Case study

Here you go ... they are taken from the sample paper

CASE STUDY – III According to an educational board survey, it was observed that class XII students apply at least one to four weeks ahead of colleges application deadline. Let X represent the week when an average student applies ahead of a college’s application deadline and the probability of student to get admission in the college 𝑃(𝑋 = 𝑥) is given as follows: 𝑃(𝑋 = 𝑥) = { 𝑘𝑥 6 𝑤ℎ𝑒𝑛 𝑥 = 0, 1 𝑜𝑟 2 (1 − 𝑘)𝑥 6 𝑘𝑥 2 𝑤ℎ𝑒𝑛 𝑥 = 3 𝑤ℎ𝑒𝑛 𝑥 = 4 0 𝑤ℎ𝑒𝑛 𝑥 > 4 Where k is a real number. Based on the above information, answer the following questions. Show steps to support your answers. a) Find the value of k. 1 b) What is the probability that Sonali will get admission in the college, given that she applied at least 2 weeks ahead of application deadline? 1 c) Calculate the mathematical expectation of number of weeks taken by a student to apply ahead of a college’s application deadline. OR To promote early admissions, the college is offering scholarships to the students for applying ahead of deadline as follows: ₹ 50000 for applying 4 weeks early, ₹ 20000 for applying 3 weeks early, ₹ 12000 for applying 2 weeks early, and ₹ 9600 for applying 1 week early What is the expected scholarship offered by the college?

CASE STUDY – I An overhead water tank has three pipes A, B and C attached to it (as shown in figure (II)). The inlet pipes A and B can fill the empty tank independently in 15 hours and 12 hours respectively. The outlet pipe C alone can empty a full tank in 20 hours. Based on the above information, answer the following questions. a) For a routine cleaning of the tank, the tank needs to be emptied. If pipes A and B are closed at the time when the tank is filled to two-fifth of its total capacity, how long will pipe C take to empty the tank completely? 1 b) How long will it take for the empty tank to fill completely, if all the three pipes are opened simultaneously? 1 c) On a given day, pipes A, B and C are opened (in order) at 5 am, 8 am and 9 am respectively, to fill the empty tank. In how many hours will the tank be filled completely? OR Given that the tank is half-full, only pipe C is opened at 6 AM, to empty the tank. After closing the pipe C and an hour’s cleaning time, tank is filled completely by pipe A and B together. What is the total time taken in the whole process?

When observed over a long period of time, a time series data can predict trend that can forecast increase or decrease or stagnation of a variable under consideration. Such analytical studies can benefit a business for forecasting or prediction of future estimated sales or production. Mathematically, for finding a line of best-fit to represent a trend, many methods are available. Methods like moving-averages and least-squares squares are some of the techniques to predict such trends. Mrs. Shamita runs a bread factory and the record of her sales of bakery items for the period of 2015 - 2019 is as follows: Year 2015 2016 2017 2018 2019 Sales (in ₹ thousands) 35 42 46 41 48 Based on the above information, answer the following questions. Show steps to support your answers. a) By taking year 2017 as origin, use method of least-squares to find the best-fit trend line equation for Mrs. Shamita’s business. Show the steps of your working. OR Demonstrate the technique to fit the best-suited straight-line trend by the method of 3-years moving averages. Also draw the trend line. 2 b) What are the estimated sales for Mrs. Shamita’s business for year 2022? 1 c) Mrs Shamita wishes to grow her business to yearly sale of ₹ 67400. In which year will she be able to reach her target?