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 step-by-step solutions for each question: 6. Solve the equation: (3X-1)/(3) - (2X-1)/(4) = 2 Step 1: Find a common denominator for the fractions, which is 12. Multiply the entire equation by 12. 12 ( (3X-1)/(3) ) - 12 ( (2X-1)/(4) ) = 12(2) Step 2: Simplify the terms. 4(3X-1) - 3(2X-1) = 24 Step 3: Distribute the numbers into the parentheses. 12X - 4 - 6X + 3 = 24 Step 4: Combine like terms on the left side. (12X - 6X) + (-4 + 3) = 24 6X - 1 = 24 Step 5: Add 1 to both sides of the equation. 6X = 24 + 1 6X = 25 Step 6: Divide by 6 to solve for X. X = (25)/(6) The solution is (25)/(6). 7. The angle of elevation of the top of a cliff 30 metres high from a landmark is 60^. How far is the landmark from the foot of the cliff? Leave your answer in surd form. Step 1: Define the variables. Let h be the height of the cliff, h = 30 m. Let d be the distance from the landmark to the foot of the cliff. The angle of elevation is = 60^. Step 2: Use the tangent function, as it relates the opposite side (height) to the adjacent side (distance) in a right-angled triangle. () = oppositeadjacent (60^) = (h)/(d) Step 3: Substitute the known values. We know (60^) = sqrt(3). sqrt(3) = (30)/(d) Step 4: Solve for d. d = (30)/(sqrt(3)) Step 5: Rationalize the denominator by multiplying the numerator and denominator by sqrt(3). d = (30)/(sqrt(3)) × sqrt(3)sqrt(3) d = 30sqrt(3)3 d = 10sqrt(3) m Comparing this with the options, option (b) is 30/sqrt(3) m, which simplifies to 10sqrt(3) m. The distance is (b) 30/sqrt(3) m. 8. For what value of X is the expression (6X-1)/(X^2-X-6) not defined? Step 1: An algebraic expression is undefined when its denominator is equal to zero. Set the denominator to zero. X^2-X-6 = 0 Step 2: Factor the quadratic equation. We need two numbers that multiply to -6 and add to -1. These numbers are -3 and 2. (X-3)(X+2) = 0 Step 3: Set each factor equal to zero and solve for X. X-3 = 0 X = 3 X+2 = 0 X = -2 The expression is not defined for X=3 and X=-2. The correct option is (b) 3, -2. 9. AOC is the diameter of a circle, if ADB = 32^, calculate CAB. Step 1: Identify properties of angles in a circle. Since AOC is the diameter, the angle subtended by the diameter at any point on the circumference is a right angle. Therefore, ABC = 90^. Step 2: Angles subtended by the same arc at the circumference are equal. ADB and ACB are both subtended by arc AB. ACB = ADB = 32^ Step 3: Consider ABC. The sum of angles in a triangle is 180^. CAB + ABC + ACB = 180^ Step 4: Substitute the known angles into the equation. CAB + 90^ + 32^ = 180^ CAB + 122^ = 180^ Step 5: Solve for CAB. CAB = 180^ - 122^ CAB = 58^ The correct option is (d) 58^. 10. ABCD is a cyclic quadrilateral. If ABD = 18^, ACB = 40^, calculate DAB. Step 1: Identify properties of angles in a circle. Angles subtended by the same arc at the circumference are equal. ADB and ACB are both subtended by arc AB. Therefore, ADB = ACB = 40^. ACD and ABD are both subtended by arc AD. Therefore, ACD = ABD = 18^. Step 2: Consider ABD. The sum of angles in a triangle is 180^. DAB + ABD + ADB = 180^ Step 3: Substitute the known angles into the equation. DAB + 18^ + 40^ = 180^ DAB + 58^ = 180^ Step 4: Solve for DAB. DAB = 180^ - 58^ DAB = 122^ The correct option is (d) 122^.