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12 STD BUSINESS MATHEMATICS 10 MARK FAQ’S: CHAPTER : 1. APPLICATION OF MATRICES AND DETERMINANTS 1 1 1 1. If A 1 2 3 verify that A AdjA AdjAA A I . (M’10) 2 1 3
2. Show that the equations 2x – y + z = 7, 3x + y – 5z = 13, x + y + z = 0 are consistent and have
unique solution. (O’09) 3. NON – TEXTUAL: Solve by matrix method the equations 3x – y – z = -2; x + y + z = 6 ; x – 2y + 4z = 9 (J’09) 4. Solve by using matrix inversion method: 2 x 8 y 5z 5, x y z 2, x 2 y z 2. (J’07 ; M’09) 5. Solve by matrix method the equations x 2 y 3 z 1 ; 3 x y 4 z 3 ; 2 x y 2 z 1. (M’06 ; J’08 ; O’10)
6. Solve by Cramer’s rule : 2 x 2 y z 1, x y z 0, 3x 2 y 3z 1. (J’06 ; M’07 ; M’08 ; O’08; J’11; M’12) 7. Solve by Cramer’s rule : x y 2, y z 6, z x 4. (O’07) 8. Solve the equations x + 2y + 5z = 23 ; 3x + y + 4z = 26 ; 6x + y + 7z = 47 by determinant method. (J’10 ; M’11; O’11) 9. A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission. Sales of Units Total commission Month drawn (in Rs.) A B C January 90 100 20 800 February 130 50 40 900 March 60 100 30 850 Find out the rates of commission on the items A, B and C, Solve by Cramer’s rule. (O’06) 10. The data below are about an economy of two industries P and Q. The values are in lakhs of rupees. User
Final Total demand output P Q P 16 12 12 40 Q 12 8 4 24 Find the technology matrix and test whether the system is viable as per Hawkins – Simon conditions.(O’08) Producer
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11. In an economy there are two industries P and Q and the following table gives the supply and demand positions in crores of rupees: User
Producer
P 10 20
P Q
Final demand 15 10
Q 25 30
Total output 50 60
Determine the outputs when the final demand changes to 35 for P and 42 for Q. (J’07 ; M’08 ; J’08 ; M’10 ; O’10 ; J’11 ; M’12) 12. In an economy of two industries P and Q the following table gives the supply and demand positions in crores of rupees: Producer P Q
User P 16 8
Q 20 40
Final demand
Total output
4 32
40 80
Find the outputs when the final demand changes to 18 for P and 44 for Q. (J’06 ; O’06 ; J’09) 13. The data below are about an economy of two industries P and Q. The values are in crores of rupees: Producer P Q
User P 50 100
Q 75 50
Final demand
Total output
75 50
200 200
Find the outputs when the final demand changes to 300 for P and 600 for Q. (M’07 ; J’10) 14. In an economy of two industries P and Q the following table gives the supply and demand positions in millions of rupees. Producer P Q
User P Q 14 6 7 18
Final Demand
Total Output
8 11
28 36
Find the outputs when the final demand changes to 20 for P and 30 for Q. (O’09) 15. Suppose that the inter-relationship between the production of two industries P and Q in a year (in millions of rupees) Producer P Q
User P Q 16 20 8 40
Final Demand
Total Output
4 32
40 80
Find the outputs when the final demand changes
(i) 12 for P and 18 for Q (ii) 8 for P and 12 for Q. (O’11)
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16. Two products A and B currently share the market with shares 60% and 40% each respectively. Each week some brand switching takes place. Of those who bought A the previous week, 70% buy it again whereas 30% switch over to B. Of those who bought B the previous week, 80% buy it again whereas 20% switch over to A. Find their shares after one week and after two weeks. If the price war continues, when is the equilibrium reached? (O’07) 17. Two products P and Q share the market currently with shares 70% and 30% each respectively. Each week some brand switching takes place. Of those who bought P in the previous week, 80% buy it again whereas 20% switch over to Q. Of those who bought Q in the previous week, 40% buy it again whereas 60% switch over to P. Find their shares after two weeks. If the price war continues, when is the equilibrium reached? (M’06 ; M’09) 18. The newspapers A and B are published in a city. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year 55% buy it again and 45% switch over to A. Find their market shares after two years. (M’11)
CHAPTER : 2. ANALYTICAL GEOMETRY 1. Find the centre, vertices, eccentricity, foci and latus rectum and directrices of the ellipse 9 x 2 16 y 2 36 x 32 y 92 0. (M’08 ; J’10 ; J’11)
2. Find the centre, vertices, eccentricity, foci and latus rectum and directrices of the ellipse 7 x 2 4 y 2 14 x 40 y 79 0. (O’07 ; O’10) 3. Find the centre, eccentricity, foci and directrices of the ellipse 3 x 2 4 y 2 6 x 8 y 5 0. (M’06 ; O’08 ; J’09) 4. Find the centre, eccentricity, foci and directrices of the hyperbola 12 x 2 4 y 2 24 x 32 y 127 0. (J’07) 5. Find the centre, eccentricity, foci and latusrectum of the hyperbola 9 x 2 16 y 2 18 x 64 y 199 0. (O’06 ; J’08) 6. Find the equation to the hyperbola which has the lines x + 4y – 5 = 0 and 2x – 3y + 1 = 0 for its asymptotes and which passes through the point (1,2). (M’11) 7. Find the equation to the hyperbola which has 3x – 4y + 7 = 0 and 4x + 3y + 1 = 0 for asymptotes and which passes through the origin. (M’12)
–
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8. Find the equations of the asymptotes of the hyperbola 2 x 2 5 xy 2 y 2 11 x 7 y 4 0. (J’06 ; M’07 ; M’09) 9. Find the equations of the asymptotes of the hyperbola 8 x 2 10 xy 3 y 2 2 x 4 y 2 0. (M’10 ; O’11) 10. Find the equations of the asymptotes of the hyperbola 3x 2 5 xy 2 y 2 17 x y 14 0. (O’09)
CHAPTER : 3. APPLICATION OF DIFFERENTIATION - I 1 1. A firm produces x tones of output at a total cost C ( x) Rs. x 3 4 x 2 25 x 8 . Find (i) Average cost 2 (ii) Average Variable Cost (iii) Average Fixed Cost. Also find the value of each of the above when the output level is 10 tonnes. (O’10)
2. Find the elasticity of demand, when the demand is q
20 and p = 3. Interpret the result. (M’10) p 1
3. If AR and MR denote the average and marginal revenues at any output level, show that elasticity of AR . Verify this for the linear demand law p a bx , where p is price and x is demand is equal to AR MR the quantity. (M’07 ; J’08 ; J’11) 4. Prove that for the cost function C 100 x 2 x 2 , where x is the output, 1 the slope of AC curve = MC AC . (MC is the marginal cost and AC is the average cost) (O’07) x 5. Determine the coefficients a and b so that the curve y = ax2 – 6x + b may pass through the point (0,2) and have its tangent parallel to the x-axis at x = 1.5. (J’09) 6. Find the equation of the tangent and normal to the demand curve y x 2 x 2 at x 6. (O’06) 7. Prove that the curves y = x2 – 3x + 1 and x(y + 3) = 4 intersect at right angles at the point (2,-1).(J’06) 8. Find the equations of the tangent and normal to the curve y x 2x 3 x 7 0 at the point where it cuts the x-axis. (M’06 ; M’09 ; M’11 ; O’11 ; M’12) 9. Find the equations of the tangent and normal at the point a sec ,b tan on the hyperbola x2 y2 1. (M’08 ; O’08) a 2 b2
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10. Find the point on the curve y = (x – 1)(x – 2) at which the tangent makes an angle 135 with the positive direction of x – axis. (J’10) 11. At what points on the circle x2 + y2 – 2x - 4y + 1 = 0, the tangent is parallel to (i) x-axis (ii) y-axis. (J’07 ; O’09)
CHAPTER : 4. APPLICATION OF DIFFERENTIATION - II 1. Find the maximum and minimum values of the function x 3 6 x 2 9 x 15. (O’06) 2. Investigate the maxima and minima of the function 2 x 3 3 x 2 36 x 10. (M’08 ; O’10) 3. Find the maximum and minimum values of the function 2 x 3 15x 2 24x 15. (M’06 ; J’11) 4. NON – TEXTUAL: Investigate the maxima and minima of the function 2 x 3 9 x 2 12x 15. (O’09) 5. Show that the maximum value of the function f ( x ) x 3 27 x 108 is 108 more than the minimum value. (J’08) 6. For the cost function C = 2000 + 1800x - 75x2 + x3 find when the total cost (C) is increasing and when it is decreasing . Also discuss the behavior of the marginal cost (MC) (J’09) 7. A certain manufacturing concern has total cost function C = 15 + 9x – 6x2 + x3. Find x, when the total cost is minimum. (M’10) 1 3 x 5 x 2 10x 5. At what level of output will 10 the marginal cost and the average variable cost attain their respective minimum? (J’07)
8. A firm produces x tonnes of output at a total cost C
x3 3 x 2 9 x 16 are respectively the sales revenue and cost function of x units 3 sold. Find the following: (i) At what output is the revenue maximum? What is the total revenue at this point? (ii) What is the marginal cost at a minimum? (iii) What output will maximise the profit? (M’09 ; M’11)
9. R = 21x – x2 and C
x 10. A Radio manufacturer finds that he can sell x radios per week at Rs.p each, where p 2100 . His 4 2 x cost of production of x radios per week is Rs. 120x . Show that his profit is maximum when the 2
production is 40 radios per week. Find also his maximum profit per week. (O’08)
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11. The total cost and total revenue of a firm are given by C = x3 – 12x2 + 48x + 11 and R = 83x – 4x2 – 21. Find the output (i) when the revenue is maximum (ii) when the profit is maximum? (J’10) 12. Find EOQ for the data given below. Also verify that carrying costs is equal to ordering costs at EOQ. (J’06) Item
Monthly Requirements
A B C
9000 25000 8000
Ordering cost per order Rs.200 Rs.648 RS.100
Carrying cost per unit Rs.3.60 Rs.10.00 Rs.0.60
13. A manufacturer has to supply his customer with 600 units of his products per year. Shortages are not allowed and storage cost amounts to 60 paise per unit per year. When the set up cost is Rs. 80 find, (i) The economic order quantity (ii) The minimum average yearly cost (O’07) 14. Calculate EOQ in units and total variable cost for the following item, assuming an ordering cost of Rs.5 and a holding cost of 10%. Item A Annual demand 460 units Unit price Re. 1 (M’12) 15. The annual demand for an item is 3200 units. The unit cost is Rs.6 and inventory carrying charges 25% per annum. If the cost of one procurement Rs.150, determine (i) Economic order quantity (ii) Time between two consecutive orders. (M’07) 16. If u log x 2 y 2 z 2 , then prove that
2u 2u 2u 1 2 2 2 . (J’08 ; O’08 ; O10) 2 x y z x y2 z2
x2 y2 u 1 u then prove that x sin 2u by Euler’s theorem. (M’08) y 17. If u tan 1 y 2 x x y 18. If z e x
3
y3
, then prove that x
z z y 3 z log z. (Use Euler’s theorem). (M’10 ; J’10 ;O’11) x y
x y then 19. NON – TEXTUAL: Prove using Euler’s theorem if u cos 1 x y u u 1 x y cot u 0. (O’06) x y 2
20. NON – TEXTUAL: Prove using Euler’s theorem if u
x3 y3 x y
then x
u u 5 y u. (J’07) x y 2
21. The demand for a commodity A is q1 240 p1 6 p2 p1 p2 . Find the partial elasticities Eq1 Eq1 and when p1 = 5 and p2 = 4.(M’06 ; J’06 ; M’07 ; O’09 ; M’11; O’11 ; M’12) Ep1 Ep 2 2
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22. The demand for a commodity A is q1 16 3 p1 2 p 2 . Find (i) the partial elasticities 2
Eq1 Eq1 and Ep1 Ep 2
(ii) the partial elasticities when p1 2 and p2 1 .(O’07 ; M’08 ; J’09) 23. The demand for a commodity A is q1 12 p1 p1 p 2 . Find (i) the partial elasticities 2
Eq1 Eq1 and Ep1 Ep 2
(ii) the partial elasticities when p1 10 and p 2 4 .(J’11)
CHAPTER : 5. APPLICATIONS OF INTEGRATION
1. Evaluate:
3
dx . (J’06 ; M’09 ; J’10 ; M’11;J’11;O’11) tan x
1 6
2. Evaluate:
3
1
dx
6
cot x
. (M’07 ; M’08 ; O’10)
sin 3 x
2
3. Evaluate:
dx. (M’06)
sin 3 x cos 3 x
0
2
4. Evaluate:
0
a sin x b cos x dx (J’07 ; O’08) sin x cos x
2
4. Evaluate:
0
x x 2 x
dx. (J’09 ; M’12) 3
5. NON – TEXTUAL: Evaluate:
0
x x 3 x
dx. (M’10)
6. Evaluate:
x sin 0
2
xdx. (O’09)
7. NON – TEXTUAL: Find the area of one loop of the curve y 2 x 2 9 x 2 between x 0 and
x 3. (J’08)
8. NON – TEXTUAL: Find the area of one loop of the curve a 2 y 2 x 2 a 2 x 2 between x 0 and
x a. (O’08)
9. NON – TEXTUAL: Find the area of one loop of the curve y 2 x 2 4 x 2 between x 0 and x 2. (O’06) 10. Find the area of the ellipse
x2 y2 1. (J’09) a2 b2
3 x , x 3. Find the demand function and the x revenue function when the price is 2 and the demand is 1. (M’11)
11. The elasticity of demand (x) with respect to price p is
12. Find the consumers’ surplus for the demand function p = 25 – x – x2 when p0 = 19. (O’08)
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13. Find the consumers’ surplus and producers’ surplus under market equilibrium if the demand function is pd 20 3x x 2 and the supply function is p s x 1. (M’06 ; O’07;J’11)
16 x and ps . Find the consumers’ surplus and x4 2 producers’ surplus at market equilibrium price. (J’08)
14. The demand and supply curves are pd
15. The demand and supply function for a commodity are given by pd 15 x and p s 0.3x 2. Find the consumers’ surplus and producers’ surplus at market equilibrium price. (M’07) 16. The demand and supply law under a pure competition are given by pd = 23 – x2 and ps = 2x2 – 4. Find the consumers’ surplus and producers’ surplus at the market equilibrium price. (O’06 ; O’10) 17. Under pure competition The demand and supply laws for commodity and pd = 56 – x2 and x2 p s 8 . Find the consumers’ surplus and producers’ surplus at the market equilibrium price. 3 (J’07;O’11) 18. In a perfect competition The demand and supply curves of a commodity are given by pd = 40 – x2 and p s 3x 2 8 x 8. Find the consumers’ surplus and producers’ surplus at the market equilibrium price. (O’09 ; J’10) . The demand and supply functions under pure competition are pd 16 x 2 and p s 2 x 2 4. Find the consumers’ surplus and producers’ surplus at market equilibrium price. (J’06 ; M’08 ; M’09 ; M’10 ; M’12)
CHAPTER : 6. DIFFERENTIAL EQUATIONS 1. The net profit p and quantity x satisfy the differential equation
dp 2 p 3 x 3 . Find the relationship dx 3 xp 2
between net profit and demand given that p 20 when x 10.(M’06) dy xy Solve : 2 . (M’09 ; M’10) 2. dx x y 2
dy y 2 2 xy Solve : . (O’10) 3. dx x 2 2 xy
4. Solve :
dy y y 2 . (O’11) dx x x 2
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5. The rate of increase in the cost c of ordering and holding as the size q of the order increases is given by the differential equation
dC c 2 q 2 . Find the relationship between c and q, if c = 4 and q = 2. dq 2cq
(J’11) 6. Suppose that Qd 30 5P 2
dP d 2 P 2 and Qs 6 3P, where P denotes price. Find the dt dt
equilibrium price for market clearance. (J’06 ; M’07 ; J’ 07 ; M’08 ; O’08 ; M’12) dP d 2 P 2 and Qs 6 8P, where P denotes price. Find the dt dt equilibrium price for market clearance. (J’09 ; M’11)
7. Suppose that Qd 42 4 P 4
8. Solve : D 2 13 D 12 y e 2 x 5e x . (O’06 ; O’09 ; J’10) 9. Solve : D 2 5D 12 y e x 3e 2 x . (J’08) 10. Solve : D 2 14 D 49 y 3 e 7 x . (O’07)
CHAPTER : 7. INTERPOLATION AND FITTING A STRAIGHT LINE 1. From the following data calculate the value of e1.75 (O’09)
x: ex:
1.7 5.474
1.8 6.050
1.9 6.686
2.0 7.389
2.1 8.166
2. From the following data, find the number of students whose height in between 80 cm and 90 cm: (M’08) Height in cm x: No. of students y:
40 – 60 250
60 - 80 120
80 – 100 100
100 - 120 70
120 - 140 50
3. Find the number of men getting wages between Rs.30 and Rs.35 from the following table. (O’11) Wages x: No. of men y:
20 – 30 9
30 – 40 30
40 – 50 35
50 – 60 42
4. Find y when x 0.2 given that (O’10) x: y:
0 176
1 185
2 194
3 202
4 212
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5. Using Gregory-Newton’s formula, find y(22.4) (M’11) X: Y:
19 91
20 100
21 110
22 120
23 131
6. Using Lagrange’s formula find y when x = 4 from the following table (J’08 ; M’10)
x: y:
0 276
3 460
5 414
6 343
8 110
7. Fit a straight line to the following data: (J’06 ; O’06 ; O’07 ; J’11 ; M’12) 4
x: y:
8 9
7
12 13
16 17
20 21
24 25
8. Fit a straight line y ax b to the following data by the method of least squares: (M’06) 100 90.2
x: y:
200 92.3
300 94.2
400 96.3
500 98.2
600 100.3
9. Fit a straight line y ax b to the following data by the method of least squares: (M’07 ; J’07) x: y:
0 1
1 3
3 2
6 5
8 4
10. Fit a straight line to the data given below. Also estimate the value of y at x = 3.5 : (M’09 ; J’10) x: y:
0 1
1 1.8
2 3.3
3 4.5
4 6.3
11. A group of 5 students took tests before and after training and obtained the following scores. (O’08) Scores before training Scores after training
3 4
4 5
4 6
6 8
8 10
Find by the method of least squares the line of best fit.
CHAPTER : 8. PROBABILITY DISTRIBUTION kx(1 x) 1. Given the p.d.f of a continuous random variable X as follows f ( x) 0 Find k and c.d.f. (J’09)
for0 x 1 otherwise
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2. Suppose that the life in hours of a certain part of radio tube is a continuous random variable X with 100 when x 100 ; p.d.f given by f ( x) x 2 elsewhere 0 (i) What is the probability that all of three such tubes in a given radio 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 50 hours of operation? (M’10) 3. A random variable X has the following probability probability distribution. x: 0 1 2 3 4 5 6 7 8 P(x) a 3a 5a 7a 9a 11a 13a 15a 17a (i) Determine the value of a (ii) Find P(X < 3) , P(X > 3) and P(0 < X < 5) (J’07 ; O’09)
kx2 , 0 x 10 A continuous random variable has the following p.d.f: f ( x ) 0 otherwise 4. Determine k and evaluate (i) P0.2 x 0.5 (ii) P x 3. (M’08) 1 1 x 1 , 5. Let X be a continuous random variable with p.d.f f ( x) 2 otherwise 0 Find (i) E(X) (ii) E(X2) (iii) Var(X) (O’06 ; O’10)
2e 2 x , x 0 6. Find the mean and variance for the probability distribution: f ( x ) x0 0 (M’06 ; J’06 ; O’08 ; M’09 ; M’11 ; J’11) NON – TEXTUAL: Find the mean and variance for the probability distribution 1 0 x 1 , f ( x) 2 x (O’07) otherwise 0, 7. NON – TEXTUAL: In a continuous distribution, whose probability density function is given 3 0 x2 x(2 x) Show that the arithmetic mean of the distribution is 1 and the by f ( x) 4 , otherwise 0, 1 variance is . (M’07) 5 8. Ten coins are thrown simultaneously. Find the probability of getting at least 7 heads. (M’06;O’10;O’11) 9. For a binomial distribution with parameters n = 5 and p = 0.3 find the probabilities of getting (i) at least 3 successes (ii) at most 3 successes. (J’10)
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10. Find the probability that at most 5 defective bolts will be found in a box of 200 bolts, If it is known that 2% of such bolts are expected to be defective. (e-4 = 0.01832) (M’10 ; M’12) 11. 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 are defective. (e-4 = 0.01832) (J’09) 12. NON – TEXTUAL: The number of accidents in a year attributed to taxi drivers in a city follows Poisson distribution with mean 3.Out of 500 taxi drivers, find the approximate number of drivers with (i) no accident in a year (ii) more than 2 accidents in a year. (e-3 = 0.04979) (M’11) 13. A sample of 1000 candidates the mean of certain test is 45 and S.D 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? (O’09 ; J’11) 14. The I.Q (intelligence quotient) of a group of 1000 children has mean 96 and the standard deviation 12. Assuming the distribution as normal, find approximately the number of children having I.Q. (i) less than 72. (ii) between 80 and 120. (M’07 ; J’07 ; M’08) 15. NON – TEXTUAL: The distribution of marks obtained by 1000 students in an examination is normally distributed with mean 34 and S.D 16. (i) Find the number of students scoring between 30 and 60 marks and (ii) Find the number of students scoring above 70 marks. (J’06) 16. In a normal distribution 20% of items are less than 100 and 30% are over 200. Find the mean and S.D of the distribution. (J’08 ; O’11 ; M’12) Z Area
0.84 0.3
0.525 0.2
17. The mean yield for one-acre plot is 663 kg with an S.D of 32 kg. Assuming normal distribution, how many one-acre plots in a batch 1000 plots would you expect to have yield (i) over 700 kg? (ii) below 650 kg? (M’09)
18. 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 54.8” and 68.8”. (O’06 ; J’10) 19. The diameter of shafts produced in a factory conforms to normal distribution. 31% of the shafts have a diameter less than 45 mm and 8% have more than 64 mm. Find the mean and standard deviation of the diameter of shafts.(O’07 ; O’08)
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CHAPTER : 9. SAMPLING TECHNIQUES AND STATISTICAL INFERENCE 1. A sample of 100 students are drawn from a school. The mean weight and variance of the sample are 67.45 kg and 9 kg. respectively. Find (a) 95% and (b) 99% confidence intervals for estimating the mean weight of the students. (J’07) 2. Out of 1000 TV viewers, 320 watched a particular programme. Find 95% confidence limits for TV watched this programme.(O’07) 3. A sample of five measurements of the diameter of a sphere were recorded by a scientist as 6.33, 6.37, 6.32 , 6.36 and 6.37 mm. Determine the point estimate of (a) mean, (b) variance. (O’09 ; J’10) 4. The mean life time of 50 electric bulbs produced by a manufacturing company is estimated to be 825 hours with a standard deviation of 110 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. (M’07;M’09;O’11) 5. A company markets car tyres. Their lives are normally distributed with a mean of 50000 kilometers and standard deviation of 2000 kilometers. A test sample of 64 tyres has a mean life of 51250 kms. Can you conclude that the sample mean differs significantly from the population mean? (Test at 5% level) (O’06) 6. A sample of 400 students is found to have a mean height of 171.38 cm. Can it reasonably be regarded as a sample from a large population with mean height of 171.17 cm and standard deviation of 3.3 cm? (Test at 5% level). (M’06 ; O’08 ; J’11) 7. 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 of significance. (J’06 ; M’11) 8. 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. 100 and standard deviation 15? ( Test at 5% level of significance ) (J’08 ; J’09 ; M’10 ; O’10 ; M’12) 9. The income distribution of the population of a village has a mean of Rs.6,000 and a variance of Rs. 32,400. Could a sample of 64 persons with a mean income of Rs. 5,950 belong to this population? (Test at both 5% and 1% levels of significance) (M’08)
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CHAPTER : 10. APPLIED STATISTICS 1. Solve graphically: Minimize Z 20 x1 40 x 2 Subject to 36 x1 6 x2 108 ; 3 x1 12 x2 36 ; 20 x1 10 x2 100 ; x1 , x2 0 ( J’07 ; M’08) 2. Solve the following, using graphical method: Maximize Z 3x1 4 x2 subject to the constraints 2 x1 x2 40 ; 2 x1 5x2 180 ; x1 , x 2 0. (J’06 ; J’10 ; O’11) 3. Solve the following, using graphical method: Minimize Z 3x1 2 x 2 subject to the constraints 5 x1 x2 10 ; 2 x1 2 x2 12 ; x1 4 x2 12 ; x1 , x 2 0. (O’09 ; J’11) 4. Solve the following, using graphical method: Maximize Z 45 x1 80 x2 subject to the constraints 5x1 20 x2 400 ;
10 x1 15 x2 450 ;
x1 , x 2 0. (O’10 ; M’12)
5. NON – TEXTUAL: Solve graphically: Maximize Z 5 x1 3 x 2 Subject to 2 x1 x2 1000 ; 0 x1 400 ; 0 x2 700 ; x1 , x2 0. (M’09) 6. NON – TEXTUAL: Solve the following using graphical method: Maximize Z 5 x1 6 x 2 Subject to the constraints, 3 x1 2 x2 120 ; 4 x1 6 x 2 260 ; x1 , x2 0 (M’06) 7. Find the co-efficient of correlation for the data given below: (O’06 ; J’08 ; J’09) X:
10
12
18
24
23
27
Y:
13
18
12
25
30
10
8. Obtain the two regression lines from the following: (M’10) X:
6
2
10
4
8
Y:
9
11
5
8
7
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9. From the data given below calculate Seasonal Indices. (O’09 ; J’10) Quarter I II III IV
1984 40 35 38 40
1985 42 37 39 38
Year 1986 41 35 38 40
1987 45 36 36 41
1988 44 38 38 42
10. Calculate the seasonal indices by the method of simple average for the following data: (M’08) Year 1985 1986 1987
Quarters II III 60 61 55 66 60 63
I 65 68 68
IV 63 61 67
11. Calculate the seasonal indices for the following data using average method: (M’06 ; M’11) Year 1982 1983 1984 1985 1986
Quarters II III 80 68 82 70 84 66 84 74 86 74
I 72 76 74 76 78
IV 70 74 80 78 82
12. Calculate Fisher’s ideal index from the following data: (O’07) Price Commodity A B C D E
1985 8 2 1 2 1
1986 20 6 2 5 5
Quantity 1985 1986 50 60 15 10 20 25 10 8 40 30
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13. Compute (i) Laspeyre’s (ii) Paasche’s (iii) Fisher’s index numbers for the year 2000 from the following: (O’06 ; J’08) Price
Quantity
Commodity
1980
1990
1980
1990
A
2
4
8
6
B
5
6
10
5
C
4
5
14
10
D
2
2
19
13
14. From the following data calculate the price index number by (a) Laspeyre’s method, (b) Paasche’s method and (c) Fisher’s method: (M’07) Base year Commodity A B C D
Price 5 10 3 6
Quantity 25 5 40 30
Current Year Price 6 15 2 8
Quantity 30 4 50 35
15. From the following data calculate the price index number by (a) Laspeyre’s method, (b) Paasche’s method and (c) Fisher’s method: (M’10) Base year Commodity A B C D E
Price 2 4 6 8 10
Quantity 40 50 20 10 10
Current Year Price 6 8 9 6 5
Quantity 50 40 30 20 20
16. From the following data calculate the price index number by (a) Laspeyre’s method, (b) Paasche’s method and (c) Fisher’s method: (M’12) Base year Commodity A B C D
Price 2 4 6 10
Quantity 40 50 20 10
Current Year Price 6 8 9 5
Quantity 50 40 30 20
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17. NON – TEXTUAL: Compute (i) Laspeyre’s (ii) Paasche’s (iii) Fisher’s index numbers for the year 2000 from the following: (M’09) Price
Quantity
Commodity
1990
2000
1990
2000
A
2
4
8
6
B
5
6
10
5
C
4
5
14
10
D
2
2
19
13
18. From the following data, construct Fisher’s Ideal index and show that it satisfies factor Reversal test and Time Reversal test: (J’06 ; J’11) Base year Commodity A B C D E F
Price 10 8 12 20 5 2
Quantity 10 12 12 15 8 10
Current Year Price 12 8 15 25 8 4
Quantity 8 13 8 10 8 6
19. From the following data, construct Fisher’s Ideal index and show that it satisfies factor Reversal test and Time Reversal test: (O’10) Base year Commodity A B C D
Price 10 7 5 16
Quantity 12 15 24 5
Current Year Price 12 5 9 14
Quantity 15 20 20 5
20. From the following data, construct Fisher’s Ideal index and show that it satisfies factor Reversal test and Time Reversal test: (J’07 ; O’08) Base year Commodity A B C D E
Price 6 2 4 10 8
Quantity 10 2 6 12 12
Current Year Price 50 100 60 30 40
Quantity 56 120 60 24 36
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21. Compute (i) Laspeyre’s (ii) Paasche’s (iii) Fisher’s index numbers for the following data: (O’11) Price Commodity
Quantity
Base year
Current year
Base year
Current year
6 2 4 10
10 2 6 12
50 100 60 30
50 120 60 25
A B C D
22. Calculate the cost of living Index Number using Family Budget method: (J’09) Commodity Quantity in base year (unit) Price in Base year(Rs.) Price in current year(Rs.)
A
B
C
D
E
F
G
H
20
50
50
20
40
50
60
40
10
30
40
200
25
100
20
150
12
35
50
300
50
150
25
180
23. The following data shows the value of sample mean X and the range R for ten samples of size 6 each. Calculate the values for central line and control limits for mean chart and range chart and determine whether the process is in control. (M’07) Sample No.
1
2
3
4
5
6
7
8
9
10
Mean X Range R
681 118
586 167
651 134
641 171
680 490
639 200
665 236
604 188
569 309
629 257
(Given for n = 6 , A2 = 0.483 , D3 = 0 , D4 = 2.004 ) 24. The following data shows the value of sample mean X 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. (O’07 ; O’08 ; M’11) Sample No. Mean X
1
2
3
4
5
6
7
8
9
10
11.2 11.8 10.8 11.6 11.0 9.6 10.4 9.6 10.6 10.0
Range R 7 4 8 5 7 4 8 (Given for n = 5 , A2 = 0.577 , D3 = 0 , D4 = 2.115 )
4
7
9
