MATH 500 Auckland Institute Arithmetic & Geometric Sequence Questions

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MATH500 S1 2020
Student ID:
Question 1: [8 marks]
a) A and B are two sets where A is a subset of B , i.e. A ⊆ B . Indicate whether the statements
in the following table are True or False.
Statement
True or False
A⊂B
n (A) ≤ n (B)
n (A) < n (B) A∪B =A A∩B =A B 0 ⊆ A0 b) A coin is tossed three times. Find the probability that the coin lands heads at least once. Hint: use the complement rule. Page 2 of 13 MATH500 S1 2020 Student ID: Question 2: [10 marks] A survey of 2000 women found that 680 were smokers and 50 had the lung disease emphysema. Of those who had emphysema, 42 were also smokers. Let S be the event that a participant in the survey is a smoker and let E be the event that the participant has emphysema. a) Complete the following table for the data in the survey. E E0 Total S S0 Total 2000 b) Present the survey data as a Venn diagram. c) Find the probability P (S ∩ E 0 ) and describe this event in words. d) Find the probability P (S ∪ E) and describe its meaning in words. Page 3 of 13 MATH500 S1 2020 Student ID: e) Find the probability P (S|E) and describe its meaning in words. f) Show whether or not the results indicate that having emphysema and being a smoker are independent events. g) Fill-in the probabilities on the following tree-diagram. E S E0 E S 0 E0 Page 4 of 13 MATH500 S1 2020 Student ID: Question 3: [7 marks] a) A company considers the following two formats for its customer IDs. • Format 1: three letters followed by three digits; repetition of letters or digits not allowed. • Format 2: two letters followed by four digits; repetition of letters or digits is allowed. (a) How many customer IDs can be generated from format 1? (b) How many customer IDs can be generated from format 2? (c) Which format generates the most customer IDs? b) How many ways to select a president, secretary and treasurer from 16 people? c) How many ways to select a committee of 3 people from 16 people, when the order within each committee is not important? Page 5 of 13 MATH500 S1 2020 Student ID: Question 4: [7 marks] a) The second term in a sequence is 2 and the fth term is 54. Find the 200th term if it is an arithmetic sequence. b) Calculate the sum of the following geometric sequence: 1 1 1 1 − + − + ··· + 2 4 8 256 Page 6 of 13 MATH500 S1 2020 Student ID: Question 5: [8 marks] Here is a quadratic function and its graph, a parabola, y (x) = 3x2 − 2x + 1 where x is along the horizontal axis and y is along the vertical axis. a) Find the coordinates of the parabola's x-intercepts (if any) by inspecting the graph. b) Use the discriminant to verify your answer. c) Find the the parabola's y -intercept Page 7 of 13 MATH500 S1 2020 Student ID: d) Coordinates of the parabola's vertex. (a) Find the derivative of the quadratic function. (b) Find the coordinates of the vertex. Hint: begin by setting the derivative in (a) to zero. (c) Indicate the vertex on the graph. Page 8 of 13 MATH500 S1 2020 Student ID: Question 6: [6 marks] The half-life of tritium (a radioactive isotope of hydrogen) is about 12.3 years. Suppose we start with a 10 mg sample mass. An accurate model of it's mass-loss as a function of time is m (t) = m0 e−rt a) Given that the denition of a half-life is the time required for a quantity to reduce to half of its initial value, what is the decay-rate r of tritium? b) How much mass remains after 20 years? c) How many years does this model predict it takes for the sample to reach one-tenth its original mass? d) How many years does this model predict it takes for the sample to reach zero mass? Page 9 of 13 MATH500 S1 2020 Student ID: Question 7: [7 marks] Find the derivatives of the following functions. √ a) f (x) = x b) f (x) = e−4x √ c) f (x) = e−4x x Page 10 of 13 MATH500 S1 2020 Student ID: Question 8: [6 marks] Integration a) Find the indenite integral of the mass-loss function in Question 6. b) Find the area contained beneath the curve in Question 5 bounded by the x-axis and limits x = −0.5 and x = 1. Page 11 of 13 MATH500 S1 2020 Student ID: Question 9: [8 marks] The velocity v (in m/s) of an object at time t (in seconds) is given by the following function. v (t) =   2t, 0 ≤ t ≤ 10 1  − (t − 10)2 + 20, 10 < t ≤ 20 5 a) Find the acceleration function a(t) of the object. b) What is the acceleration of the object at t = 15 s? c) Find the total distance traveled by integrating the velocity function. Page 12 of 13 MATH500 S1 2020 Student ID: Formula sheet Probability P (E) = n (E) n (S) Conditional probability P (E ∩ F ) P (F ) P (E|F ) = Union rule P (E ∪ F ) = P (E) + P (F ) − P (E ∩ F ) E and F are mutually exclusive if E ∩ F = ∅ (empty set) P (E ∪ F ) = P (E) + P (F ) if E and F are mutually exclusive Complement rule P (E 0 ) = 1 − P (E) Product rule P (E ∩ F ) = P (F ) P (E|F ) = P (E) P (F |E) E and F are independent if P (E|F ) = P (E) or P (F |E) = P (F ) P (E ∩ F ) = P (F ) P (E) if E and F are independent Permutations P (n, r) = n! (n − r)! Combinations C (n, r) = P (n, r) r! Arithmetic sequence an = a + (n − 1) d Partial sum; arithmetic sequence Sn = n n (2a + d (n − 1)) = (a1 + an ) 2 2 Geometric sequence an = arn−1 Dierentiation d n d ax x = nxn−1 , e = aeax dx dx Integration R R R [f (x) + g(x)] dx = f (x)dx + g(x)dx R n R 1 eax x dx = xn+1 + c; n 6= −1, eax dx = +c n+1 a cf (x)dx = c R f (x)dx, R Page 13 of 13