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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Step 1: Solve Question 1. The conjugate of a complex number a+bi is a-bi. For the complex number 2+sqrt(3)i, the real part is 2 and the imaginary part is sqrt(3). The conjugate is 2-sqrt(3)i. The correct option is b. b. 2-sqrt(3)i Step 2: Solve Question 2. The question is _? 125 = x. Based on the options, the most likely base for 125 to result in an integer is 5. Let's assume the base is 5. _5 125 = x This means 5^x = 125. Since 125 = 5 × 5 × 5 = 5^3, we have: 5^x = 5^3 Therefore, x=3. The correct option is d. d. 3 Step 3: Solve Question 3. We need to find the value of the expression (x^2-9)/(3x-2) at x=3. Substitute x=3 into the expression: ((3)^2-9)/(3(3)-2) = (9-9)/(9-2) = (0)/(7) (0)/(7) = 0 The correct option is c. c. 0 Step 4: Solve Question 4. We need to find one of the factors of the expression x^2-8x+12. To factor a quadratic expression of the form ax^2+bx+c, we look for two numbers that multiply to c and add up to b. Here, c=12 and b=-8. The two numbers are -2 and -6, because (-2) × (-6) = 12 and (-2) + (-6) = -8. So, the expression can be factored as: x^2-8x+12 = (x-2)(x-6) One of the factors is x-2. The correct option is a. a. x-2 Step 5: Solve Question 5. We need to find the HCF (Highest Common Factor) of 4x^2-8x and x-2. First, factor the expression 4x^2-8x: 4x^2-8x = 4x(x-2) The second expression is x-2. The common factor between 4x(x-2) and (x-2) is x-2. Therefore, the HCF is x-2. The correct option is c. c. x-2 Step 6: Solve Question 6. We need to solve the linear equation (12x)/(7)-x=5 for x. Combine the terms with x: (12x)/(7) - (7x)/(7) = 5 (12x-7x)/(7) = 5 (5x)/(7) = 5 Multiply both sides by 7: 5x = 5 × 7 5x = 35 Divide both sides by 5: x = (35)/(5) x = 7 The correct option is d. d. 7 Step 7: Solve Question 7. We need to determine which line the point (-2, -7) lies on. Substitute x=-2 and y=-7 into each equation. *a) y=4x+1: -7 = 4(-2)+1 -7 = -8+1 -7 = -7 This statement is true, so the point lies on this line. Let's quickly check other options to confirm. *b) y=x+1: -7 = -2+1 -7 = -1 (False) *c) y=x+2: -7 = -2+2 -7 = 0 (False) *d) 2y=3x+2: 2(-7) = 3(-2)+2 -14 = -6+2 -14 = -4 (False) The correct option is a. a. y=4x+1 Step 8: Solve Question 8. The statement is 4+x=4 x=y. This appears to have a typo and should likely be 4+x=4 x=0 or 4+x=4+y x=y. In either case, the property used is the cancellation property with respect to addition. If a+b=a+c, then b=c. In 4+x=4, we can subtract 4 from both sides to get x=0. This is an application of the cancellation property for addition. The correct option is b. b. cancellation property w.r.t addition Step 9: Solve Question 9. The matrix is $ 0 & 0 & 0 \\ 0 &