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To find the expected value E[x^2] given that E[x^5] is known and the variance V[x] is equal to 6 (which also equals E[x^2] minus E[x] squared), we can proceed stepwise. 1. **Understanding Variance (V[x])**: Variance is defined as: V[x] = E[x^2] – (E[x])^2 Given V[x] = 6, we can express this as: E[x^2] = V[x] … Read more

To simplify the expression 7 + 3i² + i, we need to understand the value of i. In mathematics, i is defined as the imaginary unit, and it is equal to the square root of -1. When simplifying the expression, we will substitute i² with -1 since i² = -1. Let’s break down the expression … Read more

To solve the problem, we need to set up a couple of equations based on the information provided. Let’s denote the two numbers as: x = the smaller number y = the larger number According to the problem: The sum of the two numbers is 70: x + y = 70 One number is four … Read more

To compute y and dy for the given expression y = 3x + x^2 + 3x + 0.6, we first simplify the equation: Combine like terms: y = 6x + x^2 + 0.6. Next, we need to substitute the provided values of x and dx. For this example, let’s assume: x = 1 dx = … Read more

To solve the equation 4sin²(x) – 4sin(x) + 1 = 0 in the interval [0, 2π], we will first treat it as a quadratic equation in terms of sin(x). Let us substitute y = sin(x). The equation then becomes: 4y² – 4y + 1 = 0 This can be solved using the quadratic formula: y … Read more

To determine the number of points of intersection between the parabola given by the equation y = x² and the circle centered at (0, 1) with a radius of 3, we’ll start by writing the equation of the circle. The standard form of a circle’s equation is given by: (x – h)² + (y – … Read more

To solve this problem, we first need to define some key terms and relationships between the numbers involved. We know: The HCF of the two numbers is 33. The LCM of the two numbers is 264. The first number (let’s call it A) divided by 2 yields a quotient of 33. The relationship between two … Read more

To find the derivative of the inverse function, we first need to clarify a few concepts regarding inverse functions and derivatives. Let’s denote the function as f(x) = x^(1/3). To find the derivative of the inverse function, we can utilize the relationship between a function and its inverse, given by the formula: (f-1(y))’ = 1 … Read more

To find the greatest common factor (GCF) of the expressions 4k, 18k^4, and 12, we first need to determine the factors of each term. Step 1: Factor Each Term 4k: The factors are 4 and k, which can be broken down to 2 × 2 and includes the variable k. 18k^4: The factors are 18 … Read more

To find the length of AJ, we first need to clarify the relationships between the segments provided. Assuming AB, JK, and AJ are parts of the same geometric configuration, we can derive the value of AJ based on the information given. If AB measures 8 inches, and we denote other lengths as follows: JK = … Read more