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 the questions from the image. Question 22: A man is five times as old as his son. If the son's age is y, then the man's current age is 5y. In four years' time: Son's age will be y+4. Man's age will be 5y+4. The product of their ages in four years will be 340. Step 1: Set up the equation for the product of their ages in four years. (y+4)(5y+4) = 340 Step 2: Expand the left side of the equation. y(5y) + y(4) + 4(5y) + 4(4) = 340 5y^2 + 4y + 20y + 16 = 340 5y^2 + 24y + 16 = 340 Step 3: Rearrange the equation to the form Ax^2 + Bx + C = 0. 5y^2 + 24y + 16 - 340 = 0 5y^2 + 24y - 324 = 0 The final answer is D. 5y^2 + 24y - 324 = 0. Question 23: Simplify: (a)/(b) - (b)/(a)(b)/(a) - (c)/(b) Step 1: Simplify the numerator (a)/(b) - (b)/(a). Find a common denominator, which is ab. (a)/(b) - (b)/(a) = (a · a)/(b · a) - (b · b)/(a · b) = (a^2)/(ab) - (b^2)/(ab) = (a^2 - b^2)/(ab) Step 2: Simplify the denominator (b)/(a) - (c)/(b). Find a common denominator, which is ab. (b)/(a) - (c)/(b) = (b · b)/(a · b) - (c · a)/(b · a) = (b^2)/(ab) - (ac)/(ab) = (b^2 - ac)/(ab) Step 3: Divide the simplified numerator by the simplified denominator. (a^2 - b^2)/(ab)(b^2 - ac)/(ab) To divide fractions, multiply the numerator by the reciprocal of the denominator. (a^2 - b^2)/(ab) × (ab)/(b^2 - ac) Step 4: Cancel out the common term ab. (a^2 - b^2)/(b^2 - ac) The final answer is D. (a^2 - b^2)/(b^2 - ac). Question 24: In the diagram, XYZ is an equilateral triangle of side 6 cm. T is the midpoint of XY. Find ( XZT). Step 1: Identify properties of an equilateral triangle. Since XYZ is equilateral, all sides are equal (XY = YZ = ZX = 6 cm) and all angles are 60^ ( X = Y = Z = 60^). Step 2: Determine the length of XT. T is the midpoint of XY, so XT = (1)/(2) XY = (1)/(2)(6 cm) = 3 cm. Step 3: Identify properties of the line segment ZT. In an equilateral triangle, the line segment from a vertex to the midpoint of the opposite side is also an altitude (perpendicular to the side) and an angle bisector. Therefore, ZT XY, which means XZT is a right-angled triangle with the right angle at T. Also, ZT bisects XZY, so XZT = (1)/(2) XZY = (1)/(2)(60^) = 30^. Step 4: Calculate ( XZT). Since XZT = 30^, we need to find (30^). (30^) = (1)/(sqrt(3)) This can also be rationalized as sqrt(3)3. Alternatively, using side lengths in XZT: We have XZ = 6 cm (hypotenuse) and XT = 3 cm (opposite side to XZT). We need to find the length of ZT (adjacent side to XZT) using the Pythagorean theorem: ZT^2 + XT^2 = XZ^2 ZT^2 + 3^2 = 6^2 ZT^2 + 9 = 36 ZT^2 = 36 - 9 ZT^2 = 27 ZT = sqrt(27) = sqrt(9 × 3) = 3sqrt(3) cm Now, calculate ( XZT): ( XZT) = OppositeAdjacent = (XT)/(ZT) = (3)/(3sqrt(3)) = (1)/(sqrt(3)) The final answer is A. (1)/(sqrt(3)). Question 25: A fence 2.4 m tall is 10 m away from a tree of height 16 m. Calculate the angle of elevation of the top of the tree from the top of the fence. Step 1: Identify the relevant lengths for the right-angled triangle. The horizontal distance between the fence and the tree is the adjacent side: 10 m. The vertical height difference from the top of the fence to the top of the tree is the opposite side. Vertical height difference = Height of tree - Height of fence Vertical height difference = 16 m - 2.4 m = 13.6 m Step 2: Use the tangent function to find the angle of elevation (). () = OppositeAdjacent () = 13.6 m10 m () = 1.36 Step 3: Calculate the angle . = (1.36) Using a calculator: ≈ 53.679^ Rounding to two decimal places, ≈ 53.68^. The closest option is 53.67^. The final answer is $B. 53.6