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 are the solutions to your assignment: Section A 1. Simplify: sqrt(72) - sqrt(32) Step 1: Simplify each surd. sqrt(72) = sqrt(36 × 2) = 6sqrt(2) sqrt(32) = sqrt(16 × 2) = 4sqrt(2) Step 2: Subtract the simplified surds. 6sqrt(2) - 4sqrt(2) = (6-4)sqrt(2) = 2sqrt(2) The simplified expression is 2sqrt(2). 2. Solve: 2x^2 - 3x - 2 = 0 Step 1: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here a=2, b=-3, c=-2. x = -(-3) ± sqrt((-3)^2 - 4(2)(-2))2(2) x = 3 ± sqrt(9 + 16)4 x = 3 ± sqrt(25)4 x = (3 ± 5)/(4) Step 2: Calculate the two solutions. x_1 = (3 + 5)/(4) = (8)/(4) = 2 x_2 = (3 - 5)/(4) = (-2)/(4) = -(1)/(2) The solutions are x = 2, x = -(1)/(2). 3. Evaluate: _10(1000) - _10(10) Step 1: Evaluate each logarithm. _10(1000) = 3 (since 10^3 = 1000) _10(10) = 1 (since 10^1 = 10) Step 2: Subtract the values. 3 - 1 = 2 The evaluated expression is 2. 4. Find the gradient of y = 3x^2 - 2x at x = 1. Step 1: Find the derivative of the function to get the gradient function. (dy)/(dx) = (d)/(dx)(3x^2 - 2x) = 6x - 2 Step 2: Substitute x=1 into the gradient function. (dy)/(dx)|_x=1 = 6(1) - 2 = 6 - 2 = 4 The gradient at x=1 is 4. 5. Find the distance between points (1,2) and (4,6). Step 1: Use the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Let (x_1, y_1) = (1,2) and (x_2, y_2) = (4,6). d = sqrt((4 - 1)^2 + (6 - 2)^2) d = sqrt((3)^2 + (4)^2) d = sqrt(9 + 16) d = sqrt(25) d = 5 The distance between the points is 5 units. 6. Find the determinant of matrix 3 & 2 \\ 1 & 4 . Step 1: For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. det = (3)(4) - (2)(1) det = 12 - 2 det = 10 The determinant of the matrix is 10. 7. Solve: x = sqrt(3)/2 for 0^ x 360^. Step 1: Find the reference angle. ^-1(sqrt(3)2) = 60^ Step 2: Identify the quadrants where x is positive (Quadrant I and Quadrant II). In Quadrant I: x_1 = 60^ In Quadrant II: x_2 = 180^ - 60^ = 120^ The solutions are x = 60^, 120^. 8. Expand: (x+2)(x-3) Step 1: Use the FOIL method (First, Outer, Inner, Last). x · x + x · (-3) + 2 · x + 2 · (-3) x^2 - 3x + 2x - 6 Step 2: Combine like terms. x^2 - x - 6 The expanded expression is x^2 - x - 6. 9. Find the mean of 2, 4, 6, 8. Step 1: Sum the numbers. Sum = 2 + 4 + 6 + 8 = 20 Step 2: Divide the sum by the count of numbers (which is 4). Mean = (20)/(4) = 5 The mean is 5. 10. Integrate: 2x \,dx Step 1: Use the power rule for integration x^n \,dx = x^n+1n+1 + C. 2x^1 \,dx = 2 · x^1+11+1 + C = 2 · (x^2)/(2) + C = x^2 + C The integral is x^2 + C. Section B 11. a) Solve the equation: x^2 + 5x + 6 = 0 Step 1: Factor the quadratic equation. Find two numbers that multiply to 6 and add to 5 (these are 2 and 3). (x + 2)(x + 3) = 0 Step 2: Set each factor to zero and solve for x. x + 2 = 0 x = -2 x + 3 = 0 x = -3 The solutions are x = -2, x = -3. b) Hence find the sum and product of the roots (3 marks) Step 1: Sum of the roots. Sum = (-2) + (-3) = -5 Alternatively, using the formula for sum of roots (-(b)/(a)): Sum = -(5)/(1) = -5 Step 2: Product of the roots. Product = (-2) × (-3) = 6 Alternatively, using the formula for product of roots ((c)/(a)): Product = (6)/(1) = 6 The sum of the roots is -5 and the product of the roots is 6. 12. a) Simplify: (3x^2y)(2xy^3) Step 1: Multiply the coefficients. 3 × 2 = 6 Step 2: Multiply the x terms using x^a · x^b = x^a+b. x^2 · x^1 = x^2+1 = x^3 Step 3: Multiply the y terms using y^a · y^b = y^a+b. y^1 · y^3 = y^1+3 = y^4 Step 4: Combine the results. 6x^3y^4 The simplified expression is 6x^3y^4. b) Express your answer in index form (2 marks) The answer from part (a) is already in index form: 6x^3y^4. 13. a) Solve the simultaneous equations: x + y = 5 (1) x^2 + y^2 = 13 (2) Step 1: From equation (1), express y in terms of x. y = 5 - x (3) Step 2: Substitute equation (3) into equation (2). x^2 + (5 - x)^2 = 13 x^2 + (25 - 10x + x^2) = 13 2x^2 - 10x + 25 = 13 Step 3: Rearrange into a standard quadratic equation. 2x^2 - 10x + 12 = 0 Step 4: Divide by 2 to simplify. x^2 - 5x + 6 = 0 Step 5: Factor the quadratic equation. (x - 2)(x - 3) = 0 Step 6: Solve for x. x - 2 = 0 x = 2 x - 3 = 0 x = 3 Step 7: Substitute the values of x back into equation (3) to find y. If x = 2: y = 5 - 2 = 3 If x = 3: y = 5 - 3 = 2 The solutions are x=2, y=3 and x=3, y=2. b) Find xy (2 marks) Step 1: Use one of the solution pairs from part (a). If x=2, y=3: xy = 2 × 3 = 6 If x=3, y=2: xy = 3 × 2 = 6 The value of xy is 6. 14. a) Find the equation of a line passing through (2,3) and (4,7) (3 marks) Step 1: Find the gradient (m) of the line. m = (y_2 - y_1)/(x_2 - x_1) = (7 - 3)/(4 - 2) = (4)/(2) = 2 Step 2: Use the point-slope form of a linear equation y - y_1 = m(x - x_1). Using point (2,3) and m=2. y - 3 = 2(x - 2) y - 3 = 2x - 4 y = 2x - 1 The equation of the line is y = 2x - 1. b) Find the midpoint of the line (2 marks) Step 1: Use the midpoint formula M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)). M = ((2 + 4)/(2), (3 + 7)/(2)) M = ((6)/(2), (10)/(2)) M = (3, 5) The midpoint of the line is (3, 5). 3 done, 2 left today. 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