## 100 Math Problems with Step-by-Step Explanations and Correct Answers

Mathematics is a subject that requires logical thinking, problem-solving skills, and a deep understanding of various concepts. In this article, we present 100 math problems covering different topics, including algebra, geometry, calculus, and more. Each problem is followed by a step-by-step explanation of how to solve it, along with the correct answer. Let's dive in!

Problem 1:

Solve the equation: 3x + 5 = 17

Solution:

Step 1: Subtract 5 from both sides: 3x = 17 - 5

Step 2: Simplify: 3x = 12

Step 3: Divide both sides by 3: x = 12 ÷ 3

Step 4: Simplify: x = 4

Answer: x = 4

Problem 2:

Find the value of x in the equation: 2(3x - 1) = 10

Solution:

Step 1: Distribute the 2: 6x - 2 = 10

Step 2: Add 2 to both sides: 6x = 10 + 2

Step 3: Simplify: 6x = 12

Step 4: Divide both sides by 6: x = 12 ÷ 6

Step 5: Simplify: x = 2

Answer: x = 2

Problem 3:

Solve the following inequality: 2x + 3 > 9

Solution:

Step 1: Subtract 3 from both sides: 2x > 9 - 3

Step 2: Simplify: 2x > 6

Step 3: Divide both sides by 2 (since 2 is positive): x > 6 ÷ 2

Step 4: Simplify: x > 3

Answer: x > 3

Problem 4:

Evaluate the expression: 4 + 2 × 3 - 1

Solution:

Step 1: Perform multiplication: 4 + 6 - 1

Step 2: Perform addition and subtraction (left to right): 10 - 1

Step 3: Simplify: 9

Answer: 9

Problem 5:

Factorize the quadratic expression: x² - 4x - 12

Solution:

Step 1: Find two numbers that multiply to -12 and add up to -4. In this case, -6 and 2 satisfy these conditions.

Step 2: Rewrite the middle term: x² - 6x + 2x - 12

Step 3: Group the terms and factor by grouping: x(x - 6) + 2(x - 6)

Step 4: Factor out the common binomial: (x + 2)(x - 6)

Answer: (x + 2)(x - 6)

Problem 6:

Find the area of a rectangle with length 8 cm and width 5 cm.

Solution:

Step 1: Use the formula for the area of a rectangle: Area = length × width

Step 2: Substitute the given values: Area = 8 cm × 5 cm

Step 3: Multiply: Area = 40 cm²

Answer: 40 cm²

Problem 7:

Calculate the volume of a sphere with radius 3 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a sphere: Volume = (4/3)πr³

Step 2: Substitute the given values: Volume = (4/3) × 3.14 × (3 cm)³

Step 3: Simplify: Volume = (4/3) × 3.14 × 27 cm³

Step 4: Multiply: Volume ≈ 113.04 cm³ (rounded to two decimal places)

Answer: Volume ≈ 113.04 cm³

Problem 8:

Solve the system of equations:

2x + 3y = 10

4x - 5y = 8

Solution:

Step 1: Multiply the first equation by 2: 4x + 6y = 20

Step 2: Add the second equation: (4x + 6y) + (4x - 5y) = 20 + 8

Step 3: Simplify: 8x + y = 28

Step 4: Solve for y: y = 28 - 8x

Step 5: Substitute the value of y into the first equation: 2x + 3(28 - 8x) = 10

Step 6: Simplify: 2x + 84 - 24x = 10

Step 7: Combine like terms: -22x + 84 = 10

Step 8: Subtract 84 from both sides: -22x = 10 - 84

Step 9: Simplify: -22x = -74

Step 10: Divide both sides by -22: x = -74 ÷ -22

Step 11: Simplify: x = 3.36 (rounded to two decimal places)

Step 12: Substitute the value of x into the equation y = 28 - 8x: y = 28 - 8(3.36)

Step 13: Simplify: y = 28 - 26.88

Step 14: Simplify further: y ≈ 1.12 (rounded to two decimal places)

Answer: x ≈ 3.36 and y ≈ 1.12

Problem 9:

Find the derivative of the function f(x) = 3x² + 2x - 1.

Solution:

Step 1: Use the power rule for derivatives: f'(x) = 2(3x²) + 1(2x) + 0

Step 2: Simplify: f'(x) = 6x² + 2x

Answer: f'(x) = 6x² + 2x

Problem 10:

Simplify the expression: √(64) - √(16)

Solution:

Step 1: Evaluate the square roots: 8 - 4

Step 2: Simplify: 4

Answer: 4

Problem 11:

Solve the trigonometric equation: sin(x) = 0.5

Solution:

Step 1: Take the inverse sine of both sides: x = arcsin(0.5)

Step 2: Use a calculator or reference table to find the angle: x ≈ 30°

Answer: x ≈ 30°

Problem 12:

Calculate the factorial of 6 (written as 6!).

Solution:

Step 1: Multiply 6 by all positive integers less than it: 6! = 6 × 5 × 4 × 3 × 2 × 1

Step 2: Simplify: 6! = 720

Answer: 6! = 720

Problem 13:

Determine the perimeter of a triangle with side lengths 5 cm, 7 cm, and 9 cm.

Solution:

Step 1: Add the lengths of all three sides: Perimeter = 5 cm + 7 cm + 9 cm

Step 2: Simplify: Perimeter = 21 cm

Answer: Perimeter = 21 cm

Problem 14:

Solve the logarithmic equation: log₂(x) + log₂(2x + 6) = 3

Solution:

Step 1: Combine the logarithms using the product rule: log₂(x(2x + 6)) = 3

Step 2: Simplify the product inside the logarithm: log₂(2x² + 6x) = 3

Step 3: Rewrite the equation in exponential form: 2x² + 6x = 2³

Step 4: Simplify the exponent: 2x² + 6x = 8

Step 5: Rearrange the equation: 2x² + 6x - 8 = 0

Step 6: Factorize: (x - 1)(2x + 8) = 0

Step 7: Set each factor equal to zero and solve: x - 1 = 0 or 2x + 8 = 0

Step 8: Solve for x: x = 1 or x = -4

Answer: x = 1 or x = -4

Problem 15:

Evaluate the integral: ∫(4x² + 3x + 2) dx

Solution:

Step 1: Apply the power rule for integration: ∫(4x² + 3x + 2) dx = (4/3)x³ + (3/2)x² + 2x + C

Answer: ∫(4x² + 3x + 2) dx = (4/3)x³ + (3/2)x² + 2x + C

Problem 16:

Find the equation of a line passing through the points (2, 5) and (4, 9).

Solution:

Step 1: Calculate the slope using the formula: m = (y₂ - y₁) / (x₂ - x₁)

Step 2: Substitute the coordinates: m = (9 - 5) / (4 - 2) = 4 / 2 = 2

Step 3: Use the point-slope form of a line: y - y₁ = m(x - x₁)

Step 4: Substitute the slope and one point: y - 5 = 2(x - 2)

Step 5: Simplify: y - 5 = 2x - 4

Step 6: Rearrange the equation: y = 2x + 1

Answer: The equation of the line is y = 2x + 1

Problem 17:

Solve the quadratic equation: 2x² - 5x - 3 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Since factorizing is not possible in this case, we'll use the quadratic formula.

Step 2: Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values from the given equation: x = (5 ± √(5² - 4(2)(-3))) / (2(2))

Step 3: Simplify: x = (5 ± √(25 + 24)) / 4

x = (5 ± √(49)) / 4

x = (5 ± 7) / 4

For the positive root:

x₁ = (5 + 7) / 4

x₁ = 12 / 4

x₁ = 3

For the negative root:

x₂ = (5 - 7) / 4

x₂ = -2 / 4

x₂ = -0.5

Answer: x = 3 or x = -0.5

Problem 18:

Simplify the complex number expression: (2 + 3i) + (4 - 2i)

Solution:

Step 1: Add the real parts: 2 + 4 = 6

Step 2: Add the imaginary parts: 3i - 2i = i

Answer: (2 + 3i) + (4 - 2i) = 6 + i

Problem 19:

Find the median of the following set of numbers: 5, 3, 9, 1, 7

Solution:

Step 1: Arrange the numbers in ascending order: 1, 3, 5, 7, 9

Step 2: Determine the middle value: The median is the middle number, which is 5.

Answer: The median is 5.

Problem 20:

Calculate the probability of rolling a 6 on a fair six-sided die.

Solution:

Step 1: Determine the number of favorable outcomes: There is only one favorable outcome, rolling a 6.

Step 2: Determine the total number of possible outcomes: There are six possible outcomes, one for each side of the die.

Step 3: Calculate the probability: Probability = favorable outcomes / total outcomes = 1/6

Answer: The probability of rolling a 6 on a fair six-sided die is 1/6.

Problem 21:

Simplify the expression: log₅(125)

Solution:

Step 1: Determine the exponent that gives 125 when raised to the base 5: 5³ = 125

Step 2: Simplify: log₅(125) = 3

Answer: log₅(125) = 3

Problem 22:

Solve the exponential equation: 2^(x + 1) = 16

Solution:

Step 1: Rewrite 16 as a power of 2: 16 = 2^4

Step 2: Set the exponents equal to each other: x + 1 = 4

Step 3: Solve for x: x = 4 - 1

x = 3

Answer: x = 3

Problem 23:

Find the area of a trapezoid with bases of length 6 cm and 10 cm, and a height of 8 cm.

Solution:

Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h

Step 2: Substitute the given values: Area = (1/2) × (6 cm + 10 cm) × 8 cm

Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm

Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 24:

Solve the matrix equation: [2 1] [x] = [5]

[3 4] [y] [8]

Solution:

Step 1: Multiply the matrix and the column vector: 2x + y = 5 and 3x + 4y = 8

Step 2: Solve the system of equations using any appropriate method (substitution, elimination, etc.):

Multiply the first equation by 3: 6x + 3y = 15

Multiply the second equation by 2: 6x + 8y = 16

Subtract the first equation from the second equation: 6x + 8y - (6x + 3y) = 16 - 15

Simplify: 6x + 8y - 6x - 3y = 1

Simplify further: 5y = 1

Solve for y: y = 1/5

Substitute the value of y into the first equation: 2x + (1/5) = 5

Simplify: 2x + 1/5 = 5

Subtract 1/5 from both sides: 2x = 5 - 1/5

Simplify: 2x = 25/5 - 1/5

Simplify further: 2x = 24/5

Divide both sides by 2: x = (24/5) / 2

Simplify: x = 24/10

Simplify further: x = 12/5

Answer: x = 12/5 and y = 1/5

Problem 25:

Evaluate the limit: lim(x → 0) (3x² + 2x + 1) / x

Solution:

Step 1: Substitute the value of x into the expression: (3(0)² + 2(0) + 1) / 0

Step 2: Simplify: (0 + 0 + 1) / 0

Step 3: Since the denominator is zero, the limit is undefined.

Answer: The limit does not exist.

Problem 26:

Simplify the expression: 4x - 2(3x + 5)

Solution:

Step 1: Distribute the -2: 4x - 6x - 10

Step 2: Combine like terms: -2x - 10

Answer: -2x - 10

Problem 27:

Find the value of x in the equation: 2(x - 3) + 5 = 17

Solution:

Step 1: Distribute the 2: 2x - 6 + 5 = 17

Step 2: Combine like terms: 2x - 1 = 17

Step 3: Add 1 to both sides: 2x = 18

Step 4: Divide both sides by 2: x = 9

Answer: x = 9

Problem 28:

Solve the inequality: 3x + 7 > 4x - 5

Solution:

Step 1: Subtract 3x from both sides: 7 > x - 5

Step 2: Add 5 to both sides: 12 > x

Answer: x < 12

Problem 29:

Evaluate the expression: 3(2² - 1) + 4

Solution:

Step 1: Simplify the exponent: 3(4 - 1) + 4

Step 2: Simplify the parentheses: 3(3) + 4

Step 3: Multiply: 9 + 4

Answer: 13

Problem 30:

Factorize the quadratic expression: x² + 7x + 10

Solution:

Step 1: Find two numbers that multiply to 10 and add up to 7. In this case, 2 and 5 satisfy these conditions.

Step 2: Rewrite the middle term: x² + 2x + 5x + 10

Step 3: Group the terms and factor by grouping: (x² + 2x) + (5x + 10)

Step 4: Factor out the common binomial: x(x + 2) + 5(x + 2)

Step 5: Combine like terms: (x + 2)(x + 5)

Answer: (x + 2)(x + 5)

Problem 31:

Calculate the perimeter of a square with side length 9 cm.

Solution:

Step 1: Use the formula for the perimeter of a square: Perimeter = 4 × side length

Step 2: Substitute the given value: Perimeter = 4 × 9 cm

Answer: Perimeter = 36 cm

Problem 32:

Determine the volume of a cylinder with radius 5 cm and height 10 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a cylinder: Volume = πr²h

Step 2: Substitute the given values: Volume = 3.14 × (5 cm)² × 10 cm

Step 3: Simplify: Volume = 3.14 × 25 cm² × 10 cm

Step 4: Multiply: Volume = 785 cm³

Answer: Volume = 785 cm³

Problem 33:

Solve the system of equations:

2x + 3y = 10

5x - 2y = 7

Solution:

Step 1: Multiply the first equation by 2: 4x + 6y = 20

Step 2: Multiply the second equation by 3: 15x - 6y = 21

Step 3: Add the equations: (4x + 6y) + (15x - 6y) = 20 + 21

Step 4: Simplify: 19x = 41

Step 5: Divide both sides by 19: x = 41/19

Step 6: Substitute the value of x into the first equation: 2(41/19) + 3y = 10

Step 7: Simplify: 82/19 + 3y = 10

Step 8: Subtract 82/19 from both sides: 3y = 10 - 82/19

Step 9: Simplify: 3y = 190/19 - 82/19

Step 10: Combine like terms: 3y = 108/19

Step 11: Divide both sides by 3: y = 108/57

Step 12: Simplify: y = 6/19

Answer: x = 41/19 and y = 6/19

Problem 34:

Find the derivative of the function f(x) = 4x³ + 2x² - 1.

Solution:

Step 1: Take the derivative of each term using the power rule:

f'(x) = 4(3x²) + 2(2x) - 0

Step 2: Simplify: f'(x) = 12x² + 4x

Answer: f'(x) = 12x² + 4x

Problem 35:

Simplify the expression: √(81) - √(25)

Solution:

Step 1: Evaluate the square roots: 9 - 5

Step 2: Simplify: 4

Answer: 4

Problem 36:

Solve the trigonometric equation: cos(x) = 0.5

Solution:

Step 1: Take the inverse cosine of both sides: x = arccos(0.5)

Step 2: Use a calculator or reference table to find the angle: x = π/3 or x = 2π/3

Answer: x = π/3 or x = 2π/3

Problem 37:

Calculate the probability of rolling a 6 on a fair six-sided die.

Solution:

Step 1: Determine the number of favorable outcomes: There is only one favorable outcome, rolling a 6.

Step 2: Determine the total number of possible outcomes: There are six possible outcomes, one for each side of the die.

Step 3: Calculate the probability: Probability = favorable outcomes / total outcomes = 1/6

Answer: The probability of rolling a 6 on a fair six-sided die is 1/6.

Problem 38:

Simplify the expression: log₅(125)

Solution:

Step 1: Determine the exponent that gives 125 when raised to the base 5: 5³ = 125

Step 2: Simplify: log₅(125) = 3

Answer: log₅(125) = 3

Problem 39:

Solve the exponential equation: 2^(x + 1) = 16

Solution:

Step 1: Rewrite 16 as a power of 2: 16 = 2^4

Step 2: Set the exponents equal to each other: x + 1 = 4

Step 3: Solve for x: x = 4 - 1

x = 3

Answer: x = 3

Problem 40:

Find the area of a trapezoid with bases of length 6 cm and 10 cm, and a height of 8 cm.

Solution:

Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h

Step 2: Substitute the given values: Area = (1/2) × (6 cm + 10 cm) × 8 cm

Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm

Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 41:

Solve the matrix equation: [2 1] [x] = [5]

[3 4] [y] [8]

Solution:

Step 1: Multiply the inverse of the matrix on both sides to isolate the variables.

[2 1]⁻¹ [2 1] [x] = [2 1]⁻¹ [5]

[3 4] [y] [3 4] [8]

[x] = [2 1]⁻¹ [5]

[y] [3 4] [8]

Step 2: Calculate the inverse of the matrix [2 1]

[3 4]

The inverse of the matrix is:

[ 4 -1]

[-3 2]

Step 3: Multiply the inverse of the matrix by the right-hand side:

[x] = [ 4 -1] [5]

[y] [-3 2] [8]

Step 4: Simplify:

[x] = [4(5) + (-1)(8)]

[y] [-3(5) + 2(8)]

[x] = [20 - 8]

[y] [-15 + 16]

[x] = [12]

[y] [ 1]

Answer: x = 12 and y = 1

Problem 42:

Evaluate the limit: lim(x → 0) (3x² + 2x + 1) / x

Solution:

Step 1: Substitute the value of x into the expression: (3(0)² + 2(0) + 1) / 0

Step 2: Simplify: (0 + 0 + 1) / 0

Step 3: Since the denominator is zero, the limit is undefined.

Answer: The limit does not exist.

Problem 43:

Solve the logarithmic equation: log₂(x) + log₂(2x + 6) = 3

Solution:

Step 1: Combine the logarithms using the product rule: log₂(x(2x + 6)) = 3

Step 2: Simplify the product inside the logarithm: log₂(2x² + 6x) = 3

Step 3: Rewrite the equation in exponential form: 2x² + 6x = 2³

Step 4: Simplify the exponent: 2x² + 6x = 8

Step 5: Rearrange the equation: 2x² + 6x - 8 = 0

Step 6: Factorize: (x - 1)(2x + 4) = 0

Step 7: Set each factor equal to zero and solve: x - 1 = 0 or 2x + 4 = 0

Step 8: Solve for x: x = 1 or x = -2

Answer: x = 1 or x = -2

Problem 44:

Find the area of a circle with a radius of 6 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the area of a circle: Area = πr²

Step 2: Substitute the given value: Area = 3.14 × (6 cm)²

Step 3: Simplify: Area = 3.14 × 36 cm²

Step 4: Multiply: Area = 113.04 cm²

Answer: The area of the circle is 113.04 cm².

Problem 45:

Solve the equation: 2sin(x) = 1

Solution:

Step 1: Divide both sides by 2: sin(x) = 1/2

Step 2: Use a calculator or reference table to find the angles where sin(x) = 1/2: x = π/6 or x = 5π/6

Answer: x = π/6 or x = 5π/6

Problem 46:

Simplify the expression: (3x + 2y) - (2x - y)

Solution:

Step 1: Distribute the negative sign: 3x + 2y - 2x + y

Step 2: Combine like terms: (3x - 2x) + (2y + y)

Step 3: Simplify: x + 3y

Answer: x + 3y

Problem 47:

Find the value of x in the equation: 5(x - 2) = 3(2x + 1)

Solution:

Step 1: Distribute the 5 and 3: 5x - 10 = 6x + 3

Step 2: Subtract 5x and 6x from both sides: -10 - 3 = 6x - 5x

Step 3: Simplify: -13 = x

Answer: x = -13

Problem 48:

Solve the inequality: 2x + 3 < 5x - 2

Solution:

Step 1: Subtract 2x from both sides: 3 < 3x - 2

Step 2: Add 2 to both sides: 5 < 3x

Step 3: Divide both sides by 3: 5/3 < x

Answer: x > 5/3

Problem 49:

Evaluate the expression: 2² + 3³ - 4⁴

Solution:

Step 1: Evaluate the exponents: 2² + 3³ - 4⁴ = 4 + 27 - 256

Step 2: Simplify: 4 + 27 - 256 = -225

Answer: -225

Problem 50:

Factorize the quadratic expression: 2x² + 5x + 3

Solution:

Step 1: Find two numbers that multiply to 6 and add up to 5. In this case, 2 and 3 satisfy these conditions.

Step 2: Rewrite the middle term: 2x² + 2x + 3x + 3

Step 3: Group the terms and factor by grouping: (2x² + 2x) + (3x + 3)

Step 4: Factor out the common binomial: 2x(x + 1) + 3(x + 1)

Step 5: Combine like terms: (2x + 3)(x + 1)

Answer: (2x + 3)(x + 1)

Problem 51:

Calculate the perimeter of a rectangle with length 12 cm and width 5 cm.

Solution:

Step 1: Use the formula for the perimeter of a rectangle: Perimeter = 2(length + width)

Step 2: Substitute the given values: Perimeter = 2(12 cm + 5 cm)

Step 3: Simplify: Perimeter = 2(17 cm)

Step 4: Multiply: Perimeter = 34 cm

Answer: The perimeter of the rectangle is 34 cm.

Problem 52:

Determine the volume of a cone with radius 4 cm and height 8 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a cone: Volume = (1/3) × πr²h

Step 2: Substitute the given values: Volume = (1/3) × 3.14 × (4 cm)² × 8 cm

Step 3: Simplify: Volume = (1/3) × 3.14 × 16 cm² × 8 cm

Step 4: Multiply: Volume = (1/3) × 3.14 × 128 cm³

Step 5: Simplify further: Volume ≈ 134.19 cm³ (rounded to two decimal places)

Answer: The volume of the cone is approximately 134.19 cm³.

Problem 53:

Solve the system of equations:

3x + 2y = 10

4x - 5y = -3

Solution:

Step 1: Multiply the first equation by 4: 12x + 8y = 40

Step 2: Multiply the second equation by 3: 12x - 15y = -9

Step 3: Subtract the second equation from the first equation: (12x + 8y) - (12x - 15y) = 40 - (-9)

Step 4: Simplify: 12x + 8y - 12x + 15y = 40 + 9

Step 5: Combine like terms: 23y = 49

Step 6: Divide both sides by 23: y = 49/23

Step 7: Substitute the value of y into the first equation: 3x + 2(49/23) = 10

Step 8: Simplify: 3x + 98/23 = 10

Step 9: Subtract 98/23 from both sides: 3x = 10 - 98/23

Step 10: Simplify: 3x = 230/23 - 98/23

Step 11: Combine like terms: 3x = 132/23

Step 12: Divide both sides by 3: x = (132/23) / 3

Step 13: Simplify: x = 132/69

Step 14: Simplify further: x = 44/23

Answer: x = 44/23 and y = 49/23

Problem 54:

Find the derivative of the function f(x) = sin(x) + cos(x).

Solution:

Step 1: Take the derivative of each term using the sum rule:

f'(x) = cos(x) - sin(x)

Answer: f'(x) = cos(x) - sin(x)

Problem 55:

Simplify the expression: √(36) + √(49)

Solution:

Step 1: Evaluate the square roots: 6 + 7

Step 2: Simplify: 13

Answer: 13

Problem 56:

Solve the trigonometric equation: tan(x) = 1

Solution:

Step 1: Take the inverse tangent of both sides: x = arctan(1)

Step 2: Use a calculator or reference table to find the angle: x = π/4

Answer: x = π/4

Problem 57:

Calculate the factorial of 7 (written as 7!).

Solution:

Step 1: Multiply 7 by all positive integers less than it: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Step 2: Simplify: 7! = 5040

Answer: 7! = 5040

Problem 58:

Determine the area of a parallelogram with base length 6 cm and height 8 cm.

Solution:

Step 1: Use the formula for the area of a parallelogram: Area = base × height

Step 2: Substitute the given values: Area = 6 cm × 8 cm

Step 3: Multiply: Area = 48 cm²

Answer: The area of the parallelogram is 48 cm².

Problem 59:

Solve the logarithmic equation: log₃(x) + log₃(x - 4) = 2

Solution:

Step 1: Combine the logarithms using the product rule: log₃(x(x - 4)) = 2

Step 2: Simplify the product inside the logarithm: log₃(x² - 4x) = 2

Step 3: Rewrite the equation in exponential form: 3² = x² - 4x

Step 4: Simplify the exponent: 9 = x² - 4x

Step 5: Rearrange the equation: x² - 4x - 9 = 0

Step 6: Solve the quadratic equation using factoring, completing the square, or the quadratic formula:

Factoring: (x - 3)(x + 3) = 0

Setting each factor equal to zero: x - 3 = 0 or x + 3 = 0

Solving for x: x = 3 or x = -3

Answer: x = 3 or x = -3

Problem 60:

Evaluate the integral: ∫(2x + 5) dx

Solution:

Step 1: Apply the power rule for integration: ∫(2x + 5) dx = x² + 5x + C

Answer: ∫(2x + 5) dx = x² + 5x + C (where C is the constant of integration)

Problem 61:

Find the equation of the line passing through the points (-1, 3) and (2, -4).

Solution:

Step 1: Calculate the slope using the formula: m = (y₂ - y₁) / (x₂ - x₁)

Step 2: Substitute the coordinates: m = (-4 - 3) / (2 - (-1)) = -7 / 3

Step 3: Use the point-slope form of a line: y - y₁ = m(x - x₁)

Step 4: Substitute the slope and one point: y - 3 = (-7/3)(x - (-1))

Step 5: Simplify: y - 3 = (-7/3)(x + 1)

Step 6: Rearrange the equation: y = (-7/3)x - 7/3 + 9/3

Step 7: Simplify: y = (-7/3)x + 2/3

Answer: The equation of the line is y = (-7/3)x + 2/3

Problem 62:

Solve the quadratic equation: x² - 6x + 5 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (x - 5)(x - 1) = 0

Setting each factor equal to zero: x - 5 = 0 or x - 1 = 0

Solving for x: x = 5 or x = 1

Answer: x = 5 or x = 1

Problem 63:

Simplify the expression: 2(x - 3) - 3(2x + 1)

Solution:

Step 1: Distribute the 2 and -3: 2x - 6 - 6x - 3

Step 2: Combine like terms: (2x - 6x) + (-6 - 3)

Step 3: Simplify: -4x - 9

Answer: -4x - 9

Problem 64:

Find the value of x in the equation: 4(x + 2) = 3(x - 1) + 5

Solution:

Step 1: Distribute the 4 and 3: 4x + 8 = 3x - 3 + 5

Step 2: Combine like terms: 4x + 8 = 3x + 2

Step 3: Subtract 3x from both sides: x + 8 = 2

Step 4: Subtract 8 from both sides: x = 2 - 8

Step 5: Simplify: x = -6

Answer: x = -6

Problem 65:

Solve the inequality: 3x + 5 ≥ 2x - 3

Solution:

Step 1: Subtract 2x from both sides: x + 5 ≥ -3

Step 2: Subtract 5 from both sides: x ≥ -8

Answer: x ≥ -8

Problem 66:

Evaluate the expression: 5² - 4(3 + 1)

Solution:

Step 1: Simplify within parentheses: 5² - 4(4)

Step 2: Evaluate the exponent: 5² = 25

Step 3: Multiply: 25 - 4(4)

Step 4: Simplify further: 25 - 16

Step 5: Subtract: 9

Answer: 9

Problem 67:

Factorize the quadratic expression: 3x² + 7x - 2

Solution:

Step 1: Find two numbers that multiply to -6 and add up to 7. In this case, 8 and -1 satisfy these conditions.

Step 2: Rewrite the middle term: 3x² + 8x - x - 2

Step 3: Group the terms and factor by grouping: (3x² + 8x) + (-x - 2)

Step 4: Factor out the common binomial: x(3x + 8) - 1(3x + 8)

Step 5: Combine like terms: (x - 1)(3x + 8)

Answer: (x - 1)(3x + 8)

Problem 68:

Calculate the perimeter of a triangle with side lengths 4 cm, 6 cm, and 7 cm.

Solution:

Step 1: Add the lengths of all three sides: 4 cm + 6 cm + 7 cm

Step 2: Add: 17 cm

Answer: The perimeter of the triangle is 17 cm.

Problem 69:

Determine the volume of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

Solution:

Step 1: Use the formula for the volume of a rectangular prism: Volume = length × width × height

Step 2: Substitute the given values: Volume = 8 cm × 5 cm × 3 cm

Step 3: Multiply: Volume = 120 cm³

Answer: The volume of the rectangular prism is 120 cm³.

Problem 70:

Solve the system of equations:

2x + y = 5

3x - 2y = 4

Solution:

Step 1: Multiply the first equation by 2: 4x + 2y = 10

Step 2: Add the equations: (4x + 2y) + (3x - 2y) = 10 + 4

Step 3: Simplify: 7x = 14

Step 4: Divide both sides by 7: x = 2

Step 5: Substitute the value of x into the first equation: 2(2) + y = 5

Step 6: Simplify: 4 + y = 5

Step 7: Subtract 4 from both sides: y = 5 - 4

Step 8: Simplify: y = 1

Answer: x = 2 and y = 1

Problem 71:

Find the derivative of the function f(x) = e^x + 3x².

Solution:

Step 1: Take the derivative of each term using the sum rule and the power rule:

f'(x) = e^x + 6x

Answer: f'(x) = e^x + 6x

Problem 72:

Simplify the expression: log₄(16) + log₄(2)

Solution:

Step 1: Evaluate the logarithms: 2 + 1

Step 2: Simplify: 3

Answer: 3

Problem 73:

Solve the trigonometric equation: sin(x) = 0.5

Solution:

Step 1: Take the inverse sine of both sides: x = arcsin(0.5)

Step 2: Use a calculator or reference table to find the angle: x = π/6 or x = 5π/6

Answer: x = π/6 or x = 5π/6

Problem 74:

Calculate the factorial of 6 (written as 6!).

Solution:

Step 1: Multiply 6 by all positive integers less than it: 6! = 6 × 5 × 4 × 3 × 2 × 1

Step 2: Simplify: 6! = 720

Answer: 6! = 720

Problem 75:

Determine the area of a trapezoid with bases of length 10 cm and 6 cm, and a height of 8 cm.

Solution:

Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h

Step 2: Substitute the given values: Area = (1/2) × (10 cm + 6 cm) × 8 cm

Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm

Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 76:

Simplify the expression: cos²(x) - sin²(x)

Solution:

Step 1: Use the trigonometric identity cos²(x) - sin²(x) = cos(2x).

Step 2: Simplify the expression: cos²(x) - sin²(x) = cos(2x).

Answer: cos²(x) - sin²(x) simplifies to cos(2x).

Problem 77:

Find the equation of the line perpendicular to the line 2x - 3y = 5 and passing through the point (4, 2).

Solution:

Step 1: Find the slope of the given line by rearranging it in slope-intercept form: 2x - 3y = 5 -> -3y = -2x + 5 -> y = (2/3)x - 5/3

Step 2: Determine the slope of the perpendicular line by taking the negative reciprocal of (2/3): m = -3/2

Step 3: Use the point-slope form of a line with the given point: y - 2 = (-3/2)(x - 4)

Step 4: Simplify: y - 2 = (-3/2)x + 6

Step 5: Rearrange the equation: y = (-3/2)x + 8

Answer: The equation of the line perpendicular to 2x - 3y = 5 and passing through the point (4, 2) is y = (-3/2)x + 8.

Problem 78:

Solve the quadratic equation: 2x² + 5x - 3 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (2x - 1)(x + 3) = 0

Setting each factor equal to zero: 2x - 1 = 0 or x + 3 = 0

Solving for x: x = 1/2 or x = -3

Answer: x = 1/2 or x = -3

Problem 79:

Evaluate the integral: ∫(3x² - 2x + 4) dx

Solution:

Step 1: Apply the power rule for integration: ∫(3x² - 2x + 4) dx = x³ - x² + 4x + C

Answer: ∫(3x² - 2x + 4) dx = x³ - x² + 4x + C (where C is the constant of integration)

Problem 80:

Find the equation of the circle with center (-2, 3) and radius 5.

Solution:

Step 1: Use the standard form of the equation for a circle: (x - h)² + (y - k)² = r²

Step 2: Substitute the given values: (x - (-2))² + (y - 3)² = 5²

Step 3: Simplify: (x + 2)² + (y - 3)² = 25

Answer: The equation of the circle is (x + 2)² + (y - 3)² = 25.

Problem 81:

Simplify the expression: 3! + 4! - 2!

Solution:

Step 1: Evaluate the factorials: 3! = 3 × 2 × 1 = 6, 4! = 4 × 3 × 2 × 1 = 24, 2! = 2 × 1 = 2

Step 2: Simplify: 6 + 24 - 2 = 28

Answer: 3! + 4! - 2! simplifies to 28.

Problem 82:

Find the value of x in the equation: log₂(x) = 5

Solution:

Step 1: Rewrite the equation in exponential form: 2⁵ = x

Step 2: Evaluate the exponent: 32 = x

Answer: x = 32

Problem 83:

Solve the trigonometric equation: cos²(x) + sin²(x) = 1

Solution:

Step 1: Use the Pythagorean identity for trigonometric functions: cos²(x) + sin²(x) = 1

Answer: The equation cos²(x) + sin²(x) = 1 holds true for all values of x.

Problem 84:

Calculate the perimeter of a square with side length 10 cm.

Solution:

Step 1: Use the formula for the perimeter of a square: Perimeter = 4 × side length

Step 2: Substitute the given value: Perimeter = 4 × 10 cm

Step 3: Multiply: Perimeter = 40 cm

Answer: The perimeter of the square is 40 cm.

Problem 85:

Determine the volume of a sphere with radius 3 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a sphere: Volume = (4/3) × π × radius³

Step 2: Substitute the given value: Volume = (4/3) × 3.14 × (3 cm)³

Step 3: Simplify: Volume = (4/3) × 3.14 × 27 cm³

Step 4: Multiply: Volume ≈ 113.04 cm³ (rounded to two decimal places)

Answer: The volume of the sphere is approximately 113.04 cm³.

Problem 86:

Solve the system of equations:

2x + 3y = 7

4x - 5y = 1

Solution:

Step 1: Multiply the first equation by 2: 4x + 6y = 14

Step 2: Multiply the second equation by 3: 12x - 15y = 3

Step 3: Add the equations: (4x + 6y) + (12x - 15y) = 14 + 3

Step 4: Simplify: 16x - 9y = 17

Step 5: Solve for y in terms of x: -9y = 17 - 16x

Step 6: Divide both sides by -9: y = (16x - 17)/9

Step 7: Substitute the value of y into the first equation: 2x + 3[(16x - 17)/9] = 7

Step 8: Simplify: 2x + (48x - 51)/9 = 7

Step 9: Multiply through by 9 to eliminate the denominator: 18x + 48x - 51 = 63

Step 10: Combine like terms: 66x - 51 = 63

Step 11: Add 51 to both sides: 66x = 114

Step 12: Divide both sides by 66: x = 114/66

Step 13: Simplify: x = 19/11

Answer: x = 19/11, y = (16(19/11) - 17)/9

Problem 87:

Find the derivative of the function f(x) = 5x³ - 2x² + 3x - 1.

Solution:

Step 1: Take the derivative of each term using the power rule:

f'(x) = (3)(5x²) - (2)(2x) + 3

Step 2: Simplify: f'(x) = 15x² - 4x + 3

Answer: f'(x) = 15x² - 4x + 3

Problem 88:

Simplify the expression: √(64) - √(9)

Solution:

Step 1: Evaluate the square roots: 8 - 3

Step 2: Simplify: 5

Answer: 5

Problem 89:

Solve the equation: 2x + 5 = 3x - 1

Solution:

Step 1: Subtract 2x from both sides: 5 = x - 1

Step 2: Add 1 to both sides: 6 = x

Answer: x = 6

Problem 90:

Find the equation of the line parallel to the line 3x + 2y = 10 and passing through the point (1, 4).

Solution:

Step 1: Rewrite the given equation in slope-intercept form: 2y = -3x + 10 -> y = (-3/2)x + 5

Step 2: Determine the slope of the given line: The coefficient of x is -3/2, so the slope is -3/2.

Step 3: The line parallel to this line will have the same slope, -3/2.

Step 4: Use the point-slope form of a line with the given point: y - 4 = (-3/2)(x - 1)

Step 5: Simplify: y - 4 = (-3/2)x + 3/2

Step 6: Rearrange the equation: y = (-3/2)x + 3/2 + 4

Step 7: Simplify further: y = (-3/2)x + 11/2

Answer: The equation of the line parallel to 3x + 2y = 10 and passing through the point (1, 4) is y = (-3/2)x + 11/2.

Problem 91:

Solve the quadratic equation: x² - 8x + 12 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (x - 2)(x - 6) = 0

Setting each factor equal to zero: x - 2 = 0 or x - 6 = 0

Solving for x: x = 2 or x = 6

Answer: x = 2 or x = 6

Problem 92:

Evaluate the integral: ∫(2cos(x) - 3sin(x)) dx

Solution:

Step 1: Apply the integral rules for cosine and sine functions: ∫(2cos(x) - 3sin(x)) dx = 2∫cos(x) dx - 3∫sin(x) dx

Step 2: Evaluate the integrals using the antiderivatives: 2sin(x) + 3cos(x) + C

Answer: ∫(2cos(x) - 3sin(x)) dx = 2sin(x) + 3cos(x) + C (where C is the constant of integration)

Problem 93:

Find the equation of the parabola with vertex (-2, 3) and focus (-2, 5).

Solution:

Step 1: The parabola is vertical, so the equation takes the form (x - h)² = 4p(y - k), where (h, k) is the vertex.

Step 2: Substitute the given values: (x + 2)² = 4p(y - 3)

Step 3: Since the focus is two units above the vertex, the distance between the focus and vertex is 2p = 2.

Step 4: Solve for p: p = 1

Step 5: Substitute the value of p into the equation: (x + 2)² = 4(y - 3)

Step 6: Simplify: (x + 2)² = 4y - 12

Step 7: Expand and rearrange the equation: x² + 4x + 4 = 4y

Step 8: Divide both sides by 4: (1/4)x² + x + 1 = y

Answer: The equation of the parabola is y = (1/4)x² + x + 1.

Problem 94:

Simplify the expression: 3³ - 2² + 5¹

Solution:

Step 1: Evaluate the exponents and addition: 27 - 4 + 5

Step 2: Simplify: 28

Answer: 28

Problem 95:

Find the value of x in the equation: 4(x - 1) = 3(x + 2)

Solution:

Step 1: Distribute the 4 and 3: 4x - 4 = 3x + 6

Step 2: Subtract 3x from both sides: x - 4 = 6

Step 3: Add 4 to both sides: x = 10

Answer: x = 10

Problem 96:

Solve the inequality: 2x + 5 > 3x - 1

Solution:

Step 1: Subtract 2x from both sides: 5 > x - 1

Step 2: Add 1 to both sides: 6 > x

Answer: x < 6

Problem 97:

Evaluate the expression: 2³ + 3² - 4¹

Solution:

Step 1: Evaluate the exponents and addition: 8 + 9 - 4

Step 2: Simplify: 13

Answer: 13

Problem 98:

Factorize the quadratic expression: x² - 9

Solution:

Step 1: Recognize the difference of squares pattern: x² - 9 = (x - 3)(x + 3)

Answer: (x - 3)(x + 3)

Problem 99:

Calculate the area of a triangle with base length 5 cm and height 8 cm.

Solution:

Step 1: Use the formula for the area of a triangle: Area = (1/2) × base × height

Step 2: Substitute the given values: Area = (1/2) × 5 cm × 8 cm

Step 3: Multiply: Area = 20 cm²

Answer: The area of the triangle is 20 cm².

Problem 100:

Solve the logarithmic equation: log₄(x - 1) + log₄(x + 1) = 2

Solution:

Step 1: Combine the logarithms using the product rule: log₄((x - 1)(x + 1)) = 2

Step 2: Simplify the product inside the logarithm: log₄(x² - 1) = 2

Step 3: Rewrite the equation in exponential form: 4² = x² - 1

Step 4: Evaluate the exponent: 16 = x² - 1

Step 5: Add 1 to both sides: 17 = x²

Step 6: Take the square root of both sides: x = ±√17

Answer: x = ±√17

Problem 1:

Solve the equation: 3x + 5 = 17

Solution:

Step 1: Subtract 5 from both sides: 3x = 17 - 5

Step 2: Simplify: 3x = 12

Step 3: Divide both sides by 3: x = 12 ÷ 3

Step 4: Simplify: x = 4

Answer: x = 4

Problem 2:

Find the value of x in the equation: 2(3x - 1) = 10

Solution:

Step 1: Distribute the 2: 6x - 2 = 10

Step 2: Add 2 to both sides: 6x = 10 + 2

Step 3: Simplify: 6x = 12

Step 4: Divide both sides by 6: x = 12 ÷ 6

Step 5: Simplify: x = 2

Answer: x = 2

Problem 3:

Solve the following inequality: 2x + 3 > 9

Solution:

Step 1: Subtract 3 from both sides: 2x > 9 - 3

Step 2: Simplify: 2x > 6

Step 3: Divide both sides by 2 (since 2 is positive): x > 6 ÷ 2

Step 4: Simplify: x > 3

Answer: x > 3

Problem 4:

Evaluate the expression: 4 + 2 × 3 - 1

Solution:

Step 1: Perform multiplication: 4 + 6 - 1

Step 2: Perform addition and subtraction (left to right): 10 - 1

Step 3: Simplify: 9

Answer: 9

Problem 5:

Factorize the quadratic expression: x² - 4x - 12

Solution:

Step 1: Find two numbers that multiply to -12 and add up to -4. In this case, -6 and 2 satisfy these conditions.

Step 2: Rewrite the middle term: x² - 6x + 2x - 12

Step 3: Group the terms and factor by grouping: x(x - 6) + 2(x - 6)

Step 4: Factor out the common binomial: (x + 2)(x - 6)

Answer: (x + 2)(x - 6)

Problem 6:

Find the area of a rectangle with length 8 cm and width 5 cm.

Solution:

Step 1: Use the formula for the area of a rectangle: Area = length × width

Step 2: Substitute the given values: Area = 8 cm × 5 cm

Step 3: Multiply: Area = 40 cm²

Answer: 40 cm²

Problem 7:

Calculate the volume of a sphere with radius 3 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a sphere: Volume = (4/3)πr³

Step 2: Substitute the given values: Volume = (4/3) × 3.14 × (3 cm)³

Step 3: Simplify: Volume = (4/3) × 3.14 × 27 cm³

Step 4: Multiply: Volume ≈ 113.04 cm³ (rounded to two decimal places)

Answer: Volume ≈ 113.04 cm³

Problem 8:

Solve the system of equations:

2x + 3y = 10

4x - 5y = 8

Solution:

Step 1: Multiply the first equation by 2: 4x + 6y = 20

Step 2: Add the second equation: (4x + 6y) + (4x - 5y) = 20 + 8

Step 3: Simplify: 8x + y = 28

Step 4: Solve for y: y = 28 - 8x

Step 5: Substitute the value of y into the first equation: 2x + 3(28 - 8x) = 10

Step 6: Simplify: 2x + 84 - 24x = 10

Step 7: Combine like terms: -22x + 84 = 10

Step 8: Subtract 84 from both sides: -22x = 10 - 84

Step 9: Simplify: -22x = -74

Step 10: Divide both sides by -22: x = -74 ÷ -22

Step 11: Simplify: x = 3.36 (rounded to two decimal places)

Step 12: Substitute the value of x into the equation y = 28 - 8x: y = 28 - 8(3.36)

Step 13: Simplify: y = 28 - 26.88

Step 14: Simplify further: y ≈ 1.12 (rounded to two decimal places)

Answer: x ≈ 3.36 and y ≈ 1.12

Problem 9:

Find the derivative of the function f(x) = 3x² + 2x - 1.

Solution:

Step 1: Use the power rule for derivatives: f'(x) = 2(3x²) + 1(2x) + 0

Step 2: Simplify: f'(x) = 6x² + 2x

Answer: f'(x) = 6x² + 2x

Problem 10:

Simplify the expression: √(64) - √(16)

Solution:

Step 1: Evaluate the square roots: 8 - 4

Step 2: Simplify: 4

Answer: 4

Problem 11:

Solve the trigonometric equation: sin(x) = 0.5

Solution:

Step 1: Take the inverse sine of both sides: x = arcsin(0.5)

Step 2: Use a calculator or reference table to find the angle: x ≈ 30°

Answer: x ≈ 30°

Problem 12:

Calculate the factorial of 6 (written as 6!).

Solution:

Step 1: Multiply 6 by all positive integers less than it: 6! = 6 × 5 × 4 × 3 × 2 × 1

Step 2: Simplify: 6! = 720

Answer: 6! = 720

Problem 13:

Determine the perimeter of a triangle with side lengths 5 cm, 7 cm, and 9 cm.

Solution:

Step 1: Add the lengths of all three sides: Perimeter = 5 cm + 7 cm + 9 cm

Step 2: Simplify: Perimeter = 21 cm

Answer: Perimeter = 21 cm

Problem 14:

Solve the logarithmic equation: log₂(x) + log₂(2x + 6) = 3

Solution:

Step 1: Combine the logarithms using the product rule: log₂(x(2x + 6)) = 3

Step 2: Simplify the product inside the logarithm: log₂(2x² + 6x) = 3

Step 3: Rewrite the equation in exponential form: 2x² + 6x = 2³

Step 4: Simplify the exponent: 2x² + 6x = 8

Step 5: Rearrange the equation: 2x² + 6x - 8 = 0

Step 6: Factorize: (x - 1)(2x + 8) = 0

Step 7: Set each factor equal to zero and solve: x - 1 = 0 or 2x + 8 = 0

Step 8: Solve for x: x = 1 or x = -4

Answer: x = 1 or x = -4

Problem 15:

Evaluate the integral: ∫(4x² + 3x + 2) dx

Solution:

Step 1: Apply the power rule for integration: ∫(4x² + 3x + 2) dx = (4/3)x³ + (3/2)x² + 2x + C

Answer: ∫(4x² + 3x + 2) dx = (4/3)x³ + (3/2)x² + 2x + C

Problem 16:

Find the equation of a line passing through the points (2, 5) and (4, 9).

Solution:

Step 1: Calculate the slope using the formula: m = (y₂ - y₁) / (x₂ - x₁)

Step 2: Substitute the coordinates: m = (9 - 5) / (4 - 2) = 4 / 2 = 2

Step 3: Use the point-slope form of a line: y - y₁ = m(x - x₁)

Step 4: Substitute the slope and one point: y - 5 = 2(x - 2)

Step 5: Simplify: y - 5 = 2x - 4

Step 6: Rearrange the equation: y = 2x + 1

Answer: The equation of the line is y = 2x + 1

Problem 17:

Solve the quadratic equation: 2x² - 5x - 3 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Since factorizing is not possible in this case, we'll use the quadratic formula.

Step 2: Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values from the given equation: x = (5 ± √(5² - 4(2)(-3))) / (2(2))

Step 3: Simplify: x = (5 ± √(25 + 24)) / 4

x = (5 ± √(49)) / 4

x = (5 ± 7) / 4

For the positive root:

x₁ = (5 + 7) / 4

x₁ = 12 / 4

x₁ = 3

For the negative root:

x₂ = (5 - 7) / 4

x₂ = -2 / 4

x₂ = -0.5

Answer: x = 3 or x = -0.5

Problem 18:

Simplify the complex number expression: (2 + 3i) + (4 - 2i)

Solution:

Step 1: Add the real parts: 2 + 4 = 6

Step 2: Add the imaginary parts: 3i - 2i = i

Answer: (2 + 3i) + (4 - 2i) = 6 + i

Problem 19:

Find the median of the following set of numbers: 5, 3, 9, 1, 7

Solution:

Step 1: Arrange the numbers in ascending order: 1, 3, 5, 7, 9

Step 2: Determine the middle value: The median is the middle number, which is 5.

Answer: The median is 5.

Problem 20:

Calculate the probability of rolling a 6 on a fair six-sided die.

Solution:

Step 1: Determine the number of favorable outcomes: There is only one favorable outcome, rolling a 6.

Step 2: Determine the total number of possible outcomes: There are six possible outcomes, one for each side of the die.

Step 3: Calculate the probability: Probability = favorable outcomes / total outcomes = 1/6

Answer: The probability of rolling a 6 on a fair six-sided die is 1/6.

Problem 21:

Simplify the expression: log₅(125)

Solution:

Step 1: Determine the exponent that gives 125 when raised to the base 5: 5³ = 125

Step 2: Simplify: log₅(125) = 3

Answer: log₅(125) = 3

Problem 22:

Solve the exponential equation: 2^(x + 1) = 16

Solution:

Step 1: Rewrite 16 as a power of 2: 16 = 2^4

Step 2: Set the exponents equal to each other: x + 1 = 4

Step 3: Solve for x: x = 4 - 1

x = 3

Answer: x = 3

Problem 23:

Find the area of a trapezoid with bases of length 6 cm and 10 cm, and a height of 8 cm.

Solution:

Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h

Step 2: Substitute the given values: Area = (1/2) × (6 cm + 10 cm) × 8 cm

Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm

Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 24:

Solve the matrix equation: [2 1] [x] = [5]

[3 4] [y] [8]

Solution:

Step 1: Multiply the matrix and the column vector: 2x + y = 5 and 3x + 4y = 8

Step 2: Solve the system of equations using any appropriate method (substitution, elimination, etc.):

Multiply the first equation by 3: 6x + 3y = 15

Multiply the second equation by 2: 6x + 8y = 16

Subtract the first equation from the second equation: 6x + 8y - (6x + 3y) = 16 - 15

Simplify: 6x + 8y - 6x - 3y = 1

Simplify further: 5y = 1

Solve for y: y = 1/5

Substitute the value of y into the first equation: 2x + (1/5) = 5

Simplify: 2x + 1/5 = 5

Subtract 1/5 from both sides: 2x = 5 - 1/5

Simplify: 2x = 25/5 - 1/5

Simplify further: 2x = 24/5

Divide both sides by 2: x = (24/5) / 2

Simplify: x = 24/10

Simplify further: x = 12/5

Answer: x = 12/5 and y = 1/5

Problem 25:

Evaluate the limit: lim(x → 0) (3x² + 2x + 1) / x

Solution:

Step 1: Substitute the value of x into the expression: (3(0)² + 2(0) + 1) / 0

Step 2: Simplify: (0 + 0 + 1) / 0

Step 3: Since the denominator is zero, the limit is undefined.

Answer: The limit does not exist.

Problem 26:

Simplify the expression: 4x - 2(3x + 5)

Solution:

Step 1: Distribute the -2: 4x - 6x - 10

Step 2: Combine like terms: -2x - 10

Answer: -2x - 10

Problem 27:

Find the value of x in the equation: 2(x - 3) + 5 = 17

Solution:

Step 1: Distribute the 2: 2x - 6 + 5 = 17

Step 2: Combine like terms: 2x - 1 = 17

Step 3: Add 1 to both sides: 2x = 18

Step 4: Divide both sides by 2: x = 9

Answer: x = 9

Problem 28:

Solve the inequality: 3x + 7 > 4x - 5

Solution:

Step 1: Subtract 3x from both sides: 7 > x - 5

Step 2: Add 5 to both sides: 12 > x

Answer: x < 12

Problem 29:

Evaluate the expression: 3(2² - 1) + 4

Solution:

Step 1: Simplify the exponent: 3(4 - 1) + 4

Step 2: Simplify the parentheses: 3(3) + 4

Step 3: Multiply: 9 + 4

Answer: 13

Problem 30:

Factorize the quadratic expression: x² + 7x + 10

Solution:

Step 1: Find two numbers that multiply to 10 and add up to 7. In this case, 2 and 5 satisfy these conditions.

Step 2: Rewrite the middle term: x² + 2x + 5x + 10

Step 3: Group the terms and factor by grouping: (x² + 2x) + (5x + 10)

Step 4: Factor out the common binomial: x(x + 2) + 5(x + 2)

Step 5: Combine like terms: (x + 2)(x + 5)

Answer: (x + 2)(x + 5)

Problem 31:

Calculate the perimeter of a square with side length 9 cm.

Solution:

Step 1: Use the formula for the perimeter of a square: Perimeter = 4 × side length

Step 2: Substitute the given value: Perimeter = 4 × 9 cm

Answer: Perimeter = 36 cm

Problem 32:

Determine the volume of a cylinder with radius 5 cm and height 10 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a cylinder: Volume = πr²h

Step 2: Substitute the given values: Volume = 3.14 × (5 cm)² × 10 cm

Step 3: Simplify: Volume = 3.14 × 25 cm² × 10 cm

Step 4: Multiply: Volume = 785 cm³

Answer: Volume = 785 cm³

Problem 33:

Solve the system of equations:

2x + 3y = 10

5x - 2y = 7

Solution:

Step 1: Multiply the first equation by 2: 4x + 6y = 20

Step 2: Multiply the second equation by 3: 15x - 6y = 21

Step 3: Add the equations: (4x + 6y) + (15x - 6y) = 20 + 21

Step 4: Simplify: 19x = 41

Step 5: Divide both sides by 19: x = 41/19

Step 6: Substitute the value of x into the first equation: 2(41/19) + 3y = 10

Step 7: Simplify: 82/19 + 3y = 10

Step 8: Subtract 82/19 from both sides: 3y = 10 - 82/19

Step 9: Simplify: 3y = 190/19 - 82/19

Step 10: Combine like terms: 3y = 108/19

Step 11: Divide both sides by 3: y = 108/57

Step 12: Simplify: y = 6/19

Answer: x = 41/19 and y = 6/19

Problem 34:

Find the derivative of the function f(x) = 4x³ + 2x² - 1.

Solution:

Step 1: Take the derivative of each term using the power rule:

f'(x) = 4(3x²) + 2(2x) - 0

Step 2: Simplify: f'(x) = 12x² + 4x

Answer: f'(x) = 12x² + 4x

Problem 35:

Simplify the expression: √(81) - √(25)

Solution:

Step 1: Evaluate the square roots: 9 - 5

Step 2: Simplify: 4

Answer: 4

Problem 36:

Solve the trigonometric equation: cos(x) = 0.5

Solution:

Step 1: Take the inverse cosine of both sides: x = arccos(0.5)

Step 2: Use a calculator or reference table to find the angle: x = π/3 or x = 2π/3

Answer: x = π/3 or x = 2π/3

Problem 37:

Calculate the probability of rolling a 6 on a fair six-sided die.

Solution:

Step 1: Determine the number of favorable outcomes: There is only one favorable outcome, rolling a 6.

Step 2: Determine the total number of possible outcomes: There are six possible outcomes, one for each side of the die.

Step 3: Calculate the probability: Probability = favorable outcomes / total outcomes = 1/6

Answer: The probability of rolling a 6 on a fair six-sided die is 1/6.

Problem 38:

Simplify the expression: log₅(125)

Solution:

Step 1: Determine the exponent that gives 125 when raised to the base 5: 5³ = 125

Step 2: Simplify: log₅(125) = 3

Answer: log₅(125) = 3

Problem 39:

Solve the exponential equation: 2^(x + 1) = 16

Solution:

Step 1: Rewrite 16 as a power of 2: 16 = 2^4

Step 2: Set the exponents equal to each other: x + 1 = 4

Step 3: Solve for x: x = 4 - 1

x = 3

Answer: x = 3

Problem 40:

Find the area of a trapezoid with bases of length 6 cm and 10 cm, and a height of 8 cm.

Solution:

Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h

Step 2: Substitute the given values: Area = (1/2) × (6 cm + 10 cm) × 8 cm

Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm

Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 41:

Solve the matrix equation: [2 1] [x] = [5]

[3 4] [y] [8]

Solution:

Step 1: Multiply the inverse of the matrix on both sides to isolate the variables.

[2 1]⁻¹ [2 1] [x] = [2 1]⁻¹ [5]

[3 4] [y] [3 4] [8]

[x] = [2 1]⁻¹ [5]

[y] [3 4] [8]

Step 2: Calculate the inverse of the matrix [2 1]

[3 4]

The inverse of the matrix is:

[ 4 -1]

[-3 2]

Step 3: Multiply the inverse of the matrix by the right-hand side:

[x] = [ 4 -1] [5]

[y] [-3 2] [8]

Step 4: Simplify:

[x] = [4(5) + (-1)(8)]

[y] [-3(5) + 2(8)]

[x] = [20 - 8]

[y] [-15 + 16]

[x] = [12]

[y] [ 1]

Answer: x = 12 and y = 1

Problem 42:

Evaluate the limit: lim(x → 0) (3x² + 2x + 1) / x

Solution:

Step 1: Substitute the value of x into the expression: (3(0)² + 2(0) + 1) / 0

Step 2: Simplify: (0 + 0 + 1) / 0

Step 3: Since the denominator is zero, the limit is undefined.

Answer: The limit does not exist.

Problem 43:

Solve the logarithmic equation: log₂(x) + log₂(2x + 6) = 3

Solution:

Step 1: Combine the logarithms using the product rule: log₂(x(2x + 6)) = 3

Step 2: Simplify the product inside the logarithm: log₂(2x² + 6x) = 3

Step 3: Rewrite the equation in exponential form: 2x² + 6x = 2³

Step 4: Simplify the exponent: 2x² + 6x = 8

Step 5: Rearrange the equation: 2x² + 6x - 8 = 0

Step 6: Factorize: (x - 1)(2x + 4) = 0

Step 7: Set each factor equal to zero and solve: x - 1 = 0 or 2x + 4 = 0

Step 8: Solve for x: x = 1 or x = -2

Answer: x = 1 or x = -2

Problem 44:

Find the area of a circle with a radius of 6 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the area of a circle: Area = πr²

Step 2: Substitute the given value: Area = 3.14 × (6 cm)²

Step 3: Simplify: Area = 3.14 × 36 cm²

Step 4: Multiply: Area = 113.04 cm²

Answer: The area of the circle is 113.04 cm².

Problem 45:

Solve the equation: 2sin(x) = 1

Solution:

Step 1: Divide both sides by 2: sin(x) = 1/2

Step 2: Use a calculator or reference table to find the angles where sin(x) = 1/2: x = π/6 or x = 5π/6

Answer: x = π/6 or x = 5π/6

Problem 46:

Simplify the expression: (3x + 2y) - (2x - y)

Solution:

Step 1: Distribute the negative sign: 3x + 2y - 2x + y

Step 2: Combine like terms: (3x - 2x) + (2y + y)

Step 3: Simplify: x + 3y

Answer: x + 3y

Problem 47:

Find the value of x in the equation: 5(x - 2) = 3(2x + 1)

Solution:

Step 1: Distribute the 5 and 3: 5x - 10 = 6x + 3

Step 2: Subtract 5x and 6x from both sides: -10 - 3 = 6x - 5x

Step 3: Simplify: -13 = x

Answer: x = -13

Problem 48:

Solve the inequality: 2x + 3 < 5x - 2

Solution:

Step 1: Subtract 2x from both sides: 3 < 3x - 2

Step 2: Add 2 to both sides: 5 < 3x

Step 3: Divide both sides by 3: 5/3 < x

Answer: x > 5/3

Problem 49:

Evaluate the expression: 2² + 3³ - 4⁴

Solution:

Step 1: Evaluate the exponents: 2² + 3³ - 4⁴ = 4 + 27 - 256

Step 2: Simplify: 4 + 27 - 256 = -225

Answer: -225

Problem 50:

Factorize the quadratic expression: 2x² + 5x + 3

Solution:

Step 1: Find two numbers that multiply to 6 and add up to 5. In this case, 2 and 3 satisfy these conditions.

Step 2: Rewrite the middle term: 2x² + 2x + 3x + 3

Step 3: Group the terms and factor by grouping: (2x² + 2x) + (3x + 3)

Step 4: Factor out the common binomial: 2x(x + 1) + 3(x + 1)

Step 5: Combine like terms: (2x + 3)(x + 1)

Answer: (2x + 3)(x + 1)

Problem 51:

Calculate the perimeter of a rectangle with length 12 cm and width 5 cm.

Solution:

Step 1: Use the formula for the perimeter of a rectangle: Perimeter = 2(length + width)

Step 2: Substitute the given values: Perimeter = 2(12 cm + 5 cm)

Step 3: Simplify: Perimeter = 2(17 cm)

Step 4: Multiply: Perimeter = 34 cm

Answer: The perimeter of the rectangle is 34 cm.

Problem 52:

Determine the volume of a cone with radius 4 cm and height 8 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a cone: Volume = (1/3) × πr²h

Step 2: Substitute the given values: Volume = (1/3) × 3.14 × (4 cm)² × 8 cm

Step 3: Simplify: Volume = (1/3) × 3.14 × 16 cm² × 8 cm

Step 4: Multiply: Volume = (1/3) × 3.14 × 128 cm³

Step 5: Simplify further: Volume ≈ 134.19 cm³ (rounded to two decimal places)

Answer: The volume of the cone is approximately 134.19 cm³.

Problem 53:

Solve the system of equations:

3x + 2y = 10

4x - 5y = -3

Solution:

Step 1: Multiply the first equation by 4: 12x + 8y = 40

Step 2: Multiply the second equation by 3: 12x - 15y = -9

Step 3: Subtract the second equation from the first equation: (12x + 8y) - (12x - 15y) = 40 - (-9)

Step 4: Simplify: 12x + 8y - 12x + 15y = 40 + 9

Step 5: Combine like terms: 23y = 49

Step 6: Divide both sides by 23: y = 49/23

Step 7: Substitute the value of y into the first equation: 3x + 2(49/23) = 10

Step 8: Simplify: 3x + 98/23 = 10

Step 9: Subtract 98/23 from both sides: 3x = 10 - 98/23

Step 10: Simplify: 3x = 230/23 - 98/23

Step 11: Combine like terms: 3x = 132/23

Step 12: Divide both sides by 3: x = (132/23) / 3

Step 13: Simplify: x = 132/69

Step 14: Simplify further: x = 44/23

Answer: x = 44/23 and y = 49/23

Problem 54:

Find the derivative of the function f(x) = sin(x) + cos(x).

Solution:

Step 1: Take the derivative of each term using the sum rule:

f'(x) = cos(x) - sin(x)

Answer: f'(x) = cos(x) - sin(x)

Problem 55:

Simplify the expression: √(36) + √(49)

Solution:

Step 1: Evaluate the square roots: 6 + 7

Step 2: Simplify: 13

Answer: 13

Problem 56:

Solve the trigonometric equation: tan(x) = 1

Solution:

Step 1: Take the inverse tangent of both sides: x = arctan(1)

Step 2: Use a calculator or reference table to find the angle: x = π/4

Answer: x = π/4

Problem 57:

Calculate the factorial of 7 (written as 7!).

Solution:

Step 1: Multiply 7 by all positive integers less than it: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Step 2: Simplify: 7! = 5040

Answer: 7! = 5040

Problem 58:

Determine the area of a parallelogram with base length 6 cm and height 8 cm.

Solution:

Step 1: Use the formula for the area of a parallelogram: Area = base × height

Step 2: Substitute the given values: Area = 6 cm × 8 cm

Step 3: Multiply: Area = 48 cm²

Answer: The area of the parallelogram is 48 cm².

Problem 59:

Solve the logarithmic equation: log₃(x) + log₃(x - 4) = 2

Solution:

Step 1: Combine the logarithms using the product rule: log₃(x(x - 4)) = 2

Step 2: Simplify the product inside the logarithm: log₃(x² - 4x) = 2

Step 3: Rewrite the equation in exponential form: 3² = x² - 4x

Step 4: Simplify the exponent: 9 = x² - 4x

Step 5: Rearrange the equation: x² - 4x - 9 = 0

Step 6: Solve the quadratic equation using factoring, completing the square, or the quadratic formula:

Factoring: (x - 3)(x + 3) = 0

Setting each factor equal to zero: x - 3 = 0 or x + 3 = 0

Solving for x: x = 3 or x = -3

Answer: x = 3 or x = -3

Problem 60:

Evaluate the integral: ∫(2x + 5) dx

Solution:

Step 1: Apply the power rule for integration: ∫(2x + 5) dx = x² + 5x + C

Answer: ∫(2x + 5) dx = x² + 5x + C (where C is the constant of integration)

Problem 61:

Find the equation of the line passing through the points (-1, 3) and (2, -4).

Solution:

Step 1: Calculate the slope using the formula: m = (y₂ - y₁) / (x₂ - x₁)

Step 2: Substitute the coordinates: m = (-4 - 3) / (2 - (-1)) = -7 / 3

Step 3: Use the point-slope form of a line: y - y₁ = m(x - x₁)

Step 4: Substitute the slope and one point: y - 3 = (-7/3)(x - (-1))

Step 5: Simplify: y - 3 = (-7/3)(x + 1)

Step 6: Rearrange the equation: y = (-7/3)x - 7/3 + 9/3

Step 7: Simplify: y = (-7/3)x + 2/3

Answer: The equation of the line is y = (-7/3)x + 2/3

Problem 62:

Solve the quadratic equation: x² - 6x + 5 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (x - 5)(x - 1) = 0

Setting each factor equal to zero: x - 5 = 0 or x - 1 = 0

Solving for x: x = 5 or x = 1

Answer: x = 5 or x = 1

Problem 63:

Simplify the expression: 2(x - 3) - 3(2x + 1)

Solution:

Step 1: Distribute the 2 and -3: 2x - 6 - 6x - 3

Step 2: Combine like terms: (2x - 6x) + (-6 - 3)

Step 3: Simplify: -4x - 9

Answer: -4x - 9

Problem 64:

Find the value of x in the equation: 4(x + 2) = 3(x - 1) + 5

Solution:

Step 1: Distribute the 4 and 3: 4x + 8 = 3x - 3 + 5

Step 2: Combine like terms: 4x + 8 = 3x + 2

Step 3: Subtract 3x from both sides: x + 8 = 2

Step 4: Subtract 8 from both sides: x = 2 - 8

Step 5: Simplify: x = -6

Answer: x = -6

Problem 65:

Solve the inequality: 3x + 5 ≥ 2x - 3

Solution:

Step 1: Subtract 2x from both sides: x + 5 ≥ -3

Step 2: Subtract 5 from both sides: x ≥ -8

Answer: x ≥ -8

Problem 66:

Evaluate the expression: 5² - 4(3 + 1)

Solution:

Step 1: Simplify within parentheses: 5² - 4(4)

Step 2: Evaluate the exponent: 5² = 25

Step 3: Multiply: 25 - 4(4)

Step 4: Simplify further: 25 - 16

Step 5: Subtract: 9

Answer: 9

Problem 67:

Factorize the quadratic expression: 3x² + 7x - 2

Solution:

Step 1: Find two numbers that multiply to -6 and add up to 7. In this case, 8 and -1 satisfy these conditions.

Step 2: Rewrite the middle term: 3x² + 8x - x - 2

Step 3: Group the terms and factor by grouping: (3x² + 8x) + (-x - 2)

Step 4: Factor out the common binomial: x(3x + 8) - 1(3x + 8)

Step 5: Combine like terms: (x - 1)(3x + 8)

Answer: (x - 1)(3x + 8)

Problem 68:

Calculate the perimeter of a triangle with side lengths 4 cm, 6 cm, and 7 cm.

Solution:

Step 1: Add the lengths of all three sides: 4 cm + 6 cm + 7 cm

Step 2: Add: 17 cm

Answer: The perimeter of the triangle is 17 cm.

Problem 69:

Determine the volume of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

Solution:

Step 1: Use the formula for the volume of a rectangular prism: Volume = length × width × height

Step 2: Substitute the given values: Volume = 8 cm × 5 cm × 3 cm

Step 3: Multiply: Volume = 120 cm³

Answer: The volume of the rectangular prism is 120 cm³.

Problem 70:

Solve the system of equations:

2x + y = 5

3x - 2y = 4

Solution:

Step 1: Multiply the first equation by 2: 4x + 2y = 10

Step 2: Add the equations: (4x + 2y) + (3x - 2y) = 10 + 4

Step 3: Simplify: 7x = 14

Step 4: Divide both sides by 7: x = 2

Step 5: Substitute the value of x into the first equation: 2(2) + y = 5

Step 6: Simplify: 4 + y = 5

Step 7: Subtract 4 from both sides: y = 5 - 4

Step 8: Simplify: y = 1

Answer: x = 2 and y = 1

Problem 71:

Find the derivative of the function f(x) = e^x + 3x².

Solution:

Step 1: Take the derivative of each term using the sum rule and the power rule:

f'(x) = e^x + 6x

Answer: f'(x) = e^x + 6x

Problem 72:

Simplify the expression: log₄(16) + log₄(2)

Solution:

Step 1: Evaluate the logarithms: 2 + 1

Step 2: Simplify: 3

Answer: 3

Problem 73:

Solve the trigonometric equation: sin(x) = 0.5

Solution:

Step 1: Take the inverse sine of both sides: x = arcsin(0.5)

Step 2: Use a calculator or reference table to find the angle: x = π/6 or x = 5π/6

Answer: x = π/6 or x = 5π/6

Problem 74:

Calculate the factorial of 6 (written as 6!).

Solution:

Step 1: Multiply 6 by all positive integers less than it: 6! = 6 × 5 × 4 × 3 × 2 × 1

Step 2: Simplify: 6! = 720

Answer: 6! = 720

Problem 75:

Determine the area of a trapezoid with bases of length 10 cm and 6 cm, and a height of 8 cm.

Solution:

Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h

Step 2: Substitute the given values: Area = (1/2) × (10 cm + 6 cm) × 8 cm

Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm

Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 76:

Simplify the expression: cos²(x) - sin²(x)

Solution:

Step 1: Use the trigonometric identity cos²(x) - sin²(x) = cos(2x).

Step 2: Simplify the expression: cos²(x) - sin²(x) = cos(2x).

Answer: cos²(x) - sin²(x) simplifies to cos(2x).

Problem 77:

Find the equation of the line perpendicular to the line 2x - 3y = 5 and passing through the point (4, 2).

Solution:

Step 1: Find the slope of the given line by rearranging it in slope-intercept form: 2x - 3y = 5 -> -3y = -2x + 5 -> y = (2/3)x - 5/3

Step 2: Determine the slope of the perpendicular line by taking the negative reciprocal of (2/3): m = -3/2

Step 3: Use the point-slope form of a line with the given point: y - 2 = (-3/2)(x - 4)

Step 4: Simplify: y - 2 = (-3/2)x + 6

Step 5: Rearrange the equation: y = (-3/2)x + 8

Answer: The equation of the line perpendicular to 2x - 3y = 5 and passing through the point (4, 2) is y = (-3/2)x + 8.

Problem 78:

Solve the quadratic equation: 2x² + 5x - 3 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (2x - 1)(x + 3) = 0

Setting each factor equal to zero: 2x - 1 = 0 or x + 3 = 0

Solving for x: x = 1/2 or x = -3

Answer: x = 1/2 or x = -3

Problem 79:

Evaluate the integral: ∫(3x² - 2x + 4) dx

Solution:

Step 1: Apply the power rule for integration: ∫(3x² - 2x + 4) dx = x³ - x² + 4x + C

Answer: ∫(3x² - 2x + 4) dx = x³ - x² + 4x + C (where C is the constant of integration)

Problem 80:

Find the equation of the circle with center (-2, 3) and radius 5.

Solution:

Step 1: Use the standard form of the equation for a circle: (x - h)² + (y - k)² = r²

Step 2: Substitute the given values: (x - (-2))² + (y - 3)² = 5²

Step 3: Simplify: (x + 2)² + (y - 3)² = 25

Answer: The equation of the circle is (x + 2)² + (y - 3)² = 25.

Problem 81:

Simplify the expression: 3! + 4! - 2!

Solution:

Step 1: Evaluate the factorials: 3! = 3 × 2 × 1 = 6, 4! = 4 × 3 × 2 × 1 = 24, 2! = 2 × 1 = 2

Step 2: Simplify: 6 + 24 - 2 = 28

Answer: 3! + 4! - 2! simplifies to 28.

Problem 82:

Find the value of x in the equation: log₂(x) = 5

Solution:

Step 1: Rewrite the equation in exponential form: 2⁵ = x

Step 2: Evaluate the exponent: 32 = x

Answer: x = 32

Problem 83:

Solve the trigonometric equation: cos²(x) + sin²(x) = 1

Solution:

Step 1: Use the Pythagorean identity for trigonometric functions: cos²(x) + sin²(x) = 1

Answer: The equation cos²(x) + sin²(x) = 1 holds true for all values of x.

Problem 84:

Calculate the perimeter of a square with side length 10 cm.

Solution:

Step 1: Use the formula for the perimeter of a square: Perimeter = 4 × side length

Step 2: Substitute the given value: Perimeter = 4 × 10 cm

Step 3: Multiply: Perimeter = 40 cm

Answer: The perimeter of the square is 40 cm.

Problem 85:

Determine the volume of a sphere with radius 3 cm. (Use π ≈ 3.14)

Solution:

Step 1: Use the formula for the volume of a sphere: Volume = (4/3) × π × radius³

Step 2: Substitute the given value: Volume = (4/3) × 3.14 × (3 cm)³

Step 3: Simplify: Volume = (4/3) × 3.14 × 27 cm³

Step 4: Multiply: Volume ≈ 113.04 cm³ (rounded to two decimal places)

Answer: The volume of the sphere is approximately 113.04 cm³.

Problem 86:

Solve the system of equations:

2x + 3y = 7

4x - 5y = 1

Solution:

Step 1: Multiply the first equation by 2: 4x + 6y = 14

Step 2: Multiply the second equation by 3: 12x - 15y = 3

Step 3: Add the equations: (4x + 6y) + (12x - 15y) = 14 + 3

Step 4: Simplify: 16x - 9y = 17

Step 5: Solve for y in terms of x: -9y = 17 - 16x

Step 6: Divide both sides by -9: y = (16x - 17)/9

Step 7: Substitute the value of y into the first equation: 2x + 3[(16x - 17)/9] = 7

Step 8: Simplify: 2x + (48x - 51)/9 = 7

Step 9: Multiply through by 9 to eliminate the denominator: 18x + 48x - 51 = 63

Step 10: Combine like terms: 66x - 51 = 63

Step 11: Add 51 to both sides: 66x = 114

Step 12: Divide both sides by 66: x = 114/66

Step 13: Simplify: x = 19/11

Answer: x = 19/11, y = (16(19/11) - 17)/9

Problem 87:

Find the derivative of the function f(x) = 5x³ - 2x² + 3x - 1.

Solution:

Step 1: Take the derivative of each term using the power rule:

f'(x) = (3)(5x²) - (2)(2x) + 3

Step 2: Simplify: f'(x) = 15x² - 4x + 3

Answer: f'(x) = 15x² - 4x + 3

Problem 88:

Simplify the expression: √(64) - √(9)

Solution:

Step 1: Evaluate the square roots: 8 - 3

Step 2: Simplify: 5

Answer: 5

Problem 89:

Solve the equation: 2x + 5 = 3x - 1

Solution:

Step 1: Subtract 2x from both sides: 5 = x - 1

Step 2: Add 1 to both sides: 6 = x

Answer: x = 6

Problem 90:

Find the equation of the line parallel to the line 3x + 2y = 10 and passing through the point (1, 4).

Solution:

Step 1: Rewrite the given equation in slope-intercept form: 2y = -3x + 10 -> y = (-3/2)x + 5

Step 2: Determine the slope of the given line: The coefficient of x is -3/2, so the slope is -3/2.

Step 3: The line parallel to this line will have the same slope, -3/2.

Step 4: Use the point-slope form of a line with the given point: y - 4 = (-3/2)(x - 1)

Step 5: Simplify: y - 4 = (-3/2)x + 3/2

Step 6: Rearrange the equation: y = (-3/2)x + 3/2 + 4

Step 7: Simplify further: y = (-3/2)x + 11/2

Answer: The equation of the line parallel to 3x + 2y = 10 and passing through the point (1, 4) is y = (-3/2)x + 11/2.

Problem 91:

Solve the quadratic equation: x² - 8x + 12 = 0

Solution:

Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (x - 2)(x - 6) = 0

Setting each factor equal to zero: x - 2 = 0 or x - 6 = 0

Solving for x: x = 2 or x = 6

Answer: x = 2 or x = 6

Problem 92:

Evaluate the integral: ∫(2cos(x) - 3sin(x)) dx

Solution:

Step 1: Apply the integral rules for cosine and sine functions: ∫(2cos(x) - 3sin(x)) dx = 2∫cos(x) dx - 3∫sin(x) dx

Step 2: Evaluate the integrals using the antiderivatives: 2sin(x) + 3cos(x) + C

Answer: ∫(2cos(x) - 3sin(x)) dx = 2sin(x) + 3cos(x) + C (where C is the constant of integration)

Problem 93:

Find the equation of the parabola with vertex (-2, 3) and focus (-2, 5).

Solution:

Step 1: The parabola is vertical, so the equation takes the form (x - h)² = 4p(y - k), where (h, k) is the vertex.

Step 2: Substitute the given values: (x + 2)² = 4p(y - 3)

Step 3: Since the focus is two units above the vertex, the distance between the focus and vertex is 2p = 2.

Step 4: Solve for p: p = 1

Step 5: Substitute the value of p into the equation: (x + 2)² = 4(y - 3)

Step 6: Simplify: (x + 2)² = 4y - 12

Step 7: Expand and rearrange the equation: x² + 4x + 4 = 4y

Step 8: Divide both sides by 4: (1/4)x² + x + 1 = y

Answer: The equation of the parabola is y = (1/4)x² + x + 1.

Problem 94:

Simplify the expression: 3³ - 2² + 5¹

Solution:

Step 1: Evaluate the exponents and addition: 27 - 4 + 5

Step 2: Simplify: 28

Answer: 28

Problem 95:

Find the value of x in the equation: 4(x - 1) = 3(x + 2)

Solution:

Step 1: Distribute the 4 and 3: 4x - 4 = 3x + 6

Step 2: Subtract 3x from both sides: x - 4 = 6

Step 3: Add 4 to both sides: x = 10

Answer: x = 10

Problem 96:

Solve the inequality: 2x + 5 > 3x - 1

Solution:

Step 1: Subtract 2x from both sides: 5 > x - 1

Step 2: Add 1 to both sides: 6 > x

Answer: x < 6

Problem 97:

Evaluate the expression: 2³ + 3² - 4¹

Solution:

Step 1: Evaluate the exponents and addition: 8 + 9 - 4

Step 2: Simplify: 13

Answer: 13

Problem 98:

Factorize the quadratic expression: x² - 9

Solution:

Step 1: Recognize the difference of squares pattern: x² - 9 = (x - 3)(x + 3)

Answer: (x - 3)(x + 3)

Problem 99:

Calculate the area of a triangle with base length 5 cm and height 8 cm.

Solution:

Step 1: Use the formula for the area of a triangle: Area = (1/2) × base × height

Step 2: Substitute the given values: Area = (1/2) × 5 cm × 8 cm

Step 3: Multiply: Area = 20 cm²

Answer: The area of the triangle is 20 cm².

Problem 100:

Solve the logarithmic equation: log₄(x - 1) + log₄(x + 1) = 2

Solution:

Step 1: Combine the logarithms using the product rule: log₄((x - 1)(x + 1)) = 2

Step 2: Simplify the product inside the logarithm: log₄(x² - 1) = 2

Step 3: Rewrite the equation in exponential form: 4² = x² - 1

Step 4: Evaluate the exponent: 16 = x² - 1

Step 5: Add 1 to both sides: 17 = x²

Step 6: Take the square root of both sides: x = ±√17

Answer: x = ±√17

## About This Math Problems Set

In this article, we have covered 25 diverse math problems that span various mathematical concepts, including algebra, geometry, calculus, and more. Each problem was accompanied by a step-by-step explanation of how to solve it, ensuring a clear understanding of the solution process. By practicing these problems, you can enhance your math skills and strengthen your problem-solving abilities. Remember, practice is key to becoming proficient in mathematics.