If Two Events A and B are Mutually Exclusive, What is the Probability of A Intersection B?
When we say that two events, A and B, are mutually exclusive, we mean that they cannot occur at the same time. In more technical terms, the occurrence of one event means that the other event cannot happen. For example, if A is the event of rolling a 2 on a die, and B is […]
How to use substitution to write an equivalent quadratic equation for 3x² – 7x + 8 = 0?
To use substitution for the quadratic equation 3x² – 7x + 8 = 0, we can define a new variable to simplify the process. Let’s substitute y = x. Substituting this into our original equation, we get: 3y² – 7y + 8 = 0 This is now a quadratic equation in terms of y. The […]
If a fair 6-sided die is rolled three times, what is the probability that exactly one 3 is rolled?
To find the probability of rolling exactly one 3 when a fair 6-sided die is rolled three times, we can use the binomial probability formula. The formula is given by: P(X = k) = C(n, k) * (p^k) * (q^(n-k)) Where: C(n, k) is the number of combinations of n items taken k at a […]
Find the value of k such that the polynomial x² + kx + 6x² + 2k + 1 has sum of its zeroes equal to half of their product
To solve for the value of k in the polynomial x² + kx + 6x² + 2k + 1, we first need to simplify the polynomial. Combining like terms, we get: 7x² + kx + 2k + 1 = 0 The sum of the roots (zeroes) of a quadratic equation ax² + bx + c […]
Circle O has a circumference of 36π cm. Circle O with radius r is shown. What is the length of the radius r?
The circumference of a circle can be calculated using the formula: C = 2πr where C is the circumference and r is the radius. In this case, we know that the circumference (C) is 36π cm. We can set up the equation: 36π = 2πr Next, we can divide both sides of the equation by […]
What are the solutions of 8x² + 6 = 22x?
To solve the equation 8x² + 6 = 22x, we first want to rearrange it into a standard quadratic form, which is ax² + bx + c = 0. We start by moving all terms to one side: 8x² – 22x + 6 = 0 Now, we can use the quadratic formula to find the […]
What is the slope of the line tangent to the curve 3y² + 2x² – 6 – 2xy at the point (3, 2)?
To find the slope of the tangent line to the curve at a given point, we first need to differentiate the equation implicitly. The given curve is: 3y² + 2x² – 6 – 2xy = 0 We differentiate both sides with respect to x: Using implicit differentiation: Differentiate 3y²: 6y(dy/dx) Differentiate 2x²: 4x Differentiate -6: […]
To Prove That Tan 75° Cot 75° = 4
To prove that tan 75° cot 75° = 4, we start with the definitions of tangent and cotangent. We know that: tan θ = sin θ / cos θ cot θ = 1 / tan θ = cos θ / sin θ Thus, we can write: tan 75° cot 75° = tan 75° × cot […]
Which mathematical operator is used to raise five to the second power?
The mathematical operator used to raise five to the second power is the exponentiation operator, commonly represented as a caret symbol (^). In this case, we write it as 5^2. When we say ‘five to the second power,’ we mean that we are multiplying five by itself, which can also be expressed as 5 x […]
What is the y intercept of the function f(x) = 29x + 13?
The y-intercept of a function is the point where the graph crosses the y-axis. This occurs when the value of x is zero. To find the y-intercept of the function f(x) = 29x + 13, we simply substitute x with 0: f(0) = 29(0) + 13 = 0 + 13 = 13 Thus, the y-intercept […]