Mathematics Level B. Højere handelseksamen. Friday May 19, 2017 At hhx171-mat/b

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1 Mathematics Level B Højere handelseksamen hhx171-mat/b Friday May 19, 2017 At

2 Mathematics B The paper consists of two partial tests. The partial test without Tools consists of assignment 1 to 5, i.e. all in all 5 questions. The answer to this partial test must be handed in by 10am. The partial test with Tools consists of assignments 6 to 11, i.e. all in all 13 questions. The 18 questions each carry a weight of 5 points in the assessment of the total paper. You are only to hand in one of the assignments 11A, 11B, and 11C. If you hand in more of these assignments, only the first assignment will be assessed. During the first hour of the test tools apart from writing and drawing tools are not allowed. In the last 3 hours of the test all tools are allowed. In the assessment of the answers to the individual questions and the overall impression emphasis will be put on whether the examinee s train of thought is clearly reflected in the answers. The paper must account for the method used and documentation in the form of a sufficient number of intermediate results and/or a mathematical explanation of the use of different facilities offered by an IT program. If graphs and illustrations are used there must be a clear relationship between text and illustration. The following three data files belong to this set of exam questions: entrepreneur egg housing prices

3 Friday May 19, 2017 page 1 of 8 The partial test without tools At Exercise 1 The function f is given by f x x x x 3 2 ( ) a) Account for x 2 being a solution to the equation f() x 0. Exercise 2 The graph of a third degree polynomial f is shown in the coordinate system. a) Find the following: The domain of f The zeros/roots The extrema The monotonicity You may use attachment y f x Exercise 3 A function f has the equation f x x x 2 ( ) a) Find f ( x), and solve the equation f ( x) 0.

4 Friday May 19, 2017 page 2 of 8 Exercise 4 Below the age distribution of unemployment benefit claimants in 2012 is shown in a cumulative frequency curve. 100 relative cumulative frequency in % age a) Find the median and account for the significance of this number. You may use attachment 2. Source: statistikbanken.dk Exercise 5 For a given commodity the relationship between production and cost can be described by a linear function Cx () ax b. At a production of 100 units of the commodity the costs are DKK At a production of 120 units of the commodity the costs are DKK Production, x Costs, Cx ( ) a) Find the equation of C and decide how much the costs increase if the production is increased by 10 units. The answers to the test without tools must be turned in by 10.00

5 Friday May 19, 2017 page 3 of 8 The partial test with tools At Exercise 6 Ib Pedersen wants to create a private retirement savings account into which he pays DKK 8000 each month for 10 years. The monthly interest rate is 0.33%. a) Find the balance of the retirement savings account after 10 years. Ib Pedersen wishes to have DKK in the account when he retires. He therefore calculates for how long he must save up if the interest rate is 0.33% per month. The calculations are shown below. b) Explanations to the calculations must be given; you may use attachment 3. n (1 r) 1 An y The formula for the future value of an annuity has r been stated n n n n ln(1.0033) ln( ) n Ib Pedersen must save up for 127 months. When Ib Pedersen retires he has DKK in his retirement savings account. He wishes to be paid DKK each month. The interest rate is still 0.33% per month. c) Find the number of months in which Ib Pedersen can be paid DKK

6 Friday May 19, 2017 page 4 of 8 Exercise 7 Every year The Danish Entrepreneurs Association produces an entrepreneur survey. The survey is performed among the members and stakeholders of The Danish Entrepreneurs Association. In the 2015 survey the questions vary, among others, about age and status of entrepreneurship. The table below shows an excerpt of the data found in the file entrepreneur. Age Status years Actual entrepreneur years Actual entrepreneur < 20 years Potential entrepreneur : : a) Create a table like the one below containing data from the survey. Age Actual entrepreneur Potential entrepreneur Total < 20 years years years years years years 60+ years Total 715 The Danish Entrepreneurs Association wants to examine whether there is a difference in the age distribution between actual and potential entrepreneurs. b) State a hypothesis that can be used to test this and test the hypothesis at a significance level of 5%. c) Write to The Danish Entrepreneurs Association presenting your answers to questions a) and b). Source: The Danish Entrepreneurs Association

7 Friday May 19, 2017 page 5 of 8 Exercise 8 A company produces and sells a commodity. A model of the revenue of the commodity can be described by the function Rx ( ) 250 x x, x 0 where Rx ( ) is the revenue of the commodity at a sales of x units. The variable costs can be described by the function Cx ( ) 55 x, x 0 a) Draw the graphs of R and C in the same coordinate system and find the sales, which make the revenue equal to the variable costs. The profit can be found as profit = revenue variable costs b) Find an equation of the profit DB and find the greatest possible profit. DKK DB quantity in units

8 Friday May 19, 2017 page 6 of 8 Exercise 9 A chicken breeder sells eggs to the industry. The weight of the eggs is approximately normally distributed with a average weight of 70 grams and a spread of 12 grams. a) Find the probability that an egg chosen at random weighs between 63 and 72 grams. In a sample of 1000 eggs the weights are registered. Data can be found in the file egg. b) Find the sample s mean, standard deviation and set of quartiles. c) Find a 95% confidence interval for the mean. Weight : XL 73 g and more L 63 g - 72 g M 53 g - 62 g S 52 g and less Source:

9 Friday May 19, 2017 page 7 of 8 You may only turn in one of the exercises 11A, 11B and 11C for assessment. If you turn in more of these exercises, only the first exercise will be assessed. Exercise 10A A company produces and sells two types of bird feed, numbers of Lux-mix x and numbers of Normal-mix y. The profit per bag of Lux-mix is DKK 45 and DKK 30 per bag of Normal-mix. The function f (, x y) ax by denotes the company s total daily profit. The feed mixes contain a combination of sunflower kernels, corn and grain with different requirements to the amounts. The Lux-mix contains 2 kg of sunflower kernels, 0.5 kg of corn, and 0 kg grain. The Normal-mix contains 1 kg sunflower kernels, 1 kg corn, and 1 kg grain. The company has 600 kg of sunflower kernels, 375 kg of corn, and 350 kg grain at its disposal per day. a) Find an equation for the function f and draw the polygon area given by the constraints above. b) Find the numbers of Lux-mix and the numbers of Normal-mix the company must produce per day to obtain the greatest possible daily profit. Exercise 10B A function f is given by f x 3 x ( ) 4x 5e 6 a) Find the zeros/roots and draw the graph of f. b) Find f ( x) and find the monotonicity.

10 Friday May 19, 2017 page 8 of 8 Exercise 10C The building society registers the current price of square meters for homes sold in a specific geographical area. The square meter price in DKK on sold weekend cottages in Jammerbugt Municipality in northern Jutland in the period from the third quarter of 2009 to the first quarter of 2015 has been registered. The table below shows an excerpt of the data found in the file housing prices. Quarter Number of quarter The square meter price 3 rd qtr th qtr st qtr : : : a) Create a linear regression model s( x) ax b, where sx ( ) is the square meter price x quarters after the 2 nd quarter of b) Explain the significance of the parameter a in the equation of s and use the model to find the expected square meter price by the 1 st quarter of Source: Realkreditforeningen.dk

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