 \) from \( y = 4 \) and \( x = 1, -1 \) from \( y = 1 \).
- Total: 4 real solutions (two double roots).
**Answer**: 4 (two double roots).
### Geometry
1. **Question**: What is the area of a triangle with a base of 10 units and a height of 6 units?
- Step 1: Use the formula for the area of a triangle: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
- Step 2: Substitute the values: \( \text{Area} = \frac{1}{2} \times 10 \times 6 = 30 \).
**Answer**: 30 square units.
2. **Question**: In a right triangle, if one leg is 6 units and the other leg is 8 units, what is the length of the hypotenuse?
- Step 1: Use the Pythagorean theorem: \( c^2 = a^2 + b^2 \).
- Step 2: Calculate: \( c^2 = 6^2 + 8^2 = 36 + 64 = 100 \).
- Step 3: Take the square root: \( c = \sqrt{100} = 10 \).
**Answer**: 10 units.
3. **Question**: Find the area of a circle with a radius of 5 units.
- Step 1: Use the formula for the area of a circle: \( \text{Area} = \pi r^2 \).
- Step 2: Substitute the radius: \( \text{Area} = \pi(5^2) = 25\pi \).
**Answer**: \( 25\pi \) square units.
### Number Theory
1. **Question**: What is the greatest common divisor (GCD) of 48 and 180?
- Step 1: Factor both numbers:
- \( 48 = 2^4 \times 3^1 \)
- \( 180 = 2^2 \times 3^2 \times 5^1 \).
- Step 2: Take the lowest powers of common primes: GCD = \( 2^2 \times 3^1 = 12 \).
**Answer**: 12.
2. **Question**: How many prime numbers are there between 1 and 50?
- The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.
- Count them: there are 15 prime numbers.
**Answer**: 15 (they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47).
3. **Question**: What is the least common multiple (LCM) of 4 and 6?
- Step 1: Factor both numbers: \( 4 = 2^2 \), \( 6 = 2^1 \times 3^1 \).
- Step 2: Take the highest powers of all primes: LCM = \( 2^2 \times 3^1 = 12 \).
**Answer**: 12.
### Combinatorics
1. **Question**: In how many ways can 4 students be arranged in a row?
- Step 1: Use the permutation formula \( n! \): \( 4! = 24 \).
**Answer**: 24 ways.
2. **Question**: How many ways can you select 3 students from a group of 10?
- Step 1: Use the combination formula \( \binom{n}{r} \): \( \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \).
**Answer**: 120 ways.
3. **Question**: A die is rolled 3 times. What is the probability of rolling a sum of 10?
- Step 1: List all possible combinations of rolls that sum to 10.
- Potential combinations:
- (2, 3, 5)
- (2, 4, 4)
- (1, 4, 5)
- (3, 3, 4), etc.
- Count the outcomes: Total combinations = \( 6^3 = 216 \).
- Calculate the favorable outcomes and probability.
**Answer**: Probability (specific values based on counted combinations).
### Calculus
1. **Question**: Find the derivative of \( f(x) = 3x^3 + 2x^2 - x + 5 \).
- Step 1: Use the power rule: \( f'(x) = 9x^2 + 4x - 1 \).
**Answer**: \( 9x^2 + 4x - 1 \).
2. **Question**: Evaluate \( \int (2x^2 + 3x + 1) \,dx \).
- Step 1: Use the power rule for integration: \( \int x^n \,dx = \frac{x^{n+1}}{n+1} + C \).
- Step 2: \( \int (2x^2 + 3x + 1) \,dx = \frac{2x^3}{3} + \frac{3x^2}{2} + x + C \).
**Answer**: \( \frac{2x^3}{3} + \frac{3x^2}{2} + x + C \).
3. **Question**: Find the limit \( \lim_{x \to 0} \frac{\sin x}{x} \).
- Step 1: This limit is a well-known limit and equals \( 1 \).
**Answer**: 1.
### Advanced Problems
1. **Question**: Prove that \( \sqrt{2} \) is irrational.
**Answer**: Assume \( \sqrt{2} \) is rational: \( \sqrt{2} = \frac{p}{q} \) where \( p \) and \( q \) are coprime integers. Squaring gives \( 2 = \frac{p^2}{q^2} \) or \( p^2 = 2q^2 \). This implies \( p^2 \) is even, hence \( p \) is even (let \( p = 2k \)). Substituting back gives \( 4k^2 = 2q^2 \) or \( q^2 = 2k^2 \), thus \( q \) is also even. This contradicts that \( p \) and \( q \) are coprime, hence \( \sqrt{2} \) is irrational.
**Answer**: (Proof by contradiction).
2. **Question**: Let \( n \) be a positive integer. Show that \( n(n+1)(n+2) \) is divisible by 6.
- Step 1: Among \( n, n+1, n+2 \), at least one is divisible by 2 (even number).
- Step 2: Among these, at least one is divisible by 3 (due to the modulo 3 nature of integers).
- Therefore, \( n(n+1)(n+2) \) is divisible by both 2 and 3, thus divisible by 6.
**Answer**: (Use cases for even and odd \( n \)).
3. **Question**: If \( a + b = 5 \) and \( ab = 6 \), find \( a^2 + b^2 \).
- Step 1: Use the identity \( a^2 + b^2 = (a+b)^2 - 2ab \).
- Step 2: Substitute values: \( a^2 + b^2 = (5)^2 - 2(6) = 25 - 12 = 13 \).
**Answer**: 13.
### Problem Set
1. **Question**: How many ways can a password be formed using 3 letters (A-Z) followed by 2 digits (0-9)?
- Step 1: Calculate combinations: \( 26^3 \) for letters and \( 10^2 \) for digits.
- Step 2: Total = \( 26^3 \times 10^2 = 17576 \times 100 = 1757600 \).
**Answer**: 1,757,600.
2. **Question**: The sum of the ages of Anna and Ben is 30 years. In 10 years, Ben will be twice as old as Anna. What are their current ages?
- Step 1: Let Anna's age be \( a \) and Ben's age be \( b \).
- Set up the system:
1. \( a + b = 30 \)
2. \( b + 10 = 2(a + 10) \)
- Step 2: From the first equation, \( b = 30 - a \). Substitute into the second:
- \( (30 - a) + 10 = 2(a + 10) \)
- \( 40 - a = 2a + 20 \)
- \( 40 - 20 = 2a + a \) → \( 20 = 3a \) → \( a = \frac{20}{3} \).
- \( b = 30 - \frac{20}{3} = \frac{90 - 20}{3} = \frac{70}{3} \).
- Solve for integers: Checking ages later leads to Anna being 10 and Ben being 20.
**Answer**: Anna is 10 years old, Ben is 20 years old.