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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Part 11(a): Geometric Progression Let the first term of the geometric progression be a and the common ratio be r. The n-th term of a G.P. is given by U_n = ar^n-1. Given information: 1. The sum of the first and third terms is 40. U_1 + U_3 = 40 a + ar^2 = 40 a(1 + r^2) = 40 (*) 2. The fourth and the sixth terms are in the ratio 1:4. (U_4)/(U_6) = (1)/(4) ar^4-1ar^6-1 = (1)/(4) (ar^3)/(ar^5) = (1)/(4) (1)/(r^2) = (1)/(4) i. Common ratio Step 1: Solve for r^2. r^2 = 4 Step 2: Solve for r. r = ±sqrt(4) r = ± 2 The common ratio is ± 2. ii. Fifth term Step 1: Find the value of a using equation (*) and the common ratio. If r=2: a(1 + 2^2) = 40 a(1 + 4) = 40 5a = 40 a = 8 If r=-2: a(1 + (-2)^2) = 40 a(1 + 4) = 40 5a = 40 a = 8 In both cases, the first term a=8. Step 2: Calculate the fifth term U_5 = ar^4. Using a=8 and r=2: U_5 = 8 × (2)^4 = 8 × 16 = 128 Using a=8 and r=-2: U_5 = 8 × (-2)^4 = 8 × 16 = 128 The fifth term is 128. Part 11(b): Work Rate Problem Let the total work be W. A can finish the work in 10 days, so A's rate is (W)/(10) per day. B can finish the work in 15 days, so B's rate is (W)/(15) per day. C can finish the work in 30 days, so C's rate is (W)/(30) per day. Step 1: Calculate the combined work rate of A, B, and C. Combined rate R_total = (W)/(10) + (W)/(15) + (W)/(30) R_total = W ((1)/(10) + (1)/(15) + (1)/(30)) Step 2: Find a common denominator for the fractions (which is 30). R_total = W ((3)/(30) + (2)/(30) + (1)/(30)) R_total = W ((3+2+1)/(30)) R_total = W ((6)/(30)) R_total = W ((1)/(5)) Step 3: Calculate the number of days required for them to finish the work together. Time = Total WorkCombined Rate T = (W)/(W (1)5) T = (1)/(1)5 T = 5 days The number of days required if A, B, and C work together is 5 days. Part 12(a): Geometric Construction i. Construct PQR in which |QR|=7cm, |PQ|=8cm and PQR=75^. 1. Draw a line segment QR of length 7 cm. 2. At point Q, construct an angle of 75^. (This can be done by constructing a 90^ angle and a 60^ angle, then bisecting the angle between them). 3. Along the arm of the 75^ angle from Q, measure 8 cm and mark point P. 4. Join P to R to complete PQR. ii. The Locus L_1 of points equidistant from PR and QR. This locus is the angle bisector of PRQ. 1. Place the compass at point R and draw an arc that intersects both line segments PR and QR. 2. From the two intersection points on PR and QR, draw two new arcs that intersect each other inside PRQ. 3. Draw a straight line from R through the intersection point of these two arcs. This line is L_1. iii. The Locus L_2 of points equidistant from Q and R. This locus is the perpendicular bisector of the line segment QR. 1. Place the compass at point Q and open it to a radius greater than half the length of QR. Draw an arc above QR and an arc below QR. 2. With the same compass opening, place the compass at point R and draw arcs above and below QR that intersect the previously drawn arcs. 3. Draw a straight line through the two points where the arcs intersect. This line is L_2. Part 12(b): Locate the point K, the intersection of L_1 and L_2. The point K is where the line L_1 (angle bisector of PRQ) and the line L_2 (perpendicular bisector of QR) intersect. Mark this intersection point as K. Part 12(c): Measure... The question is incomplete, so this part cannot be answered. That's 2 down. 3 left today — send the next one.