This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: Problem 44: Evaluate _1^2 (3x^2 - 1)\,dx Step 1: Find the indefinite integral of the function. (3x^2 - 1)\,dx = 3x^2+12+1 - 1x^0+10+1 + C = x^3 - x + C Step 2: Evaluate the definite integral using the Fundamental Theorem of Calculus. [x^3 - x]_1^2 = (2^3 - 2) - (1^3 - 1) Step 3: Calculate the values. (8 - 2) - (1 - 1) = 6 - 0 = 6 The correct option is A. A. 6 Problem 45: In the diagram below, ABC is an isosceles triangle, |AC|= 12cm, |AB| = |BC| and ABC = 120^. Find |AB|. Step 1: Find the base angles of the isosceles triangle. Since |AB| = |BC|, the angles opposite these sides are equal: BAC = BCA. The sum of angles in a triangle is 180^. BAC + BCA + ABC = 180^ 2 × BAC + 120^ = 180^ 2 × BAC = 60^ BAC = BCA = 30^ Step 2: Use the Sine Rule to find the length of |AB|. The Sine Rule states (a)/( A) = (b)/( B). We want to find |AB|. We know |AC| = 12 cm. (|AB|)/(( BCA)) = (|AC|)/(( ABC)) (|AB|)/((30^)) = 12 cm(120^) Step 3: Substitute the sine values and solve for |AB|. We know (30^) = (1)/(2) and (120^) = (180^ - 60^) = (60^) = sqrt(3)2. (|AB|)/(1)2 = (12)/(sqrt(3))2 |AB| = (1)/(2) × (12)/(sqrt(3))2 |AB| = (12)/(sqrt(3)) Rationalize the denominator: |AB| = 12sqrt(3)3 = 4sqrt(3) cm Step 4: Approximate the value. |AB| ≈ 4 × 1.732 = 6.928 cm Rounding to one decimal place, this is 6.9 cm. The correct option is A. A. 6.9cm Problem 46: The probability that Saka will score for his team Arsenal is (3)/(5) and the probability that Rashford will score for his team Manchester United is (1)/(3). In a game involving the two teams, find the probability that one of the two players will score. Step 1: Define the probabilities of scoring and not scoring for each player. Let P(S) be the probability Saka scores = (3)/(5). Let P(R) be the probability Rashford scores = (1)/(3). The probability Saka does not score, P(S') = 1 - P(S) = 1 - (3)/(5) = (2)/(5). The probability Rashford does not score, P(R') = 1 - P(R) = 1 - (1)/(3) = (2)/(3). Step 2: Identify the scenarios where "one of the two players will score". This means exactly one player scores. There are two mutually exclusive scenarios: • Saka scores AND Rashford does NOT score: P(S and R') • Saka does NOT score AND Rashford scores: P(S' and R) Step 3: Calculate the probability for each scenario (assuming the events are independent). P(S and R') = P(S) × P(R') = (3)/(5) × (2)/(3) = (6)/(15) P(S' and R) = P(S') × P(R) = (2)/(5) × (1)/(3) = (2)/(15) Step 4: Add the probabilities of these two scenarios to find the total probability. P(exactly one scores) = P(S and R') + P(S' and R) = (6)/(15) + (2)/(15) = (8)/(15) The correct option is C. C. (8)/(15) Problem 47: If y = (2)/(3)x^3 - 5x^2 + 3, find (dy)/(dx) Step 1: Differentiate each term of the function with respect to x using the power rule (d)/(dx)(ax^n) = anx^n-1. For the first term, (d)/(dx)((2)/(3)x^3): (2)/(3) × 3x^3-1 = 2x^2 Step 2: Differentiate the second term. For the second term, (d)/(dx)(-5x^2): -5 × 2x^2-1 = -10x Step 3: Differentiate the third term. For the third term, (d)/(dx)(3): The derivative of a constant is 0. Step 4: Combine the derivatives of the terms to get the final derivative. (dy)/(dx) = 2x^2 - 10x + 0 (dy)/(dx) = 2x^2 - 10x The correct option is D. D. 2x^2 - 10x Last free one today — make it count tomorrow, or type /upgrade for unlimited.