Rewrite the equation in the slope-intercept form y = mx + c.

*1) Given the equation of a line: (x)/(4) + (y)/(3) - (1)/(2) = 0. a) Determine the gradient of the line. Step 1: Rewrite the equation in the slope-intercept form y = mx + c. (y)/(3) = -(x)/(4) + (1)/(2) Step 2: Multiply by 3 to isolate y. y = 3 (-(x)/(4) + (1)/(2)) y = -(3)/(4)x + (3)/(2) Step 3: Identify the gradient m. The gradient m is the coefficient of x. m = -(3)/(4) The gradient of the line is -(3)/(4). b) Determine the coordinates of the point where the line cuts the coordinate axes. Step 1: Find the x-intercept (where y=0). Substitute y=0 into the original equation: (x)/(4) + (0)/(3) - (1)/(2) = 0 (x)/(4) - (1)/(2) = 0 (x)/(4) = (1)/(2) x = 4 × (1)/(2) x = 2 The x-intercept is (2, 0). Step 2: Find the y-intercept (where x=0). Substitute x=0 into the original equation: (0)/(4) + (y)/(3) - (1)/(2) = 0 (y)/(3) - (1)/(2) = 0 (y)/(3) = (1)/(2) y = 3 × (1)/(2) y = (3)/(2) The y-intercept is (0, (3)/(2)). The coordinates where the line cuts the coordinate axes are (2, 0) and (0, (3)/(2)). c) An equation of a line passing through (a, b) and perpendicular to the line above. Step 1: Find the gradient of the given line. From part (a), the gradient of the given line is m = -(3)/(4). Step 2: Find the gradient of the perpendicular line. If two lines are perpendicular, the product of their gradients is -1. Let m_ be the gradient of the perpendicular line. m · m_ = -1 (-(3)/(4)) m_ = -1 m_ = (-1)/(-3)4 m_ = (4)/(3) Step 3: Use the point-slope form y - y_1 = m_(x - x_1) with (x_1, y_1) = (a, b). y - b = (4)/(3)(x - a) Step 4: Rearrange into a standard form (optional, but often preferred). 3(y - b) = 4(x - a) 3y - 3b = 4x - 4a 4x - 3y - 4a + 3b = 0 The equation of the perpendicular line is y - b = (4)/(3)(x - a) or 4x - 3y - 4a + 3b = 0. 2) The price tag on an article in a supermarket is 7,000 FCFA. During an auction sale, the price was reduced by 12%. a) At what price will the article be sold? Step 1: Calculate the reduction amount. Reduction = 12\% of 7,000 FCFA Reduction = (12)/(100) × 7000 Reduction = 12 × 70 Reduction = 840 \, FCFA Step 2: Calculate the selling price. Selling price = Original price - Reduction Selling price = 7000 - 840 Selling price = 6160 \, FCFA The article will be sold for 6160 \, FCFA. b) Given that there is a 1% tax on the purchase price, how much will be paid? (Leave your answer to the nearest 5FCFA). Step 1: Calculate the tax amount. The purchase price is the selling price from part (a), which is 6160 FCFA. Tax = 1\% of 6160 FCFA Tax = (1)/(100) × 6160 Tax = 61.60 \, FCFA Step 2: Calculate the total amount paid. Total paid = Selling price + Tax Total paid = 6160 + 61.60 Total paid = 6221.60 \, FCFA Step 3: Round the total amount to the nearest 5 FCFA. The nearest multiple of 5 to 6221.60 is 6220. The amount to be paid is 6220 \, FCFA. 3) Given that f(x) = x^3 - 2x^2 - 5x + 6. a) Show that (x+2) is a factor of f. Step 1: Use the Factor Theorem. According to the Factor Theorem, if (x+2) is a factor of f(x), then f(-2) must be equal to 0. Substitute x = -2 into f(x): f(-2) = (-2)^3 - 2(-2)^2 - 5(-2) + 6 f(-2) = -8 - 2(4) - (-10) + 6 f(-2) = -8 - 8 + 10 + 6 f(-2) = -16 + 16 f(-2) = 0 Since f(-2) = 0, (x+2) is a factor of f(x). b) Factorise f(x) completely. Step 1: Divide f(x) by (x+2) to find the quadratic factor. Using synthetic division with root -2: ` -2 | 1 -2 -5 6 | -2 8 -6 ------------------ 1 -4 3 0 ` The coefficients of the quotient are 1, -4, 3. So, the quadratic factor is x^2 - 4x + 3. Step 2: Factorise the quadratic expression x^2 - 4x + 3. We need two numbers that multiply to 3 and add to -4. These numbers are -1 and -3. x^2 - 4x + 3 = (x - 1)(x - 3) Step 3: Write f(x) as a product of its linear factors. f(x) = (x+2)(x-1)(x-3) The complete factorisation of f(x) is (x+2)(x-1)(x-3).