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12th Standard English medium Business Maths Reduced Syllabus Public Exam Model Question Paper With Answer Key - 2021

12th Standard

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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. The rank of m x n matrix whose elements are unity is ________.

    (a)

    0

    (b)

    1

    (c)

    m

    (d)

    n

  2. If A, B are two n x n non-singular matrices, then ___________

    (a)

    AB is non-singular

    (b)

    AB is singular

    (c)

    (AB)-1 = A-1 B-1

    (d)

    (AB)-1 does not exit

  3. \(\int \frac{\sin 5 x-\sin x}{\cos 3 x} d x\) is _______.

    (a)

    −cos 2x + c

    (b)

    −cos 2x + c

    (c)

    \(-\frac14\)cos2x + c

    (d)

    −4cos2x + c

  4. \(\int { { a }^{ 3x+2 } } \) dx = _____________ +c

    (a)

    a3x+2

    (b)

    \(\frac { { a }^{ 3x+2 } }{ 3 } \)

    (c)

    \(\frac { { a }^{ 3x+2 } }{ 3loga } \)

    (d)

    3 log a (a3x+2)

  5. For a demand function p, if \(\int \frac{d p}{p}=k \int \frac{d x}{x}\) then k is equal to ________.

    (a)

    \(\eta \)d

    (b)

    -\(\eta \)d

    (c)

    \(\frac{-1}{\eta_{d}}\)

    (d)

    \(\frac{1}{\eta_{d}}\)

  6. The area enclosed by the curve y = cos2x in [0,\(\pi\)] the lines x=0, x = \(\pi\) and the X-axis is ________sq.units.

    (a)

    2\(\pi\)

    (b)

    2\(\pi\)

    (c)

    \(\frac{2}{\pi}\)

    (d)

    \(\frac{\pi}{2}\)

  7. Solution of \(\frac { dy }{ dx } \) + Px = 0 ______.

    (a)

    x = cepy

    (b)

    x = ce−py

    (c)

    x = py + c

    (d)

    x = cy

  8. The solution of \(\frac { dp }{ dt } \) = ke-t (k is a constant) is _____________

    (a)

    c-\(\frac { k }{ { e }^{ t } } \) = p

    (b)

    p = ket+c

    (c)

    t = log\(\left( \frac { c-p }{ k } \right) \)

    (d)

    t = logp

  9. If c is a constant then Δc = _______.

    (a)

    c

    (b)

    Δ

    (c)

    Δ2

    (d)

    0

  10. For the set of values

    x 1961 1971 1981 1991 2001
    y 46 66 81 93 101
    A B
    1) Δy (a) -5
    2) Δ2y (b) 2
    3) Δ3y (c) -3
    4) Δ4y (d) 20
    (a)

    1 - a, 2 - b, 3 - c, 4 - d

    (b)

    1 - b, 2 - c, 3 - d, 4 - a

    (c)

    1 - d, 2 - a, 3 - b, 4 - c

    (d)

    1 - c, 2 - a, 3 - b, 4 - c

  11. In a discrete probability distribution the sum of all the probabilities is always equal to ________.

    (a)

    zero

    (b)

    one

    (c)

    minimum

    (d)

    maximum

  12. If X is a discrete random variable. then P(X≥a)=________.

    (a)

    P(X

    (b)

    1-P(X≤a)

    (c)

    1-P(X

    (d)

    0

  13. The weights of newborn human babies are normally distributed with a mean of 3.2 kg and a standard deviation of 1.1 kg. What is the probability that a randomly selected newborn baby weighs less than 2.0 kg?

    (a)

    0.138

    (b)

    0.428

    (c)

    0.766

    (d)

    0.262

  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. In simple random sampling from a population of N units, the probability of drawing any unit at the first draw is  ______.

    (a)

    \(\frac{n}{N}\)

    (b)

    \(\frac{1}{N}\)

    (c)

    \(\frac{N}{n}\)

    (d)

    1

  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. R is calculated using ________.

    (a)

    xmax - xmin

    (b)

    xmin - xmax

    (c)

    \(\overset{-}{x}\)max \(\overset{-}{x}\)min

    (d)

    \(\overset{=}{x}\)max \(\overset{=}{x}\)min

  18. The normal equations for estimating a and b so that the line y = ax + b may be the line of best fit are __________

    (a)

    aΣx2 + bΣx = Σxy, aΣx + nb = Σy

    (b)

    aΣx + bΣx2 = Σxy, aΣx2 + nb = Σy

    (c)

    aΣx + nb = Σxy, aΣx2 + bΣx = Σy

    (d)

    aΣx2 + nb = Σxy, aΣx + bΣx = Σy

  19. The purpose of a dummy row or column in an assignment problem is to _______.

    (a)

    prevent a solution from becoming degenerate

    (b)

    balance between total activities and total resources

    (c)

    provide a means of representing a dummy problem

    (d)

    none of the above

  20. The penalty is the difference between the ___ costs in each row and column.

    (a)

    smallest

    (b)

    biggest

    (c)

    minimum

    (d)

    least

    1. Part-II

      Answer any seven questions and Question number 30 is compulsory.


    7x 2 = 14
  21. Two newspapers A and B are published in a city . Their 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 one year

  22. Using second fundamental theorem, evaluate the following:
    \(\int_{0}^{\frac{1}{4}} \sqrt{1-4 x} \ d x\)

  23. Find the area under the curve y = 4x - x2 included between x = 0, x = 3 and the X-axis.

  24. Find the order and degree of the following differential equations.
    \(\frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } =0\)

  25. Find the missing term from the following data

    x 1 2 3 4
    f(x) 100 - 126 157
  26. Define Mathematical expectation in terms of discrete random variable.

  27. Students of a class were given an aptitude test. Marks were found to be normally distributed with mean 60 and S.D. 5. Find the percentage of students who scored more than 60 marks.

  28. What is an estimator?

  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. Write mathematical form of transportation problem.

      1. Part-III

        Answer any seven questions and Question number 40 is compulsory.

    7x 3 = 21
  31. Find the rank of each of the following matrices.
    \(\left( \begin{matrix} 1 & -2 & 3 \\ -2 & 4 & -1 \\ -1 & 2 & 7 \end{matrix}\begin{matrix} 4 \\ -3 \\ 6 \end{matrix} \right) \)

  32. Evaluate \(\int { \frac { { sec }^{ 2 }x }{ 3+tanx } } dx\)

  33. The marginal revenue function for a firm is given by MR = \(\frac { 2 }{ x+3 } -\frac { 2x }{ { \left( x+3 \right) }^{ 2 } } +5\). Show that the demand function is \(P=\frac{2}{x+3}+5\)

  34. Find the equation of the curve passing through (1, 0) and which has slope 1+ \(\frac { y }{ x } \) at (x, y).

  35. Using graphic method, find the value of y when x = 38 from the following data:

    x 10 20 30 40 50 60
    y 63 55 44 34 29 22
  36. A player tosses two unbiased coins. He wins Rs. 5 if two heads appear, Rs. 2 if one head appear and Rs.1 if no head appear. Find the expected amount to win.

  37. 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 ?

  38. 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 10. Find the 95% confidence limits for the estimate of their mean marks.

  39. You are given below the values of sample mean ( \(\bar{X}\) ) and the range ( R ) for ten samples of size 5 each. Draw mean chart and comment on the state of control of the process.

    Sample number 1 2 3 4 5 6 7 8 9 10
    \(\overset{-}{X}\) 43 49 37 44 45 37 51 46 43 47
    R 5 6 5 7 7 4 8 6 4 6

    Given the following control chart constraint for : n = 5, A= 0.58, D= 0 and D= 2.115

  40. Determine an initial basic feasible solution to the following transportation problem using North West corner rule.

      1. Part-IV

        Answer all the questions.

    7 x 5 = 35
    1. In a market survey three commodities A, B and C were considered. In finding out the index number some fixed weights were assigned to the three varieties in each of the commodities. The table below provides the information regarding the consumption of three commodities according to the three varieties and also the total weight received by the commodity

      Commodity Variety Variety Total weight
      I II III
      A 1 2 3 11
      B 2 4 5 21
      C 3 5 6 27

      Find the weights assigned to the three varieties by using Cramer’s Rule.

    2. Integrate the following with respect to x.
      \(e^{x}\left[\frac{x-1}{(x+1)^{3}}\right]\)

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

    2. Find the area of the parabola \({ y }^{ 2 }=8x\) bounded by its latus rectum.

    1. Solve: x2\(\frac { dy }{ dx } \) = y2+2xy given that y = 1, when x = 1

    2. Using Newton’s formula for interpolation estimate the population for the year 1905 from the table:

      Year 1891 1901 1911 1921 1931
      Population 98.752 1,32,285 1,68,076 1,95,690 2,46,050
    1. The probability function of a random variable X is given by
      \(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
      Evaluate the following probabilities.
      P(\(\le\))0

    2. In a particular university 40% of the students are having news paper reading habit. Nine university students are selected to find their views on reading habit. Find the probability that
      (i) none of those selected have news paper reading habit
      (ii) all those selected have news paper reading habit
      (iii) atleast two third have news paper reading habit.

    1. Marks in an aptitude test given to 800 students of a school was found to be normally distributed 10% of the students scored below 40 marks and 10% of the students scored above 90 marks. Find the number of students scored between 40 and 90?

    2. The mean life time of a sample of 169 light bulbs manufactured by a company is found to be 1350 hours with a standard deviation of 100 hours. Establish 90% confidence limits within which the mean life time of light bulbs is expected to lie.

    1. Measurements of the weights of a random sample of 200 ball bearings made by certain machine during one week showed a mean of 0.824 newtons and a S.D. of 0.042 newton's. Find
      a) 95% and
      b) 99% confidence limits for the mean weight of all the ball bearings.

    2. The following are the sample means and ranges for 10 samples, each of size 5. Calculate the control limits for the mean chart and range chart and state whether the process is in control or not.

      Sample number 1 2 3 4 5 6 7 8 9 10
      Mean 5.10 4.98 5.02 4.96 4.96 5.04 4.94 4.92 4.92 4.98
      Range 0.3 0.4 0.2 0.4 0.1 0.1 0.8 0.5 0.3 0.5
    1. A computer centre has got three expert programmers. The centre needs three application programmes to be developed. The head of the computer centre, after studying carefully the programmes to be developed, estimates the computer time in minitues required by the experts to the application programme as follows.

      Assign the programmers to the programme in such a way that the total computer time is least.

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

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