SLOW LEARNERS MATERIALS BUSINESS MATHEMATICS SIX MARKS QUESTIONS

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1 SLOW LEARNERS MATERIALS BUSINESS MATHEMATICS SIX MARKS QUESTIONS 1. Form the differential equation of the family of curves = + where a and b are parameters. 2. Find the differential equation by eliminating the arbitrary constants a and b from =a tan+sec 3 Find the differential equation of all circles + +2= 0 which pass through the origin and whose centres are on the x axis. 4. Solve : = 5. The slope of a curve at any point is the reciprocal of twice the ordinate of the point. The curve also passes through the point (4, 3). Find the equation of the curve. 6. Solve :++ =0 7. Solve the equation :1 =1. Solve : + =1 9. Solve : + =4, if y = 0 when =. Solve : 3= 11. Solve : + = 12. Solve : ++25=5 13. Solve : +4 +4=2 14. Solve : ++25= Solve :3 +1=0 16. Find the missing term from the following data : x : f(x) :

2 17. Using Lagarange s formula find y(11) from the following table x : y : From the following data, find f(3) x : f(x) : Find the missing term from the following data. x : y : From the following data estimate the export for the year 2000 Year x : Export y : Using Gregory Newton s formula, find y() from the following data x : y : Using Gregory Newton s formula, find y when x = 5 x : y : From the following data find y(25) by using Lagrange s formula x : y : In a straight line of best find x intercept when =, =16.9, =30, =47.4 and n = Fit a straight line to the following =, =19, =30. =53 and n = In a line of best fit find the slope and the y intercept if =, =25. =30, =90 and n = 5

3 27. Fit a straight line y = ax + b to the following data by the method of least squares. x : y : For the following probability distribution of X X : p(x) : Find (i) 1 (ii) 2 (iii) 0<<2 29. A random variable X has the following probability distribution Values of X, x : p(x) : Evaluate the following probabilities (a) 0(b) <0 (c)0 30. A continuous random variablex has the following p.d.f =,0 0,h Determine k and evaluate (i) (ii) f the function f(x) is defined by =,0 <. Find the value of c 32. Let X be a continuous random variable with p.d.f,0< 1,1 2 = +3,2 3 0,h (i) Determine the constant a (ii) Compute A player tosses two fair coins. He wins Rs.5 if two heads appear, Rs.2 if 1 head appears and Rs.1 if no head occurs. Find his expected amount of gain. 34 A random variable X has the probability function as follows :

4 Values of X : Probability : Evaluate (i) E(3X + 1) (ii) E(X 2 ) (iii) Var (X) 35. Find the mean, variance and the standard deviation for the following probability distribution Values of x, x : Probability p(x) : On an average if one vessel in every ten is wrecked, find the probability that out of five vessels expected to arrive, at least four will arrive safely. 37. Find the probability that atmost 5 defective fuses will be found in a box of 200 fuses if experience shows that 2 percent of such fuses are defective. = It is stated that 2% of razor blades supplied by a manufacturer are defective. A random sample of 200 blades is drawn from a lot. Find the probability that 3 or more blades defective. = A random sample of size 50 with mean 67.9 is drawn from a normal population. If it is known that the standard error of the sample mean is 0.7, find 95% confidence interval form the population mean. 40. Measurements of the weights of a random sample of 200 ball bearings made by a certain machine during one week showed mean of 0.24 newtons and a standard deviation of newtons. Find (a) 95% and (b) 99% confidence limits for the mean weight of all the ball bearings. 41. A random sample of marks in mathematics secured by 50 students out of 200 students showed a mean of 75 and a standard deviation of. Find the 95% confidence limits for the estimate of their mean marks. 42. The mean I.Q of a sample of 1600 children was 99. Is it likely that this was a random sample from a population with mean I.Q 0 and standard deviation 15? ( Test at 5% level of significance) 43. A furniture manufacturing company plans to make two products, chairs and tables from its available resources, which consist of 400 board feet of mahogany timber and 450 man-hours of labour. It knows that to make a chair requires 5 board feet and man-hours and yields a profit of Rs.45, while each table uses 20 board feet and 15 man-hours and has a profit of Rs.0. How many chairs and tables should the company make to get the maximum profit under the above resource constraints? Formulate the above as an LPP.

5 44. Calculate the correlation co-efficient from the following data X : Y : Calculate the 3 yearly Moving Averages of the production figures (in mat.tonnes) given below. Year Production Year Production Estimate the trend values using the data given below by taking 4 yearly Moving Average Year Production Year Production 47. Obtain the trend values by the method of Semi-Average Year Production From the following data calculate the price index number by (a) Laspeyre s method (b) Paasche s method (c) Fisher s method Commodity A B C D Base year Price Quantity Current year Price Quantity Using three year moving averages determine the trend values for the following data. Year Produ Tion (in tonnes)

6 MARKS QUETIONS 1. The net profit p and quantity x satisfy the differential equation =. Find the relationship between the net profit and demand given that p = 20 when x = 2. The rate of increase in the cost C of ordering and holding as the size q of the order increases is given by q if C = 1 when q = 1 =. Find the relationship between C and 3. The rate increase in the cost C of ordering and holding as the size q of the order increases is given by the differential equation relationship between C and q if C = 4 when q = 2. =. Find the 4. The total cost of production y and the level of output x are related to the marginal cost of production by the equation cost function if y = 4 when x = 2? =. What is the total 5. Suppose that the quantity demanded = and quantity supplied = 6+ where p is the price. Find the equilibrium price for market clearance. 6. Solve : 13+12= Solve : 5+6= +3. Solve : 14+49=3+ 9. Suppose that = and =6+3. Find the equilibrium price for market clearance.. If =2459, =201, =110 and =402 find 11. From the data, find the number of students whose height is between 0cm. and 90cm. Height in cmsx : No.of Students y : Using Lagrange s formula find y when x = 4 from the following table. x : y :

7 13. Using Gregory Newton formula, find y when x = 145 given that x : y : From the following data find the area of a circle of diameter 96 by using Gregory Newton formula Diameter x : Area y : Using Gregory Newton s formula, find y(22.4) x ; y : Fit a straight line to the following data : x : y : Fit a straight line to the data given below. Also estimate the value y at x = 3.5 x : y : Given the p.d.f. of a continuous random variable X as follows 1 0<<1 = 0 h Find k and c.d.f 19. Suppose that the life in hours of a certain part of a radio tube is a continuous random variable X with p.d.f is given by =,h 0 0 h (i) what is the probability that all of three such tubes in a given ratio set will have to be replaced during the first of 150 hours of operation? (ii) What is the probability that none of three of the original tubes will have to be replaced during that first 150 hours of operation? 20. A random variable X has the following probability distribution.

8 Value of X, x : p(x) : a 3a 5a 7a 9a 11a 13a 15a 17a (i) Determine the value of a (ii) Find <3,>3 and 0<<5 21. Let X be a continuous random variable with p.d.f= 1<<1 0,h Find (i) E(X) (ii) E(X 2 ) (iii) Var(x) 22. Let X be a continuous random variable with p.d.f= 2,0 2. Show that the mean and the distribution is 1 and the variance is. 23. Find the mean, variance and standard deviation of the following probability distribution. Values of X : Probability p(x) : 24. Find the mean and variance for the following probability distribution. = 2, 0 0, <0 25. The number of accidents in a year attributed to taxi drivers in a city follows poisson distribution with mean 3. Out of 00 taxi drivers, find the approximate number of drivers with (i) no accident in a year (ii) more than 3 accidents in a year. 26. In a sample of 00 candidates the mean of certain test is 45 and S.D is 15. Assuming the normality of the distribution find the following : (i) How many candidates score between 40 and 60? (ii) How many candidates score above 50? (iii) How many candidates score below 30? 27. The I.Q (intelligence quotient) of a group of 00 school children has mean 96 and the standard deviation 12. Assuming that the distribution of I.Q among school children is normal, find approximately the number of school children having I..Q (i) less than 72 (ii) between 0 and 120

9 2. In a normal distribution 20% of the items are less than 0 and 30% are over 200. Find the mean and S.D of the distribution. 29. A large number of measurements is normally distributed with a mean of 65.5 and S.D of 6.2. Find the percentage of measurements that fall between 54. and A sample of 0 students are drawn from a school. The mean weight and variance of the sample are kg and 9 kg. respectively. Find (a) 95% and (b) 99% confidence intervals for estimating the mean weight of the students. 31. A random sample of 500 apples was taken from large consignment and 45 of them were found to be bad. Find the limits at which the bad apples lie at 99% confidence level. 32. Out of 00 TV viewers, 320 watched a particular programme. Find 95% confidence limits for TV viewers who watched this programme. 33. A sample of five measurements of the diameter of a sphere were recorded by a scientist as 6.33,6.37,6.36, 6.32 and 6.37mm. Determine the point estimate of (a) mean, (b) variance. 34. The mean life time of 50 electric bulbs produced by a manufacturing company is estimated to be 25 hours with a standard deviation of 1 hours. If is the mean life time of all the bulbs produced by the company, test the hypothesis that =900 hours at 5% level of significance. 35. A company markets car tyres. Their lives are normally distributed with a mean of kilometers and standard deviation of 2000 kilometers. A test sample of 64 tyres has a mean life of kms. Can you calculate that the sample mean differes significantly from the population mean? (test at 5% level) 36. A sample of 400 students is found to have a mean height of cms. Can it reasonably be regarded as a sample from a large population with mean height of cms and standard deviation of 3.3 cms. (Test at 5% level) 37. To test the conjecture of the management that 60 percent employees favour a new bonus scheme, a sample of 150 employees was drawn and their opinion was taken whether they favoured it or not. Only 55 employees out of 150 favoured the new bonus scheme. Test the conjecture at 1% level significance. 3. Solve graphically : Minimize = Subject to 36 +6

10 , Solve the following using graphical method Maximize =3 +4 subject to the constraints ; ;, Calculate the correlation co-efficient from the data below : X : Y : Find the co-efficient of correlation for the data given below. X : Y : From the data given below, find the correlation co-efficient X : Y : Obtain the two regression lines from the following X : Y : From the data given below calculate Seasonal Indices. year Quarter I II III IV Obtain the trend values by the method of Semi-Average Year

11 Net profit (Re lakhs) 46. Calculate the trend values by four year moving averages method. Year Produ ction 47. Calculate the seasonal indices for the following data using average method Quarters Year I II III IV Compute (i) Laspeyre s (ii) Paasche s and (iii) Fisher s Index Numbers for the 2000 from the following : Commodity Price Quantity A B C D Calculate Fisher s Ideal Index from the following data and verify that it satisfies both Time Reversal and Factor Reversal Test Commodity A B C D E Price Quantity

12 50. Calculate the cost of living index by aggregate expenditure method. Commodity A B C D E F Quantity Price(Rs) Compute (i) Laspeyre s (ii) Paasche s and (iii) Fisher s index Numbers Commodity A B C D Price Base Current Year year Quantity Base Current Year year Calculate Fisher s Ideal Index from the following data and show how it satisfies time reversal test and factor reversal test. Commodity A B C D E F Base year (1997) Price Quantity Current year (199) Price Quantity Calculate the cost of living index number using Family Budget method for the following data taking the base year as 1995 Commodity Weight Price (per unit)

13 A B C D E The following data relate to the life (in hours) of samples of 6 electric bulbs each drawn at an interval of one hour from a production process. Draw the control chart for and R and comment. Sample No. Life time (in hours) (Given for n = 6, A 2 = 0.43, D 3 = 0, D 4 = 2.004) 55. The following data shows the value of sample mean and the range R for ten samples of size 5 each. Calculate the values for central line and control limits for mean chart and range chart and determine whether the process is in control Sample no. Mean Range (R)

14 56. The following are the and R values for 20 samples of 5 readings. Draw and R chart and write your conclusion Samples R Samples R Note : Students are advised to practice the similar type of problems from the text books

12 STD BUSINESS MATHEMATICS

12 STD BUSINESS MATHEMATICS STD BUSINESS MATHEMATICS www.kalvisolai.com 0 MARK FAQ S: CHAPTER :. APPLICATION OF MATRICES AND DETERMINANTS. If A verify that AAdjA AdjA A AI. (M 0). Show that the equations y + z = 7, + y 5z =, + y