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's the solution for question 6: a) i) Write down the equation of the straight line through P and Q. Step 1: Find the gradient m of the line passing through (0, 7) and (7, 0). m = (y_2 - y_1)/(x_2 - x_1) = (0 - 7)/(7 - 0) = (-7)/(7) = -1 Step 2: Use the point-slope form y - y_1 = m(x - x_1) with the point (0, 7). y - 7 = -1(x - 0) y - 7 = -x y = -x + 7 The equation of the straight line is: y = -x + 7 ii) Find the coordinates of P and Q. Step 1: Set the equation of the line equal to the equation of the curve to find the intersection points. The curve is y = (x-1)^2 + 4. The line is y = -x + 7. -x + 7 = (x-1)^2 + 4 Step 2: Expand and simplify the equation. -x + 7 = x^2 - 2x + 1 + 4 -x + 7 = x^2 - 2x + 5 0 = x^2 - 2x + x + 5 - 7 0 = x^2 - x - 2 Step 3: Factor the quadratic equation to find the x-coordinates. (x - 2)(x + 1) = 0 So, x = 2 or x = -1. Step 4: Substitute the x-values back into the line equation y = -x + 7 to find the corresponding y-coordinates. For x = 2: y = -(2) + 7 = 5 Point Q is (2, 5). For x = -1: y = -(-1) + 7 = 1 + 7 = 8 Point P is (-1, 8). The coordinates of P and Q are: P(-1, 8) and Q(2, 5) iii) Calculate the value of the shaded area. Step 1: The shaded area is the area between the line y_L = -x + 7 and the curve y_C = (x-1)^2 + 4 = x^2 - 2x + 5 from x = -1 to x = 2. The line is above the curve in this region. Area = _-1^2 (y_L - y_C)\,dx Area = _-1^2 ((-x + 7) - (x^2 - 2x + 5))\,dx Step 2: Simplify the integrand. Area = _-1^2 (-x + 7 - x^2 + 2x - 5)\,dx Area = _-1^2 (-x^2 + x + 2)\,dx Step 3: Integrate the expression. Area = [-(x^3)/(3) + (x^2)/(2) + 2x]_-1^2 Step 4: Evaluate the definite integral. Area = (-((2)^3)/(3) + ((2)^2)/(2) + 2(2)) - (-((-1)^3)/(3) + ((-1)^2)/(2) + 2(-1)) Area = (-(8)/(3) + (4)/(2) + 4) - (-(-1)/(3) + (1)/(2) - 2) Area = (-(8)/(3) + 2 + 4) - ((1)/(3) + (1)/(2) - 2) Area = (-(8)/(3) + 6) - ((1)/(3) + (1)/(2) - (4)/(2)) Area = ((-8 + 18)/(3)) - ((2 + 3 - 12)/(6)) Area = (10)/(3) - ((-7)/(6)) Area = (10)/(3) + (7)/(6) Area = (20)/(6) + (7)/(6) Area = (27)/(6) = (9)/(2) = 4.5 The value of the shaded area is: 4.5 square units That's 2 down. 3 left today — send the next one.