Question 1 Define rational and irrational numbers. Give examples. Answer 1 Rational Number: A number is said to be a rational number if we can write it as a fraction where the numerator and denominator of the fraction are both whole numbers, and the denominator cannot equal zero. Rational numbers have finite or recurring decimals. Examples: 1 4 = 0.25 , 9 8 = 1.125 4 1 =0.25, 8 9 =1.125 All numbers above are rational numbers. Irrational Number: An irrational number is a number which is not a rational number. It is a number that cannot be written as a ratio of two integers or cannot be expressed as fraction form, and irrational numbers have infinite or non-recurring decimals. Examples: 2 , 3 , 𝜋 , 𝑒 2 , 3 ,π,e Question 2 Mathematical Reasoning True or False? There exist irrational numbers with repeating decimals? There exists a natural number 𝑥 x such that 𝑦 ≥ 𝑥 y≥x for every natural number 𝑦 y? Answer 1 Step 1 To find if the statement is true or false: There exist irrational numbers with repeating decimals. By definition of a rational number, a number that can be expressed as a ratio 𝑝 𝑞 q p where 𝑝 p and 𝑞 q are integers such that 𝑞 ≠ 0 q  =0 is called a rational number. A number that cannot be expressed as a ratio of two integers is called an irrational number. Step 2 Let a number with repeating digits be 𝑥 = 0. 𝑎 𝑎 𝑎 𝑎 … x=0.aaaa… where 𝑎 a is any digit from 1 to 9. Multiply by 10 on both sides: 10 𝑥 = 𝑎 . 𝑎 𝑎 𝑎 𝑎 … 10x=a.aaaa… Now consider: 10 𝑥 − 𝑥 = 𝑎 10x−x=a Thus, we can write the repeating decimal number 𝑥 = 0. 𝑎 𝑎 𝑎 𝑎 … x=0.aaaa… as a ratio 𝑥 = 𝑎 9 x= 9 a . This means every repeating decimal number is a rational number. Therefore, the given statement "There exist irrational numbers with repeating decimals" is False. Question 3 Solve the differential equation for Newton's Law of Cooling to find the temperature function in the following case. A pot of boiling soup (100°C) is put in a cellar with a temperature of 10°C. After 30 minutes, the soup has cooled to 80°C. When will the temperature of the soup reach 30°C? Answer 1 Data Analysis We have: Initial temperature of soup, 𝑇 0 = 10 0 ∘ 𝐶 T 0 =100 ∘ C Surrounding temperature, 𝑇 𝑠 = 1 0 ∘ 𝐶 T s =10 ∘ C Final temperature, 𝑇 𝑓 = 8 0 ∘ 𝐶 T f =80 ∘ C Time taken, 𝑡 = 30 min t=30min To determine the time needed to reach 30°C under the same circumstances, apply Newton's Law of Cooling. (Solution details omitted, as further steps involve solving the differential equation using Newton's Law of Cooling.) Question 4 Adult male height in America is normally distributed, with an average of 69.3 inches and a standard deviation of 2.67 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.8 inches and a standard deviation of 2.54 inches. a) If a man is 6 feet 5 inches tall, what is their z-score? Answer 1 Step 1 Given: Men: Population mean ( 𝜇 μ) = 69.3, Standard deviation ( 𝜎 σ) = 2.67 Women: Population mean ( 𝜇 μ) = 64.8, Standard deviation ( 𝜎 σ) = 2.54 Formula: Let "X" be the person's height 𝑧 = 𝑋 − 𝜇 𝜎 z= σ X−μ Step 2 Height of a man (X) = ( 6 × 12 ) + 5 = 77 (6×12)+5=77 inches 𝑧 = 77 − 69.3 2.67 z= 2.67 77−69.3 𝑧 = 2.883395131 z=2.883395131 Round to two decimals. 𝑧 = 2.88 z=2.88 Question 5 Solving Equations Graphically Find all real solutions of the equation, rounded to two decimals. 𝑥 3 − 2 𝑥 2 − 𝑥 − 1 = 0 x 3 −2x 2 −x−1=0 Answer 1 Given, 𝑥 3 − 2 𝑥 2 − 𝑥 − 1 = 0 x 3 −2x 2 −x−1=0 To solve the equation graphically, draw the graph of the given equation and find the point at which it cuts the x-axis. (Having plotted the graph, it shows that the equation has one real solution at approximately 𝑥 = 2.547 x=2.547.) Step 2 Hence, from the graph, it is quite clear that the equation has only 1 real solution, i.e., 𝑥 ≈ 2.55 x≈2.55