Express the angles in terms of x.

Here are the solutions to the questions: 1. (a) In a triangle PQR, angle Q is three-quarter of angle P and angle R is half of angle P. If the measure of angle P is (x+5)^, form an equation in x and hence, calculate the angles of triangle PQR. Step 1: Express the angles in terms of x. Given: P = (x+5)^ Q = (3)/(4) P = (3)/(4)(x+5)^ R = (1)/(2) P = (1)/(2)(x+5)^ Step 2: Form an equation using the sum of angles in a triangle. The sum of angles in a triangle is 180^. P + Q + R = 180^ (x+5) + (3)/(4)(x+5) + (1)/(2)(x+5) = 180 Step 3: Solve the equation for x. Factor out (x+5): (x+5) (1 + (3)/(4) + (1)/(2)) = 180 (x+5) ((4)/(4) + (3)/(4) + (2)/(4)) = 180 (x+5) ((9)/(4)) = 180 Multiply both sides by (4)/(9): x+5 = 180 × (4)/(9) x+5 = 20 × 4 x+5 = 80 x = 80 - 5 x = 75 Step 4: Calculate the measure of each angle. P = (x+5)^ = (75+5)^ = 80^ Q = (3)/(4)(80^) = 3 × 20^ = 60^ R = (1)/(2)(80^) = 40^ The equation in x is (x+5) ((9)/(4)) = 180. The angles of triangle PQR are P = 80^, Q = 60^, R = 40^. 1. (b) Find the smallest number by which 240 must be multiplied to get a perfect square. Step 1: Find the prime factorization of 240. 240 = 24 × 10 = (2^3 × 3) × (2 × 5) = 2^4 × 3^1 × 5^1 Step 2: Identify prime factors with odd powers. For a number to be a perfect square, all exponents in its prime factorization must be even. In 2^4 × 3^1 × 5^1, the prime factors 3 and 5 have odd powers (1). Step 3: Determine the smallest multiplier. To make the powers of 3 and 5 even, we need to multiply by 3^1 and 5^1. The smallest number to multiply by is 3 × 5 = 15. 240 × 15 = (2^4 × 3^1 × 5^1) × (3^1 × 5^1) = 2^4 × 3^2 × 5^2 = (2^2 × 3 × 5)^2 = (60)^2 = 3600 The smallest number by which 240 must be multiplied to get a perfect square is 15. 2. (a) A car travels at a distance of 60km. Construct a table of value for speed and time. Given that the time for the journey vary as: 1(1)/(2)hrs, 2hrs, 2(1)/(2)hrs, and 3hrs. Step 1: Recall the relationship between distance, speed, and time. Distance (D) = Speed (S) × Time (T) Therefore, Speed (S) = DistanceTime Step 2: Identify the given distance and times. Distance (D) = 60 km Times (T): 1(1)/(2) hrs = 1.5 hrs, 2 hrs, 2(1)/(2) hrs = 2.5 hrs, 3 hrs. Step 3: Calculate the speed for each given time. • For T = 1.5 hrs: S = 60 km1.5 hrs = 40 km/hr • For T = 2 hrs: S = 60 km2 hrs = 30 km/hr • For T = 2.5 hrs: S = 60 km2.5 hrs = 24 km/hr • For T = 3 hrs: S = 60 km3 hrs = 20 km/hr Step 4: Construct the table of values. | Time (hrs) | Speed (km/hr) | | :--------- | :------------ | | 1(1)/(2) (1.5) | 40 | | 2 | 30 | | 2(1)/(2) (2.5) | 24 | | 3 | 20 | 2. (b) A crate containing 24 bottles of drinks is made up of 10 bottles of Coca Cola, 8 bottles of Fanta and the rest are Sprite. If one bottle is picked at random, what is the probability of picking: (i) a coca cola drink (ii) a sprite drink Step 1: Determine the total number of bottles and the number of each type of drink. Total bottles = 24 Coca Cola bottles = 10 Fanta bottles = 8 Sprite bottles = Total bottles - Coca Cola bottles - Fanta bottles Sprite bottles = 24 - 10 - 8 = 6 Step 2: Calculate the probability for each event. The formula for probability is P(event) = Number of favorable outcomesTotal number of outcomes. (i) Probability of picking a Coca Cola drink: P(Coca Cola) = Number of Coca Cola bottlesTotal bottles = (10)/(24) = (5)/(12) (ii) Probability of picking a Sprite drink: P(Sprite) = Number of Sprite bottlesTotal bottles = (6)/(24) = (1)/(4) The probability of picking a Coca Cola drink is (5)/(12). The probability of picking a Sprite drink is (1)/(4). 2. (c) Calculate the unknown angles a and b in the diagram above. Step 1: Identify parallel lines and apply angle properties. The lines marked with arrows (let's call them line X and line P) are parallel. Draw an imaginary line through the vertex where angles a and b are located, parallel to lines X and P. The angle 45^ and the part of angle a on the left are alternate interior angles. So, that part of angle a is 45^. The angle 55^ and the part of angle a on the right are alternate interior angles. So, that part of angle a is 55^. Step 2: Calculate angle a. Angle a is the sum of these two alternate interior angles. a = 45^ + 55^ a = 100^ Step 3: Calculate angle b. Angles a and b form angles around a point, so their sum is 360^. a + b = 360^ 100^ + b = 360^ b = 360^ - 100^ b = 260^ The unknown angles are a = 100^ and b = 260^. That's 2 down. 3 left today — send the next one.