Statistics II PUC Model Question Paper 2023-24 PDF

II PUC MODEL QUESTION PAPER 2023-24 STATISTICS (31) Duration: 3 Hrs 15 Min. Max. Marks: 80 Instructions: 1. Statistical table and graph sheets will be supplied on request. 2. Scientific calc...

II PUC MODEL QUESTION PAPER 2023-24 STATISTICS (31) Duration: 3 Hrs 15 Min. Max. Marks: 80 Instructions: 1. Statistical table and graph sheets will be supplied on request. 2. Scientific calculators are allowed. 3. All working steps should be clearly shown. 4. Only the first written answers will be considered for Section-A. SECTION – A I. Choose the correct answer from the choices given: 5×1=5 1. The expected number of years that a new born baby would live is called a) Cohort b) Radix c) Longevity d) Survival ratio 2. The prices of items increased by 10% in 2012 as compared to 2010. Then the index number for the year 2010 is a) 110 b) 10 c) 100 d) 0 3. Following is the probability distribution of a binomial variate: X 0 1 2 3 4 P(X) 0.0625 0.25 0.375 0.25 0.0625 The mode of the distribution is a) 4 b) 2 c) 3 d) 1 4. There are four possible decisions under the testing of null hypothesis (H0): i) Accept H0 when it is true ii) Reject H0 when it is not true iii) Accept H0 when it is not true iv) Reject H0 when it is true The correct decisions are a) i and ii b) iii and iv c) i and iii d) ii and iv 5. The cost associated with the maintenance of an inventory until they are sold or used is called a) Capital cost b) Setup cost c) Shortage cost d) Holding cost. II. Fill in the blanks by choosing the appropriate word from those given in the brackets: (Chance, Fisher’s, Balanced, Parameter, Strategy, Bell) 5×1=5 6.Both time reversal test and factor reversal tests are satisfied by __________ index number. 7.The t-curve is __________shaped. 8.A statistical constant of the population is called a ______________. 9.A small amount of variation for which no specific cause can be attributed is termed as ___________cause of variation. 10. The ____________ of a player is the pre-determined rule by which a player determines his course of action. 1 III. Match the following: 5×1=5 A B 11. N.R.R. per woman = 1.2 (a) Inventory Model II 12. Index numbers (b) X = 0, 1 13. The range of Bernoulli distribution (c) Population increases 14. H1: μ >50 (d) Economic barometers 15. Shortages are allowed (e) H0: μ = 50 (f) Historigram IV. Answer the following questions:5 × 1 = 5 16. Define fertility. 17. Which variation of time series is predictable? 18. Write the relation between mean and variance of a Bernoulli distribution. 19. Define null hypothesis. 20. When is a transportation problem balanced? SECTION – B V. Answer any FIVE of the following questions: 5 × 2 = 10 21. Diagrammatically represent ‘Business Cycle’ with stages. 22. Write two assumptions of interpolation and extrapolation. 23. Find the mean of a Hyper geometric distribution whose parameters are a = 4, b = 6 & n = 5. 24. Find the standard deviation of a chi-square distribution with 8 d.f. 25. A random sample of size 36 is drawn from a population whose standard deviation is 4. Compute standard error of the sample mean. 26. Define ‘point estimation’ and ‘interval estimation’. 27. If P' = 0.02 and n = 25, calculate upper control limit for np–chart. 28. Test whether solution for the following T.P. is non-degenerate? 10 12 2 3 15 20 4 6 14 7 SECTION – C VI. Answer any FOUR of the following questions: 4×5 = 20 29. Calculate the cost of living index number for the following data. Comment on the result. Prices (Rs.) Items Weights 2012 2018 Food 4000 6000 15 Clothing 2500 3500 08 Housing 6000 9000 12 Fuel 1000 1500 10 Others 1600 2000 15 2  30. Following is data regarding annual life insurance premium. Using binomial expansion method estimate the premium at the age 30 and 45 years. Age (in Years) 20 25 30 35 40 45 Premium (in Rs.) 1426 1581 - 1996 2256 - 31. The probability that a team winning the game is 3/5. If this team participates in 6 games, then find the probability that it wins in i) all the games ii) more than one game. 32. Fit a Poisson distributionfor the following data. Number of mistakes per page 0 1 2 3 4 and more Number of pages 20 45 30 5 0 33. Test whether there is any significant difference in the population proportions at 5% level of significance, from the following data. Size Proportion Sample I 100 0.02 Sample II 200 0.01 34. From the following 2 × 2 contingency table test whether result depends on family condition of the students are independent. (Use 1% L.O.S.) Family condition Good Bad Pass 10 15 Result Fail 15 10 35. Solve the following game using the principle of dominance. Is the game fair? Player – B B1 B2 B3 A1 4 –1 2 Player – A A22 0 3 A36 –5 1 36. The purchase price of a machine A is Rs. 5000. Its resale value and maintenance costs are as follows: Year 1 2 3 4 5 Maintenance cost (Rs.) 100 200 330 510 860 Resale value (Rs.) 3000 2500 2000 1500 1000 What would be the optimum replacement period? VII. Answer any TWO of the following questions: 2×5 = 10 37. Height of a group of candidates who attended the Agniveer army selection camp follows normal distribution with mean and S.D of height of candidates is 160cm and 3.9cm respectively. The minimum required height for Agniveer army selection is 165cm. Show that only 10% of the above group is eligible. 38. A machine is designed so as to fill the bottles with mean 1litre of cold pressed ground nut oil. A sample of 26 bottles when measured had a mean content of 998 ml. with S.D. of 5 ml. Test at 5% level of significance whether the machine is functioning properly. 39. Following table gives mean (x) and Range (R) of samples of size 5 each. Sub group number 1 2 3 4 5 6 Mean (x) 52 49 53 48 51 47 Range (R) 9 11 10 12 8 10 Find the control limits for drawing x– chart. 3  40. F1 and F2 are two types of foods are used to get minimum supplements of vitamins B and C. Using graphical method for the following Linear Programming Problem model, find how much quantities of foods are needed to optimize the cost: Minimize Z = 10x + 5y Subject to 2x + 3y ≥ 12 2x + y ≥ 8 and x ≥ 0, y ≥ 0 OR (For Visually challenged students only) There are two types of foods F1 and F2 each containing different proportions of vitamins B and C. Food F1 contains 2 units of vitamin B and 2 units of vitamin C. Food F2 contains 3 units of vitamin B and 1 unit of vitamin C. The minimum daily requirement of vitamins B and C for a person is 12 and 8 units respectively. One unit of food F1 costs Rs.10 and one unit of food F2 costs Rs. 5. Formulate the L.P.P. to optimize the expenditure. SECTION – D VIII. Answer any TWO of the following questions: 2×10 = 20 41. Calculate the standardized death rates for both Localities A and B from the following data.State which locality is healthier? Age-group Locality - A Locality - B Standard (in years) Population Deaths Population Deaths Population 0 – 10 6,000 60 7,000 84 4,000 10 – 20 10,000 80 15,000 90 16,000 20 – 60 20,000 240 25,000 250 18,000 60 & above 4,000 120 3,000 120 2,000 42. Calculate the Laspeyre’s, Paasche’s and Dorbish- Bowley’s price index numbers from the following data. Items Base Year Current Year Price (in Rs) Quantity Price (in Rs) Quantity Rice 50 10 60 08 Wheat 40 08 45 12 Dhal 100 03 160 02 Oil 80 02 120 03 43. a) Following data shows the sales figures of a company. Mention the trend by calculating three yearly moving averages. Year 2010 2011 2012 2013 2014 2015 2016 Sales (in thousand Rs.) 120 104 130 126 145 131 132 b) Following are the figures of production (in thousand quintals) of a sugar factory. Fit a straight line trend of the type Y = a + b x to this data. Year 2016 2017 2018 2019 2020 2021 2022 Production 80 90 92 83 94 99 92 ***** 4  ¢éwÃAiÀÄ ¦.AiÀÄÄ.¹. ªÀiÁzÀj ¥Àæ±Éß ¥ÀwæPÉ2023-24 «µÀAiÀÄ: ¸ÀASÁå±Á¸ÀÛç (31) ¸ÀªÀÄAiÀÄ: 3UÀAmÉ 15¤«ÄµÀ. UÀjµÀÖ CAPÀUÀ¼ÀÄ: 80 ¸ÀÆZÀ£É: 1. ¸ÁATåPÀ PÉÆõÀÖPÀ ªÀÄvÀÄÛ D¯ÉÃR PÁUÀzÀUÀ¼À£ÀÄß PÉýzÁUÀ ¤ÃqÀ¯ÁUÀĪÀÅzÀÄ. 2. ªÉÊeÁÕ¤PÀ PÁå®Ä̯ÉÃlgïUÀ¼À£ÀÄß §¼À¸À§ºÀÄzÀÄ. 3. PÁAiÀÄðzÀ J¯Áè ºÀAvÀUÀ¼À£ÀÄß ¸ÀàµÀÖªÁV vÉÆÃj¸À¨ÉÃPÀÄ. 4. A - «¨sÁUÀzÀ°è£À ¥Àæ±ÉßUÀ½UÉ ¥ÀæxÀªÀĪÁV §gÉzÀ GvÀÛgÀUÀ¼À£ÀÄß ªÀiÁvÀæ ¥ÀjUÀt¸À¯ÁUÀĪÀÅzÀÄ. «¨sÁUÀ - A I. ¸ÀjAiÀiÁzÀ GvÀg Û ÀªÀ£ÀÄß DAiÉÄ̪ÀiÁr §gɬÄj: 5× 1 = 5 1. £Àªe À ÁvÀ ²±ÀÄ«£À ¤jÃQëvÀ fëvÁªÀ¢üAiÀÄ£ÀÄß _________ JAzÀÄ PÀgA É iÀÄĪÀgÀÄ. a) vÀAqÀ b) D¢¸ÀªÀÄƺÀ c) ¢ÃWÁðAiÀÄĵÀå d) §zÀÄPÀĽAiÀÄĪÀ ¥ÀæªiÀ Át 2. 2010PÉÌ ºÉÆð¹zÁUÀ 2012gÀ°è ªÀ¸ÄÀ ÛUÀ¼À ¨É¯ÉUÀ¼ÀÄ ±ÉÃRqÁ 10gÀµÀÄÖ ºÉZÁÑVªÉ. ºÁUÁzÀgÉ 2010gÀ ¸ÀÆZÁåAPÀzÀ ¨É¯É a) 110 b) 10 c) 100 d) 0 3. MAzÀÄ ¢é¥ÀzÀ ZÀ®PÀzÀ ¸ÀA¨sÀªÀ «vÀguÀ ÉAiÀÄÄ X 0 1 2 3 4 P(X) 0.0625 0.25 0.375 0.25 0.0625 DzÁUÀ, F «vÀgÀuÉAiÀÄ §ºÀÄ®PÀzÀ ¨É¯É a) 4 b) 2 c) 3 d) 1 4. ±ÀÆ£Àå ¥ÀjPÀ®à£É (H0) C£ÀÄß ¥ÀjÃQë¸ÀĪÁUÀ ¸ÁzsÀå«gÀĪÀ £Á®ÄÌ wêÀiÁð£ÀUÀ¼ÄÀ : i) ±ÀÆ£Àå ¥ÀjPÀ®à£É (H0) ¸ÀvåÀ «zÁÝUÀ CzÀ£ÀÄß ¹éÃPÀj¸ÀĪÀÅzÀÄ. ii) ±ÀÆ£Àå ¥ÀjPÀ®à£É (H0) ¸ÀvåÀ «®èzÁUÀ CzÀ£ÀÄß wgÀ¸ÌÀ j¸ÀĪÀÅzÀÄ. iii) ±ÀÆ£Àå ¥ÀjPÀ®à£É (H0)¸ÀvåÀ «®èzÁUÀ CzÀ£ÀÄß ¹éÃPÀj¸ÀĪÀÅzÀÄ. iv) ±ÀÆ£Àå ¥ÀjPÀ®à£É (H0) ¸ÀvåÀ «zÁÝUÀ CzÀ£ÀÄß wgÀ¸ÌÀ j¸ÀĪÀÅzÀÄ. EªÀÅUÀ¼° À è ¸ÀjAiÀiÁzÀ wêÀiÁð£ÀUÀ¼ÄÀ : a) i ªÀÄvÀÄÛ ii b) iii ªÀÄvÀÄÛ iv c) i ªÀÄvÀÄÛ iii d) ii ªÀÄvÀÄÛ iv 5. ¸ÀgPÀ ÄÀ zÁ¸ÁÛ£ÀÄ G¥ÀAiÉÆÃUÀ CxÀªÁ ªÀiÁgÁl DUÀĪÀªg À ÉUÉ CªÀÅUÀ¼À ¤ªÀðºÀuÉUÉ ¸ÀA§A¢ü¹zÀ ¨É¯É a) C¸À®Ä ¨É¯É b) CtÂUÉƽ¸ÀĪÀ ªÉZÀÑ c) PÉÆgÀvÉ ªÉZÀÑ d) »qÀĪÀ½ ªÉZÀÑ II. DªÀgÀtzÀ°ègÀĪÀ ¸ÀjAiÀiÁzÀ GvÀÛgÀªÀ£ÀÄß Dj¹, ©lÖ ¸ÀܼÀ vÀÄA©j: 5× 1 = 5 (DPÀ¹äPÀ, ¦ü±ÀgÀ£À, ¸ÀªÀÄvÉÆî£À, ¥ÁæZÀ®/¤AiÀÄvÁAPÀ, vÀAvÀæ, UÀAmÉ) 6. ¸ÀªÄÀ AiÀÄ ºÁUÀÆ CA±À ªÀåwjPÀÛ ¥ÀjÃPÉëUÀ¼Égq À À£ÀÄß vÀȦۥÀr¸ÀĪÀzÄÀ ________ ¸ÀÆZÁåAPÀ. 7. t - ªÀPÀæªÅÀ ______ DPÁgÀªÁVzÉ. 8. ¸ÀªÄÀ ¶×AiÀÄ ¸ÁATåPÀ C¼ÀvA É iÀÄ£ÀÄß _________ J£ÀÄߪÀgÄÀ. 9. ¤¢ðµÀÖ PÁgÀtUÀ½®èzÉ ªÀ¸ÄÀ ÛUÀ¼À UÀÄtªÀÄlÖz° À èAiÀÄ ¸ÀtÚ ¥ÀæªiÀ Át Kj½vÀU¼ À À£ÀÄß ______ PÁgÀtzÀ Kj½vÀUÀ¼£ É ÀÄߪÀgÄÀ. 10. M§â DlUÁgÀ£ÀÄ DlQÌAvÀ ªÀÄÄAavÀªÁV gÀƦ¹zÀ vÀ£Àß ¤zsÁðgÀUÀ¼À UÀtªÉà ________. III. ºÉÆA¢¹ §gɬÄj: 5× 1 = 5 A B 11. M§â ªÀÄ»¼ÉAiÀÄ NRR = 1.2 a) ¸ÀgPÀ ÄÀ zÁ¸ÁÛ£ÀÄ ªÀiÁzÀj II 5  12. ¸ÀÆZÁåAPÀUÀ¼ÀÄ b) X = 0, 1 13. §£ÉÆÃð° ZÀ®PÀzÀ ªÁå¦Û c) d£À¸AÀ SÉå KjPÉ DUÀÄwÛzÉ. 14. H1 : μ > 50 d) DyðPÀ ªÀiÁ¥À£ÀU¼ À ÀÄ 15. PÉÆgÀvÉUÉ CªÀPÁ±À«zÉ e) H0 : μ = 50 f) »¸ÉÆÖÃjUÁæªÀiï IV. F PɼÀV£À ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹: 5× 1 = 5 16. ¥sÀ®ªÀAwPÉAiÀÄ£ÀÄß ªÁåSÁ夹. 17. PÁ® ¸Àgt À ÂAiÀÄ AiÀiÁªÀ Kj½vÀªÀ£ÀÄß H»¸À§ºÀÄzÀÄ? 18. §£ÉÆÃð° «vÀgu À ÉAiÀÄ ¸ÀgÁ¸Àj ªÀÄvÀÄÛ «ZÀ®£ÉAiÀÄ ¸ÀA§AzsÀª£ À ÀÄß §gɬÄj. 19. ±ÀÆ£Àå ¥ÀjPÀ®à£ÉAiÀÄ£ÀÄß ªÁåSÁ夹. 20. MAzÀÄ ¸ÁUÁtÂPÁ ¸ÀªÀĸÉå AiÀiÁªÁUÀ ¸ÀªÄÀ vÉÆî£ÀªÁUÀĪÀÅzÀÄ? «¨sÁUÀ - B V. F PɼÀV£À AiÀiÁªÀÅzÁzÀgÀÆ LzÀÄ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹: 5 × 2 = 10 21. ªÁå¥ÁgÀ ZÀPÀæªÀ£ÀÄß avÀæ ¸À»vÀ ªÁåSÁ夹. 22. CAvÀgÉéñÀ£À ªÀÄvÀÄÛ §»gÉéñÀ£ÀUÀ¼À JgÀqÄÀ PÀ®à£UÉÀ À¼À£ÀÄß §gɬÄj. 23. MAzÀÄ CweÁå«Äw «vÀgÀuÉAiÀÄ°è ¤AiÀÄvÁAPÀUÀ¼ÄÀ a = 4, b = 6 &n = 5DzÁUÀ, ¸ÀgÁ¸Àj PÀAqÀÄ»r¬Äj. 24. MAzÀÄ PÉʪÀUð À «vÀgu À ÉAiÀÄ°è ¤AiÀÄvÁAPÀUÀ¼ÄÀ 8 DzÁUÀ, ¤AiÀÄvÀ «ZÀ®£É PÀAqÀÄ»r¬Äj. 25. ¤AiÀÄvÀ «ZÀ®£É 4 EgÀĪÀ MAzÀÄ ¸ÀªÄÀ ¶×¬ÄAzÀ UÁvÀæ 36 EgÀĪÀ MAzÀÄ ¤zÀ±ÀðPÀªÀ£ÀÄß DAiÀÄÄÝPÆ É ¼Àî¯ÁVzÉ. ¤zÀ±ÀðPÀ ¸ÀgÁ¸ÀjAiÀÄ ¤AiÀÄvÀ zÉÆõÀª£ À ÀÄß PÀAqÀÄ»r¬Äj 26. ©AzÀÄ CAzÁf¸ÀÄ«PÉ ªÀÄvÀÄÛ CAvÀgÁ¼ÀzÀ CAzÁf¸ÀÄ«PÉU¼ À À£ÀÄß ªÁåSÁ夹. 27. P' = 0.02 ªÀÄvÀÄÛ n = 25 DzÁUÀ, np - £ÀPÀ ëÉAiÀÄ ªÉÄðßAiÀÄAvÀæt «Äw PÀAqÀÄ»r¬Äj. 28. ¸ÁUÁtÂPÁ ¸ÀªÀĸÉåAiÀÄ F ¥ÀjºÁgÀªÀÅ CªÀ£ÀwºÉÆAzÀzÀ ¥ÀjºÁgÀªÉà JAzÀÄ ¥ÀjÃQë¹. 10 12 2 3 15 20 4 6 14 7 «¨sÁUÀ - C VI. F PɼÀV£À AiÀiÁªÀÅzÁzÀgÀÆ £Á®ÄÌ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹: 4× 5 = 20 29. F zÀvÁÛA±ÀPÉÌ fêÀ£À ªÉZÑÀ ¸ÀÆZÁåAPÀ PÀAqÀÄ»rzÀÄ, ¥sÀ°vÁA±ÀzÀ §UÉÎ «ªÀIJð¹. ¨É¯É (gÀÆ.) ªÀ¸ÄÀ ÛUÀ¼ÄÀ ¨sÁgÀU¼À ÀÄ 2012 2018 DºÁgÀ 4000 6000 15 §mÉÖ 2500 3500 08 ªÀ¸w À 6000 9000 12 EAzs£ À À 1000 1500 10 EvÀgÉ 1600 2000 15 30. F zÀvÁÛA±ÀªÅÀ ªÁ¶ðPÀ «ªÀiÁ PÀAwUÉ ¸ÀA§A¢ü¹zÉ. ¢é¥z À À «¸ÀÛgÀuÁ «zsÁ£À §¼À¹ 30 ªÀÄvÀÄÛ 45£ÉAiÀÄ ªÀAiÀĹìUÉ «ªÀiÁ PÀAvÀ£ÀÄß CAzÁf¹. ªÀAiÀĸÀÄì (ªÀµÀðUÀ¼° À è) 20 25 30 35 40 45 «ªÀiÁ PÀAvÀÄ (gÀÆUÀ¼° À è) 1426 1581 - 1996 2256 - 6  31. MAzÀÄ vÀAqÀªÅÀ DlzÀ°èUÉ®ÄèªÀ ¸ÀA¨sÀª¤ À ÃAiÀÄvÉ3/5 DVzÉ. MAzÀÄ ªÉÃ¼É F vÀAqÀªÅÀ DgÀÄ DlUÀ¼° À è ¨sÁUÀª» À ¹zÀgÉ, CzÀÄi)J¯Áè DlUÀ¼° À è UÉ®ÄèªÀ ii)MAzÀQÌAvÀ ºÉZÀÄÑ DlzÀ°è UÉ®ÄèªÀ ¸ÀA¨sª À ÀvÉUÀ¼£ À ÀÄß PÀAqÀÄ»r¬Äj. 32. F zÀvÁÛA±ÀPÉÌ ¥ÉƸÁ£ï «vÀgÀuÉAiÀÄ£ÀÄß ¤AiÉÆÃf¹. ¥Àæw ¥ÀÄlzÀ°èAiÀÄ vÀ¥ÀÄàUÀ¼À ¸ÀASÉå 0 1 2 3 4 ªÀÄvÀÄÛ ºÉZÀÄÑ ¥ÀÄlUÀ¼À ¸ÀASÉå 20 45 30 5 0 33. F zÀvÁÛA±À¢AzÀ ¸ÀªÄÀ ¶× C£ÀÄ¥ÁvÀUÀ¼À°è UÀªÄÀ £ÁºÀð ªÀåvÁå¸À«zÉAiÉÄà JA§ÄzÀ£ÀÄß ±ÉÃRqÁ 5 gÀ ®PÁëºÀð ªÀÄlÖzÀ°è ¥ÀjÃQë¹. UÁvÀæ C£ÀÄ¥ÁvÀ ¤zÀ±ÀðPÀ I 100 0.02 ¤zÀ±ÀðPÀ II 200 0.01 34. F PɼÀV£À 2 × 2 ¸ÀAWÀnvÀ ¸ÁgÀtÂAiÀÄ°è «zÁåyðUÀ¼À ¥sÀ°vÁA±À CªÀgÀ PÀÄlÄA§ ¥Àg¹ À ÜwAiÀÄ ªÉÄÃ¯É CªÀ®A©vÀªÁVzÉAiÉÄà JA§ÄzÀ£ÀÄß ¥ÀjÃQë¹. (±ÉÃRqÁ 1% ®PÁëºÀðvÉ G¥ÀAiÉÆÃV¹) PÀÄlÄA§ ¥ÀgÀ¹Üw GvÀÛªÀÄ PÉlÖ GwÛÃtð 10 15 ¥sÀ°vÁA±À C£ÀÄwÛÃtð 15 10 35. F PɼV À £À QæÃqÉAiÀÄ£ÀÄß ¥Àæ§ÄvÀé «zsÁ£À¢AzÀ ©r¹. EzÀÄ ¤µÀàPëÀ¥ÁvÀzÀ DlªÉÃ? DlUÁgÀ – B B1 B2 B3 A1 4 –1 2 DlUÁgÀ – A A22 0 3 A3 6 –5 1 36. MAzÀÄ AiÀÄAvÀæzÀ PÉÆAqÀ¨¯ É É gÀÆ. 5000. EzÀgÀ ¤ªÀðºÀuÁ ªÉZÀÑ ªÀÄvÀÄÛ ªÀÄgÀÄ ªÀiÁgÁl ¨É¯ÉU¼ À ÀÄ EAwªÉ: ªÀµÀð 1 2 3 4 5 ¤ªÀðºÀuÁ ªÉZÀÑ (gÀÆ.) 100 200 330 510 860 ªÀÄgÀÄ ªÀiÁgÁl (gÀÆ.) 3000 2500 2000 1500 1000 ªÀ¸ÄÀ ÛªÀ£ÀÄß §zÀ¯Á¬Ä¸ÀĪÀ ¸ÀªÀÄÄavÀ CªÀ¢ü AiÀiÁªÀÅzÀÄ? VII. F PɼÀV£À AiÀiÁªÀÅzÁzÀgÀÆ JgÀqÀÄ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹: 2 × 5 = 10 37. CVß«ÃgÀ¸ÉãÉAiÀÄ DAiÉÄÌ ²©gÀPÉÌ ºÁdgÁzÀ C¨sÀåyðUÀ¼À UÀÄA¦£À JvÀÛgÀªÅÀ ¸ÀgÁ¸Àj 160 ¸ÉA.«ÄÃ. ªÀÄvÀÄÛ ¤AiÀÄvÀ «ZÀ®£É 3.9 ¸ÉA.«ÄÃ. EgÀĪÀ ¥Àæ¸ÁªÀiÁ£Àå «vÀgu À ÉAiÀÄ£ÀÄß ºÉÆA¢zÉ. CVß«ÃgÀ¸ÉãÉUÉ ¸ÉÃgÀ®Ä CªÀ±åÀ «gÀĪÀ PÀ¤µÀÖ JvÀÛgÀ 165 ¸ÉA.«ÄÃ. DzÀg,É ªÉÄð£À UÀÄA¦£À°è ±ÉÃ.10 gÀµÀÄÖ C¨sÀåyðUÀ¼ÀÄ ªÀiÁvÀæ CºÀðgÁVgÀÄvÁÛgÉ JAzÀÄ vÉÆÃj¹. 38. UÁt¢AzÀ vÀUÉ¢gÀĪÀ ±ÉAUÁ JuÉÚAiÀÄ£ÀÄß ¥Àæw ¨Ál¯ïUÉ 1 °Ãlgï vÀÄA§ÄªÀAvÉ MAzÀÄ AiÀÄAvÀ檣 À ÀÄß CtÂUÉƼÀ¸À¯ÁVzÉ. F AiÀÄAvÀæ¢AzÀ vÀÄA§®àlÖ 26 ¨Ál¯ïUÀ¼£ À ÀÄß DAiÀÄÄÝ ¥ÀjÃQë¹zÁUÀ, CªÀÅUÀ¼À°èAiÀÄ ±ÉAUÁ JuÉÚAiÀÄ ¸ÀgÁ¸Àj ¥ÀæªiÀ ÁtªÀÅ 998 «Ä.°Ã. ªÀÄvÀÄÛ ¤.«. 5 «Ä.°Ã. JAzÀÄ PÀAqÀħA¢zÉ. ºÁUÁzÀg,É AiÀÄAvÀæªÅÀ ¸ÀjAiÀiÁV PÉ®¸À ªÀiÁqÀÄwÛzÉAiÉÄà JA§ÄzÀ£ÀÄß ±ÉÃRqÁ 5gÀ ®PÁëºÀðvÉAiÀÄ°è ¥ÀjÃQë¹. F zÀvÁÛA±ÀªÅÀ ¥ÀæwAiÉÆAzÀÄ ¤zÀ±ÀðPÀ UÁvÀæ 5EgÀĪÀ ¸ÀgÁ¸Àj (x)ªÀÄvÀÄÛ ªÁå¦Û (R) UÀ½UÉ ¸ÀA§A¢ü¹zÉ. ªÁå¦Û G¥ÀUÄÀ A¥ÀÄ ¸ÀASÉå 1 2 3 4 5 6 ¸ÀgÁ¸Àj (x) 52 49 53 48 51 47 ªÁå¦Û (R) 9 11 10 12 8 10 x- £ÀPÉë J¼ÉAiÀÄ®Ä ¤AiÀÄAvÀæt «ÄwUÀ¼£ À ÀÄß PÀAqÀÄ»r¬Äj. 7  39. F1 ªÀÄvÀÄÛ F2 JA§ JgÀqÄÀ DºÁgÀ ¥ÀzÁxÀðUÀ½AzÀ CªÀ±åÀ «gÀĪÀ B ªÀÄvÀÄÛ C fêÀ¸v À ÀéU¼ À À£ÀÄß ¥ÀqAÉ iÀÄ®Ä G¥ÀAiÉÆÃV¹zÀ F gÉÃSÁvÀäPÀ PÀæªÀÄ«¢ü ¸ÀªÄÀ ¸ÉåAiÀÄ ªÀiÁzÀjAiÀÄ£ÀÄß D¯ÉÃR «zsÁ£À¢AzÀ ©r¹. ¨É¯É ¸ÀªÀÄÄavÀUÉƽ¸À®Ä CªÀ±åÀ «gÀĪÀ DºÁgÀ ¥ÀzÁxÀðUÀ¼À ¥ÀæªiÀ ÁtªÀ£ÀÄß PÀAqÀÄ»r¬Äj. PÀ¤µÀ×UÉƽ¹ Z = 10x + 5y ¤§AzsÀ£ÉUÀ½UÉ M¼À¥l À ÄÖ 2x + 3y ≥ 12 2x + y ≥ 8 ªÀÄvÀÄÛ x ≥ 0, y ≥ 0 CxÀªÁ (zÀ馅 «PÀ®ZÉÃvÀ£À «zÁåyðUÀ½UÉ ªÀiÁvÀæ) F1 ªÀÄvÀÄÛ F2 JA§ JgÀqÄÀ DºÁgÀ ¥ÀzÁxÀðUÀ¼ÄÀ ««zsÀ ¥ÀæªiÀ ÁtzÀ°è B ªÀÄvÀÄÛ C fêÀ¸v À ÀéU¼ À À£ÀÄß M¼ÀUÆ É ArªÉ. F1 JA§ DºÁgÀ ¥ÀzÁxÀðªÀÅ 2 ªÀiÁ£ÀzÀ BfêÀ¸v À Àé ªÀÄvÀÄÛ 2 ªÀiÁ£ÀzÀ C fêÀ¸v À Àé ºÉÆA¢zÉ. F2 JA§ DºÁgÀ ¥ÀzÁxÀðªÀÅ 3 ªÀiÁ£ÀzÀ BfêÀ¸v À Àé ªÀÄvÀÄÛ 1 ªÀiÁ£ÀzÀ C fêÀ¸v À Àé ºÉÆA¢zÉ. M§â ªÀåQÛUÉ ¢£ÀPÌÉ CªÀ±åÀ «gÀĪÀ B ªÀÄvÀÄÛ C fêÀ¸v À ÀéU¼ À À ¥ÀæªiÀ Át 12 ªÀÄvÀÄÛ 8 DVzÉ. 1 ªÀiÁ£ÀzÀ F1 DºÁgÀ ¥ÀzÁxÀðzÀ ¨É¯É gÀÆ. 10ªÀÄvÀÄÛ 1 ªÀiÁ£ÀzÀ F2 DºÁgÀ ¥ÀzÁxÀðzÀ ¨É¯É gÀÆ. 5 DVzÉ. RZÀð£ÀÄß ¸ÀªÀÄÄavÀUÉƽ¸À®Ä gÉÃSÁvÀäPÀ PÀæªÀÄ«¢ü ¸ÀªÀĸÉå gÀa¹. «¨sÁUÀ - D VIII. F PɼÀV£À AiÀiÁªÀÅzÁzÀgÀÆ JgÀqÀÄ ¥Àæ±ÉßUÀ½UÉ GvÀÛj¹: 2 × 10 = 20 40. F PɼÀV£À zÀvÁÛA±ÀPÉÌ JgÀqÀÆ ¥ÀæzÃÉ ±ÀU¼ À À ¤AiÀÄwÃPÀÈvÀ ªÀÄgÀt zÀgU À À¼£À ÀÄß PÀAqÀÄ»rzÀÄ, AiÀiÁªÀ ¥ÀæzÃÉ ±À ºÉZÀÄÑ DgÉÆÃUÀåPÀgÀªÁVzÉ JAzÀÄ w½¹. ªÀAiÉÆêÀUð À ¥ÀæzÃÉ ±À - A ¥ÀæzÃÉ ±À - B DzÀ±ð À [ªÀµÀðUÀ¼° À è] d£À¸A À SÉå ªÀÄgÀtUÀ¼ÄÀ d£À¸A À SÉå ªÀÄgÀtUÀ¼ÄÀ d£À ¸ A À SÉå 0 – 10 6,000 60 7,000 84 4,000 10 – 20 10,000 80 15,000 90 16,000 20 – 60 20,000 240 25,000 250 18,000 60 ªÀÄvÀÄÛ ºÉZÀÄÑ 4,000 120 3,000 120 2,000 41. F zÀvÁÛA±ÀPÉÌ ¯Áå¸ÉàAiÀÄgÀ£À, ¥Á±ÉA Ñ iÀÄ£À ªÀÄvÀÄÛ qÁ©ð±ï-¨Ë°AiÀÄ ¨É¯É ¸ÀÆZÁåAPÀUÀ¼£ À ÀÄß PÀAqÀÄ»r¬Äj. ªÀ¸ÄÀ Û DzsÁgÀ ªÀµÀð ¥ÀæZÀ°vÀ ªÀµÀð ¨É¯É (gÀÆ.) ¥ÀjªÀiÁt ¨É¯É (gÀÆ.) ¥ÀjªÀiÁt CQÌ 50 10 60 08 UÉÆâü 40 08 45 12 ¨ÉÃ¼É 100 03 160 02 JuÉÚ 80 02 120 03 42. a) F zÀvÁÛA±ÀªÅÀ MAzÀÄ PÀA¥À¤AiÀÄ ªÀiÁgÁlzÀ ªÀiÁ»w vÉÆÃj¸ÀÄvÀÛzÉ. vÉæöʪÁ¶ðPÀ ZÀ®£À ¸ÀgÁ¸ÀjUÀ¼À£ÀÄß ¥Àqz É ÀÄ ¥ÀæªÈÀ wÛAiÀÄ£ÀÄß w½¹. ªÀµÀð 2010 2011 2012 2013 2014 2015 2016 ªÀiÁgÁl (¸Á«gÀ gÀÆ.) 120 104 130 126 145 131 132 b) MAzÀÄ PÁSÁð£ÉAiÀÄ ¸ÀPÀÌgÉ GvÁàzÀ£É (¸Á«gÀ QéAl¯ïUÀ¼À°è) F PɼU À É ¤ÃqÀ¯ÁVzÉ. EzÀPÌÉ Y = a + b X jÃwAiÀÄ ¸Àg¼ À ÀgÃÉ SÁ ¥ÀæªÈÀ wÛgÉÃSÉAiÀÄ£ÀÄß ¤AiÉÆÃf¹. ªÀµÀð 2016 2017 2018 2019 2020 2021 2022 GvÁàzÀ£É 80 90 92 83 94 99 92 8