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12th Standard English Medium Business Maths Reduced Syllabus Annual Exam Model Question Paper - 2021

12th Standard

    Reg.No. :
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Business Maths

Time : 02:30:00 Hrs
Total Marks : 90

        Part-I

        Choose the most appropriate answer from the given four alternatives and write the option code and the corresponding answer.

    20 x 1 = 20
  1. Which of the following is not an elementary transformation?

    (a)

    \({ R }_{ i }\leftrightarrow { R }_{ j }\)

    (b)

    \({ R }_{ i }\rightarrow { 2R }_{ i }+{ 2C }_{ j }\)

    (c)

    \({ R }_{ i }\rightarrow { 2R }_{ i }-{ 4R }_{ j}\)

    (d)

    \({ C }_{ i }\rightarrow { C }_{ i }+{ 5C }_{ j }\)

  2. If A is a singular matrix, then Adj A is ___________

    (a)

    non-singular

    (b)

    singular

    (c)

    symmetric

    (d)

    not defined

  3. \(\int _{ 0 }^{ 1 }{ (2x+1) } dx\) is _______.

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  4. \(\int { { \left| x \right| }^{ 3 } } \)dx = ________________ +c

    (a)

    \(\frac { { -x }^{ 4 } }{ 4 } \)

    (b)

    \(\frac { { \left| x \right| }^{ 4 } }{ 4 } \)

    (c)

    \(\frac { { x }^{ 4 } }{ 4 } \)

    (d)

    none of these

  5. Area bounded by the curve y = x (4 − x) between the limits 0 and 4 with x − axis is ________.

    (a)

    \(\frac{30}{3}\) sq.units

    (b)

    \(\frac{31}{2}\)sq.units

    (c)

    \(\frac{32}{3}\) sq.units

    (d)

    \(\frac{15}{2}\) sq.units

  6. The area of the region bounded by the line 2y = -x + 8, X - axis and the lines x = 2 and x = 4 is ________ sq.units.

    (a)

    \(\frac{1}{5}\)

    (b)

    \(\frac{2}{5}\)

    (c)

    5

    (d)

    \(\frac{5}{2}\)

  7. The variable separable form of \(\frac { dy }{ dx } =\frac { y(x-y) }{ x(x+y) } \) by taking y = vx and \(\frac { dy }{ dx } =v+x\frac { dv }{ dx } \) is ______.

    (a)

    \(\frac { 2{ v }^{ 2 } }{ 1+v } dv=\frac { dx }{ x } \)

    (b)

    \(\frac { 2{ v }^{ 2 } }{ 1+v } dv=-\frac { dx }{ x } \)

    (c)

    \(\frac { 2{ v }^{ 2 } }{ 1-v } dv=\frac { dx }{ x } \)

    (d)

    \(\frac { 1+v }{ 2{ v }^{ 2 } } dv=-\frac { dx }{ x } \)

  8. The I.F. of \(\frac { dy }{ dx } \)- y tan x = cos x is _____

    (a)

    sec x

    (b)

    cos x

    (c)

    etanx

    (d)

    cot x

  9. ∇ f(a) = _______.

    (a)

    f (a) + f(a−h)

    (b)

    f (a) − f(a + h)

    (c)

    f (a) − f(a − h)

    (d)

    f (a)

  10. For the given set of values, the value of ∇2y is ______________

    x 75 80 85 90
    y 2459 2018 1180 402
    (a)

    402

    (b)

    -778

    (c)

    60

    (d)

    457

  11. A variable which can assume finite or countably infinite number of values is known as ________.

    (a)

    continuous

    (b)

    discrete

    (c)

    qualitative

    (d)

    none of them

  12. If X is a continuous random variable. then P(X≥a)= _________.

    (a)

    P(C

    (b)

    1-P(X>a)

    (c)

    P(X>a)

    (d)

    1-P(X≤a-1)

  13. The average percentage of failure in a certain examination is 40. The probability that out of a group of 6 candidates atleast 4 passed in the examination are ________.

    (a)

    0.5443

    (b)

    0.4543

    (c)

    0.5543

    (d)

    0.4573

  14. A coin is tossed 3 times. The probability of getting exactly 2 heads is _________

    (a)

    \(\frac{1}{2}\)

    (b)

    \(\frac{1}{8}\)

    (c)

    \(\frac{3}{8}\)

    (d)

    \(\frac{1}{4}\)

  15. A finite subset of statistical individuals in a population is called ________.

    (a)

    a sample

    (b)

    a population

    (c)

    universe

    (d)

    census

  16. Probability of rejecting null hypothesis. when it is true is _______

    (a)

    Type I error

    (b)

    Type II error

    (c)

    Sampling error

    (d)

    Standard error

  17. A typical control charts consists of ________.

    (a)

    CL, UCL

    (b)

    CL, LCL

    (c)

    CL, LCL, UCL

    (d)

    UCL, LCL

  18. Time series consists of data arranged

    (a)

    Statistical methods

    (b)

    Chronologically

    (c)

    Order oftheir occurrence

    (d)

    increasing or decreasing order

  19. Decision theory is concerned with _______.

    (a)

    analysis of information that is available

    (b)

    decision making under certainty

    (c)

    selecting optimal decisions in sequential problem

    (d)

    All of the above

  20. The least cost method is more economical than North West Corner Rule, since it starts with the ___________

    (a)

    least cost

    (b)

    minimum cost

    (c)

    maximum cost

    (d)

    lower beginning cost

    1. Part-II

      Answer any seven questions and Question number 30 is compulsory.


    7x 2 = 14
  21. Find the rank of the matrix \(\left( \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right) \)

  22. Evaluate the following:
     \(\Gamma (4)\)

  23. If the marginal revenue for a commodity is MR = 9 - 6x2 + 2x, find the total revenue function.

  24. Find the order and degree of the following differential equation
    \(\frac { { d }^{ 2 }y }{ { dx }^{ 3 } } -3{ \left( \frac { dy }{ dx } \right) }^{ 6 }+2y={ x }^{ 2 }\)

  25. Find the missing term from the following data

    x 1 2 3 4
    f(x) 100 - 126 157
  26. Prove that if E(X) = 0, then V(X) = E(X2)

  27. If the mean of the binomial distribution with 9 trial is 6, then find the variance.

  28. Define critical region.

  29. The following data shows the value of sample mean (\(\bar{X}\)) and the range R for 10 samples of size 5 each. Calculate the control limits for : mean chart and range chart.

    Sample No. 1 2 3 4 5 6 7 8 9 10
    Mean \(\bar{X}\) 11.2 11.8 10.8 11.6 11.0 9.6 10.4 9.6 10.6 10.0
    Range 7 4 8 5 7 4 8 4 7 9

    (Given for n = 5, A2 = .577, D3 = 0, D4 = 2.115)

  30. What do you mean by balanced transportation problem?

      1. Part-III

        Answer any seven questions and Question number 40 is compulsory.

    7 x 3 = 21
  31. A company has determined that the marginal cost function for a product of a particular commodity is given by MC = 125 +10x - \(\frac{x^2}{9}\) where C rupees is the cost of producing x units of the commodity. If the fixed cost is Rs.250 what is the cost of producing 15 units.

  32. Solve: cos2x dy + y.etanx dx = 0

  33. A second degree polynomial passes though the point (1,-1) (2,-1) (3,1) (4,5). Find the polynomial.

  34. If a random variable. X has the probability distribution

    X 0 1 2 3 4 5
    P(X=x) a 2a 3a 4a 5a 6a

    then find F(4)

  35. Suppose A and B are two equally strong table tennis players. Which of the following two events is more probable:
    (a) A beats B exactly in 3 games out of 4 or
    (b) A beats B exactly in 5 games out of 8 ?

  36. 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.

  37. The following data gives readings of 10 samples of size 6 each in the production of a certain product. Draw control chart for mean and range with its control limits.

    Sample 1 2 3 4 5 6 7 8 9 10
    Mean 383 508 505 582 557 337 514 614 707 753
    Range 95 128 100 91 68 65 148 28 37 80
  38. Determine an initial basic feasible solution to the following transportation problem using North West corner rule.

    1. A total of Rs. 8,600 was invested in two accounts. One account earned \(4\frac { 3 }{ 4 } %\)% annual interest and the other earned \(6\frac { 1 }{ 2 } %\)annual interest. If the total interest for one year was Rs. 431.25, how much was invested in each account? (Use determinant method).

    2. Evaluate  \(\int { \frac { { { x }^{ 4 }+{ x }^{ 4 }+1 } }{ { x }^{ 2 }-x+1 } } \)

      1. Part-IV

        Answer all the questions.

    7 x 5 = 35
    1. Show that the equations 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5 are consistent and solve them by rank method.

    2. Evaluate the following using properties of definite integrals:
      \(\int _{ 0 }^{ 1 }{ \frac { x }{ ({ 1-x) }^{ \frac { 3 }{ 4 } } } dx } \)

    1. Evaluate \(\int { \frac { 1 }{ { 3x }^{ 2 }+13x-10 } } dx\)

    2. If the marginal cost (MC) of a production of the company is directly proportional to the number of units (x) produced, then find the total cost function, when the fixed cost is Rs. 5,000 and the cost of producing 50 units is Rs. 5,625.

    1. The rate of increase in the cost Cof ordering holding as the size q of the order increases is given by the differential equation \(\frac { dc }{ dq } =\frac { { c }^{ 2 }+2cq }{ { q }^{ 2 } } \). Find the relationship between c and q if c = 1 when q = 1.

    2. The population of a certain town is as follows

      Year : x 1941 1951 1961 1971 1981 1991
      Population in lakhs:y 20 24 29 36 46 51

      Using appropriate interpolation formula, estimate the population during the period 1946.

    1. Construct the distribution function for the discrete random variable X whose probability distribution is given below. Also draw a graph of p(x) and F(x).

      X = x 1 2 3 4 5 6 7
      P(x) 0.10 0.12 0.20 0.30 0.15 0.08 0.05
    2. Forty percent of business travellers carry a laptop. In a sample of 15 business travelers,
      (i) what is the probability that 3 will have a laptop?
      (ii) what is the probability that 12 of the travelers will not have a laptop?
      (iii) what is the probability that atleast three of the travelers have a laptop?

    1. If the height of 300 students are normally distributed with mean 64.5 inches and standard deviation 3.3 inches find the height below which 99% of the student lie?

    2. The mean breaking strength of cables supplied by a manufacturer is 1,800 with a standard deviation 100. By a new technique in the manufacturing process it is claimed that the breaking strength of the cables has increased. In order to test this claim a sample of 50 cables is tested. It is found that the mean breaking strength is 1,850. Can you support the claim at 0.01 level of significance.

    1. A sample poll of 100 voters chosen at random from all voters in a given district indicated that 55% of them were in favour of a particular candidate. Find
      (a) 95% confidence limits
      (b) 99% confidence limits for the proportion to all voters in favour of this candidate.

    2. Calculate price index number for 2005 by
      (a) Laspeyre’s
      (b) Paasche’s method

      Commodity 1995 2005
      Price Quantity Price Quantity
      A 5 60 15 70
      B 4 20 8 35
      C 3 15 6 20
    1. Assign four trucks 1, 2, 3 and 4 to vacant spaces A, B, C, D, E and F so that distance travelled is minimized. The matrix below shows the distance.

    2. Obtain an initial basic feasible solution to the following transportation problem using Vogels' approximation method.

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