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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Okay, bazuunelson, let's work through these problems. 15. (a) Differentiate y = (x^2+3x)(x-2) Step 1: Expand the expression for y. y = x^2(x-2) + 3x(x-2) y = x^3 - 2x^2 + 3x^2 - 6x y = x^3 + x^2 - 6x Step 2: Differentiate y with respect to x. Using the power rule for differentiation, (d)/(dx)(x^n) = nx^n-1: (dy)/(dx) = (d)/(dx)(x^3) + (d)/(dx)(x^2) - (d)/(dx)(6x) (dy)/(dx) = 3x^2 + 2x - 6 The derivative is (dy)/(dx) = 3x^2 + 2x - 6. 15. (b) Find the gradient at x=1 Step 1: Substitute x=1 into the derivative found in part (a). (dy)/(dx) |_x=1 = 3(1)^2 + 2(1) - 6 (dy)/(dx) |_x=1 = 3(1) + 2 - 6 (dy)/(dx) |_x=1 = 3 + 2 - 6 (dy)/(dx) |_x=1 = 5 - 6 (dy)/(dx) |_x=1 = -1 The gradient at x=1 is -1. 16. (a) Integrate: (3x^2+4x) dx Step 1: Apply the power rule for integration, x^n dx = x^n+1n+1 + C. (3x^2+4x) dx = 3x^2 dx + 4x dx = 3(x^2+12+1) + 4(x^1+11+1) + C = 3((x^3)/(3)) + 4((x^2)/(2)) + C Step 2: Simplify the expression. = x^3 + 2x^2 + C The integral is x^3 + 2x^2 + C. 16. (b) Find area under curve from x=0 to x=2 Step 1: Evaluate the definite integral using the result from part (a). The area is given by _0^2 (3x^2+4x) dx. = [x^3 + 2x^2]_0^2 Step 2: Substitute the upper and lower limits. = (2^3 + 2(2^2)) - (0^3 + 2(0^2)) = (8 + 2(4)) - (0 + 0) = (8 + 8) - 0 = 16 The area under the curve is 16. 17. (a) Solve: 2^(x+1) = 16 Step 1: Express 16 as a power of 2. 16 = 2^4 Step 2: Rewrite the equation with the same base. 2^(x+1) = 2^4 Step 3: Equate the exponents. x+1 = 4 Step 4: Solve for x. x = 4-1 x = 3 The solution is x=3. 17. (b) Express your answer in simplest form The answer from part (a) is x=3. This is already in its simplest form. The answer is 3. 18. (a) A bag contains 4 red and 6 blue balls. Find probability of picking red Step 1: Determine the total number of balls. Total balls = Number of red balls + Number of blue balls Total balls = 4 + 6 = 10 Step 2: Calculate the probability of picking a red ball. Probability (Red) = Number of red ballsTotal number of balls Probability (Red) = (4)/(10) Step 3: Simplify the probability. Probability (Red) = (2)/(5) The probability of picking a red ball is (2)/(5). 18. (b) If two balls are picked, find probability both are red Step 1: Calculate the probability of the first ball being red. Probability (1st Red) = (4)/(10) Step 2: Calculate the probability of the second ball being red, given the first was red (without replacement). After picking one red ball, there are 3 red balls left and a total of 9 balls. Probability (2nd Red | 1st Red) = (3)/(9) Step 3: Multiply the probabilities to find the probability of both being red. Probability (Both Red) = Probability (1st Red) × Probability (2nd Red | 1st Red) = (4)/(10) × (3)/(9) = (2)/(5) × (1)/(3) = (2)/(15) The probability of picking two red balls is (2)/(15). 19. (a) Find the equation of a circle with center (1,-2) and radius 4 Step 1: Recall the standard equation of a circle. The equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. Step 2: Substitute the given values. Given center (h,k) = (1,-2) and radius r = 4. (x-1)^2 + (y-(-2))^2 = 4^2 (x-1)^2 + (y+2)^2 = 16 The equation of the circle is (x-1)^2 + (y+2)^2 = 16. 19. (b) Find the area of the circle Step 1: Recall the formula for the area of a circle. Area A = r^2. Step 2: Substitute the given radius r=4. A = (4)^2 A = 16 The area of the circle is 16 units^2. 20. (a) A sequence has first term 5 and common difference 3. Find the 15th term Step 1: Identify the type of sequence and the given values. This is an arithmetic sequence with first term a_1 = 5 and common difference d = 3. We need to find the 15th term, so n=15. Step 2: Use the formula for the n-th term of an arithmetic sequence. a_n = a_1 + (n-1)d a_15 = 5 + (15-1)3 a_15 = 5 + (14)3 a_15 = 5 + 42 a_15 = 47 The 15th term is 47. 20. (b) Find the sum of first 15 terms Step 1: Use the formula for the sum of the first n terms of an arithmetic sequence. We can use S_n = (n)/(2)(a_1 + a_n). We have n=15, a_1=5, and a_15=47 from part (a). S_15 = (15)/(2)(5 + 47) S_15 = (15)/(2)(52) Step 2: Calculate the sum. S_15 = 15 × 26 S_15 = 390 The sum of the first 15 terms is 390. 3 done, 2 left today. You're making progress.