Given equation: 3x2 - 4x + 2 = 0.

Here's the solution to the problem: 1. (i) a) Given that the roots of the equation 3x^2 - 4x + 2 = 0 are and . Show that ^3 + ^3 = -(8)/(27). Step 1: Identify the coefficients and find the sum and product of roots. For a quadratic equation ax^2 + bx + c = 0, the sum of roots is + = -(b)/(a) and the product of roots is = (c)/(a). Given equation: 3x^2 - 4x + 2 = 0. Here, a=3, b=-4, c=2. Sum of roots: + = -(-4)/(3) = (4)/(3) Product of roots: = (2)/(3) Step 2: Use the identity for ^3 + ^3. The identity for the sum of cubes is ^3 + ^3 = ( + )(^2 - + ^2). We can express ^2 + ^2 in terms of + and : ^2 + ^2 = ( + )^2 - 2 . Substitute this into the identity: ^3 + ^3 = ( + )(( + )^2 - 2 - ) ^3 + ^3 = ( + )(( + )^2 - 3 ) Step 3: Substitute the values of + and . ^3 + ^3 = ((4)/(3))(((4)/(3))^2 - 3((2)/(3))) ^3 + ^3 = ((4)/(3))((16)/(9) - (6)/(3)) ^3 + ^3 = ((4)/(3))((16)/(9) - (18)/(9)) ^3 + ^3 = ((4)/(3))(-(2)/(9)) ^3 + ^3 = -(8)/(27) This shows the required result. 1. (i) b) Hence, find the equation with integral coefficients whose roots are (1)/(^3) and (1)/(^3). Step 1: Define the new roots and calculate their sum. Let the new roots be p = (1)/(^3) and q = (1)/(^3). Sum of new roots: p + q = (1)/(^3) + (1)/(^3) = (^3 + ^3)/(^3 ^3) From part (a), we know ^3 + ^3 = -(8)/(27). Also, ^3 ^3 = ( )^3. Since = (2)/(3): ( )^3 = ((2)/(3))^3 = (8)/(27) Now, substitute these values into the sum of new roots: p + q = (-8)/(27)(8)/(27) = -1 Step 2: Calculate the product of the new roots. pq = ((1)/(^3))((1)/(^3)) = (1)/(^3 ^3) = (1)/(( )^3) pq = (1)/(8)27 = (27)/(8) Step 3: Form the quadratic equation with integral coefficients. The general form of a quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0. x^2 - (-1)x + (27)/(8) = 0 x^2 + x + (27)/(8) = 0 To obtain integral coefficients, multiply the entire equation by 8: 8(x^2 + x + (27)/(8)) = 8(0) 8x^2 + 8x + 27 = 0 1. (iii) Find the set of real values of x for which |(x+3)/(x-5)| = (x+3)/(x-5). Step 1: Understand the property of absolute value. The equation |A| = A holds true if and only if A 0. Therefore, we need to find the values of x for which (x+3)/(x-5) 0. Step 2: Find the critical points. The critical points are the values of x where the numerator or denominator is zero. Numerator: x+3 = 0 x = -3. Denominator: x-5 = 0 x = 5. These points divide the number line into three intervals: (-, -3), (-3, 5), and (5, ). Step 3: Test each interval. • For x < -3 (e.g., x = -4): (-4+3)/(-4-5) = (-1)/(-9) = (1)/(9). This is positive, so the inequality holds. Since the inequality is 0, x=-3 is included (as (0)/(-8)=0). So, x -3 is part of the solution. • For -3 < x < 5 (e.g., x = 0): (0+3)/(0-5) = (3)/(-5) = -(3)/(5). This is negative, so the inequality does not hold. • For x > 5 (e.g., x = 6): (6+3)/(6-5) = (9)/(1) = 9. This is positive, so the inequality holds. Note that x=5 is excluded because it makes the denominator zero, which is undefined. Step 4: Combine the valid intervals. The set of real values of x for which the inequality holds is x -3 or x > 5. In set notation: \x R x -3 or x > 5\