Here are the solutions for questions 2, 3, and 4. 2a) Solve the following system of equations: x + 2y = 8, -x + y = -2. 2b) Find the value(s) of x in (2^x)2 - 5(2^x) + 4 = 0. 3) A firm buys a machine

Hey Megastar, good to see you again. Here are the solutions for questions 2, 3, and 4. 2a) Solve the following system of equations: x + 2y = 8 \\ -x + y = -2 Step 1: Add the two equations together to eliminate x. (x + 2y) + (-x + y) = 8 + (-2) 3y = 6 y = 2 Step 2: Substitute the value of y into the first equation to find x. x + 2(2) = 8 x + 4 = 8 x = 8 - 4 x = 4 The solution is x=4 and y=2. The final answer is x=4, y=2. 2b) Find the value(s) of x in (2^x)^2 - 5(2^x) + 4 = 0. Step 1: Let u = 2^x. Substitute u into the equation to form a quadratic equation. u^2 - 5u + 4 = 0 Step 2: Factor the quadratic equation. (u-1)(u-4) = 0 This gives two possible values for u: u=1 or u=4. Step 3: Substitute back u = 2^x and solve for x. For u=1: 2^x = 1 2^x = 2^0 x = 0 For u=4: 2^x = 4 2^x = 2^2 x = 2 The values of x are 0 and 2. The final answer is x=0, 2. 3) A firm buys a machine for N32,500. It depreciates every year at the rate of 10%. Find its value after 4 years. Step 1: Identify the initial value (P_0), depreciation rate (r), and number of years (n). P_0 = N32,500 r = 10\% = 0.10 n = 4 years Step 2: Use the depreciation formula P_n = P_0 (1 - r)^n. P_4 = 32500 (1 - 0.10)^4 P_4 = 32500 (0.90)^4 Step 3: Calculate (0.90)^4. (0.90)^4 = 0.6561 Step 4: Multiply by the initial value. P_4 = 32500 × 0.6561 P_4 = 21323.25 The value of the machine after 4 years is N21,323.25. The final answer is N21,323.25. 4a) Find the coefficient of x^8 in the expansion of (x^2 + (2)/(x))^11. Step 1: Write the general term for the binomial expansion (a+b)^n, which is T_r+1 = nr a^n-r b^r. Here, a = x^2, b = (2)/(x), and n = 11. T_r+1 = 11r (x^2)^11-r ((2)/(x))^r T_r+1 = 11r x^2(11-r) 2^r x^-r T_r+1 = 11r 2^r x^22-2r-r T_r+1 = 11r 2^r x^22-3r Step 2: Set the exponent of x equal to 8 to find the value of r. 22 - 3r = 8 3r = 22 - 8 3r = 14 r = (14)/(3) Since r must be a non-negative integer for a term to exist in the binomial expansion, and (14)/(3) is not an integer, there is no term containing x^8 in this expansion. The final answer is 0. 4b) Simplify _b x^2 - _b (x-3) = _b (x+4). Step 1: Use the logarithm property A - B = (A)/(B) on the left side of the equation. _b ((x^2)/(x-3)) = _b (x+4) Step 2: Since the logarithms have the same base, equate their arguments. (x^2)/(x-3) = x+4 Step 3: Solve for x. Multiply both sides by (x-3). x^2 = (x+4)(x-3) x^2 = x^2 - 3x + 4x - 12 x^2 = x^2 + x - 12 Subtract x^2 from both sides: 0 = x - 12 x = 12 Step 4: Check the domain of the original logarithmic expressions. For _b A, A must be positive. x^2 > 0 x ≠ 0 x-3 > 0 x > 3 x+4 > 0 x > -4 The solution x=12 satisfies all these conditions. The final answer is x=12. Drop the next question! 📸