Identify the properties of the equilateral triangle PQR.

Here are the solutions to the problems: 1.a. Calculate the area of the shaded region. Step 1: Identify the properties of the equilateral triangle PQR. The side length of the equilateral triangle PQR is s = 18 cm. All angles in an equilateral triangle are 60^, so P = Q = R = 60^. Step 2: Calculate the area of the equilateral triangle PQR. The formula for the area of an equilateral triangle is sqrt(3)4 s^2. Area of PQR = sqrt(3)4 (18)^2 = sqrt(3)4 × 324 = 81sqrt(3) cm^2 Using sqrt(3) ≈ 1.73205: Area of PQR ≈ 81 × 1.73205 = 140.29605 cm^2 Step 3: Determine the radius of the sector PAB. The arc is centered at P and touches QR at M. This means PM is the radius of the sector. In an equilateral triangle, the line segment from a vertex to the midpoint of the opposite side (PM) is also an altitude. Consider the right-angled triangle PQM, where PQ = 18 cm and QM = (1)/(2) QR = (1)/(2) × 18 = 9 cm. Using the Pythagorean theorem: PM^2 + QM^2 = PQ^2 PM^2 + 9^2 = 18^2 PM^2 + 81 = 324 PM^2 = 324 - 81 = 243 PM = sqrt(243) = sqrt(81 × 3) = 9sqrt(3) cm So, the radius of the sector is r = 9sqrt(3) cm. Step 4: Calculate the area of the sector PAB. The angle of the sector is QPR = 60^. The formula for the area of a sector is ()/(360^) r^2. Using = (22)/(7): Area of sector PAB = (60)/(360) × (22)/(7) × (9sqrt(3))^2 = (1)/(6) × (22)/(7) × (81 × 3) = (1)/(6) × (22)/(7) × 243 = (11)/(3 × 7) × 243 = (11)/(21) × 243 = (11 × 81)/(7) = (891)/(7) cm^2 Area of sector PAB ≈ 127.28571 cm^2 Step 5: Calculate the area of the shaded region. The shaded region is the area of the triangle PQR minus the area of the sector PAB. Area of shaded region = Area of PQR - Area of sector PAB Area of shaded region ≈ 140.29605 - 127.28571 Area of shaded region ≈ 13.01034 cm^2 Correct to two decimal places: The area of the shaded region is 13.01 cm^2. 1.b. Solve the equation a - sqrt(a^2 + 12) + 2 = 0 Step 1: Isolate the square root term. a + 2 = sqrt(a^2 + 12) Step 2: Square both sides of the equation to eliminate the square root. (a+2)^2 = (sqrt(a^2 + 12))^2 a^2 + 4a + 4 = a^2 + 12 Step 3: Solve for a. Subtract a^2 from both sides: 4a + 4 = 12 Subtract 4 from both sides: 4a = 12 - 4 4a = 8 Divide by 4: a = (8)/(4) a = 2 Step 4: Check the solution by substituting a=2 back into the original equation. 2 - sqrt(2^2 + 12) + 2 = 0 2 - sqrt(4 + 12) + 2 = 0 2 - sqrt(16) + 2 = 0 2 - 4 + 2 = 0 0 = 0 The solution is correct. The value of a is 2. 1.c. The ratio of the age of a mother to that of the daughter is 11:3. If the mother is 55 years old, what will be the ratio of their ages in 5 years time? Step 1: Determine the current ages of the mother and daughter. Let the mother's age be M and the daughter's age be D. Given ratio M:D = 11:3. Given mother's current age M = 55 years. We can write the ratio as (M)/(D) = (11)/(3). Substitute M=55: (55)/(D) = (11)/(3) To find D, cross-multiply: 11D = 55 × 3