To estimate the best possible bounds for an integral, we can utilize a few key properties of integrals. Understanding these properties will help you assess the limitations and ranges within which the integral will fall. Here are the steps and concepts involved: 1. Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus connects differentiation and … Read more
To demonstrate that the relation for points in a plane relative to the origin is an equivalence relation, we must establish that it satisfies three essential properties: reflexivity, symmetry, and transitivity. 1. Reflexivity A relation is reflexive if every element is related to itself. In this context, for any point p in the set A, … Read more
To find the greatest possible value of the integral of the function f on the closed interval from 0 to 2, given that f(x) ≤ 4 in that interval, we can apply the properties of integrals and continuous functions. We start with the integral: ∫02 f(x) dx Since we know that the function f has … Read more
To determine the rate of interest earned on an investment of Rs. 100,000 that grows to Rs. 144,000 after two years, we can follow these steps: Identify the principal amount: The principal (P) is Rs. 100,000. Identify the final amount: The amount (A) after two years is Rs. 144,000. Calculate the total interest earned: The … Read more
To find the probability of rolling a 5 on the first roll and then a 2 on the second roll of a die, we need to consider the probabilities of each individual event. 1. **Rolling a 5**: The die has six faces, numbered from 1 to 6. The probability of rolling any specific number, including … Read more
The sine function, denoted as sin x, is a fundamental concept in trigonometry that represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. For any angle x, the value of sin x can range from -1 to 1. This means that no matter what angle you provide, … Read more
Finding the Other Number Given that: LCM = 14 * HCF LCM + HCF = 600 One number = 280 Let HCF = x. Therefore, the LCM can be expressed as: LCM = 14x From the information provided, we can form the equation: 14x + x = 600 Simplifying this gives: 15x = 600 Solving … Read more
To determine the values of b that make the function f continuous at x = 2, we need to ensure that the left-hand limit and the right-hand limit of the function at this point are equal, as well as the value of the function at that point. The function is defined in two pieces: f(x) … Read more
To determine by how much the student’s answer is greater than the correct answer, we first need to calculate both the incorrect multiplication (7236 multiplied by 65) and the correct multiplication (7236 multiplied by 56). Let’s do the math: 1. Calculate the incorrect answer: 7236 multiplied by 65: 7236 × 65 = 469740 2. Calculate … Read more
To find the 50th term of the sequence 5, 2, 9, 16, we first need to determine whether this sequence follows a specific pattern or rule. Let’s look at the sequence in detail: 1st term (n=1): 5 2nd term (n=2): 2 3rd term (n=3): 9 4th term (n=4): 16 Next, we can observe the terms … Read more