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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Step 1: Set up equations for the sum of terms in an Arithmetic Progression (AP). The sum of the first n terms of an AP is given by S_n = (n)/(2)(2a + (n-1)d), where a is the first term and d is the common difference. Given: The sum of the first five terms (S_5) is 5. S_5 = (5)/(2)(2a + (5-1)d) = 5 (5)/(2)(2a + 4d) = 5 Divide both sides by (5)/(2): 2a + 4d = 2 Divide by 2: a + 2d = 1 (*) Given: The sum of the next 10 terms is 175. This means the sum from the 6th term to the 15th term. This can be written as S_15 - S_5 = 175. So, S_15 = S_5 + 175 = 5 + 175 = 180. Now use the formula for S_15: S_15 = (15)/(2)(2a + (15-1)d) = 180 (15)/(2)(2a + 14d) = 180 Divide both sides by (15)/(2): 2a + 14d = (180 × 2)/(15) 2a + 14d = 24 Divide by 2: a + 7d = 12 (**) Step 2: Solve the system of equations to find a and d. We have two equations: 1) a + 2d = 1 2) a + 7d = 12 Subtract equation (1) from equation (2): (a + 7d) - (a + 2d) = 12 - 1 5d = 11 d = (11)/(5) Substitute d = (11)/(5) into equation (1): a + 2((11)/(5)) = 1 a + (22)/(5) = 1 a = 1 - (22)/(5) a = (5)/(5) - (22)/(5) a = -(17)/(5) Step 3: Find the sum of the first twenty terms (S_20). Using the formula S_n = (n)/(2)(2a + (n-1)d) with n=20, a=-(17)/(5), and d=(11)/(5): S_20 = (20)/(2)(2(-(17)/(5)) + (20-1)((11)/(5))) S_20 = 10(-(34)/(5) + 19((11)/(5))) S_20 = 10(-(34)/(5) + (209)/(5)) S_20 = 10((209 - 34)/(5)) S_20 = 10((175)/(5)) S_20 = 10 × 35 S_20 = 350 The sum of the first twenty terms is 350. 10. b) The gradient of a curve at the point (x,y) is given by ax^2 - 4x - 4. If the minimum point on the curve is (2,16), find (i) the value of a; (ii) the equation of the curve. b)(i) the value of a; Step 1: Use the property of a minimum point. At a minimum point, the gradient of the curve is zero. Given the gradient (dy)/(dx) = ax^2 - 4x - 4. The minimum point is (2,16), so at x=2, (dy)/(dx) = 0. Step 2: Substitute x=2 into the gradient equation and solve for a. a(2)^2 - 4(2) - 4 = 0 4a - 8 - 4 = 0 4a - 12 = 0 4a = 12 a = 3 The value of a is 3. b)(ii) the equation of the curve. Step 1: Substitute the value of a into the gradient function. Now we know a=3, so the gradient function is: (dy)/(dx) = 3x^2 - 4x - 4 Step 2: Integrate the gradient function to find the equation of the curve. y = (3x^2 - 4x - 4)\,dx y = 3x^2+12+1 - 4x^1+11+1 - 4x + C y = (3x^3)/(3) - (4x^2)/(2) - 4x + C y = x^3 - 2x^2 - 4x + C Step 3: Use the given minimum point (2,16) to find the constant of integration C. Substitute x=2 and y=16 into the equation of the curve: 16 = (2)^3 - 2(2)^2 - 4(2) + C 16 = 8 - 2(4) - 8 + C 16 = 8 - 8 - 8 + C 16 = -8 + C C = 16 + 8 C = 24 Step 4: Write the final equation of the curve. Substitute C=24 back into the equation: y = x^3 - 2x^2 - 4x + 24 The equation of the curve is y = x^3 - 2x^2 - 4x + 24. That's 2 down. 3 left today — send the next one.