11th Half Yearly Model Question

11th Standard

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Business Maths

Time : 02:30:00 Hrs
Total Marks : 90
    I. Answer all the questions write the options code and the corresponding answer: 
    20 x 1 = 20
  1. If A \(=\begin{pmatrix} -1 & 2 \\ 1 & -4 \end{pmatrix}\) then A (adj A) is 

    (a)

    \(\begin{pmatrix} -4 & -2 \\ -1 & -1 \end{pmatrix}\)

    (b)

    \(\begin{pmatrix} 4 & -2 \\ -1 & 1 \end{pmatrix}\)

    (c)

    \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

    (d)

    \(\begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}\)

  2. The value of \(\begin{vmatrix} x & x^2 & -yz & 1 \\ y & y^2 & -zx & 1 \\ z & z^2 & -xy &1 \end{vmatrix}\) is

    (a)

    1

    (b)

    0

    (c)

    -1

    (d)

    -xyz

  3. The number of ways selecting 4 players out of 5 is

    (a)

    4!

    (b)

    20

    (c)

    25

    (d)

    5

  4. The value of (5Co + 5C1) + (5C1 + 5C2) + (5C2 + 5C3) + (5C3 + 5C4) + (5C4 + 5C5 ) is

    (a)

    26-2

    (b)

    25-1

    (c)

    28

    (d)

    27

  5. (1, - 2) is the centre of the circle x2 + y2 + ax + by - 4 = 0 , then its radius

    (a)

    3

    (b)

    2

    (c)

    4

    (d)

    1

  6. The equation of directrix of the parabola y2 = - x is

    (a)

    4x+ 1 =0

    (b)

    4x - 1 = 0

    (c)

    x - 4=0

    (d)

    x + 4 = 0

  7. The radian measure of 37o30' is

    (a)

    \(\frac{5\pi}{24}\)

    (b)

    \(\frac{3\pi}{24}\)

    (c)

    \(\frac{7\pi}{24}\)

    (d)

    \(\frac{9\pi}{24}\)

  8. \(\left(\frac{\cos x}{cosec x}\right)-\sqrt{1-\sin^2x}\sqrt{1-\cos^2x}\) is

    (a)

    cos2x-sin2x

    (b)

    sin2x-cos2x

    (c)

    1

    (d)

    0

  9. \(\frac{d}{dx}(\frac{1}{x})\) is equal to

    (a)

    \(-\frac{1}{x^2}\)

    (b)

    \(-\frac{1}{x}\)

    (c)

    log x

    (d)

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

  10. \(\frac{d}{dx}(5e^x-2logx)\) is equal to

    (a)

    5ex - \(\frac{2}{x}\)

    (b)

    5ex - 2x

    (c)

    5ex - \(\frac{1}{x}\)

    (d)

    2 logx

  11. Marginal revenue of the demand function p= 20–3x is

    (a)

    20–6x

    (b)

    20–3x

    (c)

    20+6x

    (d)

    20+3x

  12. If demand and the cost function of a firm are p= 2–x and c = 2x2 +2x +7 then its profit function is

    (a)

    x2 + 7

    (b)

    x2 - 7

    (c)

    - x2 + 7

    (d)

    - x2 - 7

  13. The brokerage paid by a person on this sale of 400 shares of face value Rs.100 at 1% brokerage

    (a)

    Rs 600

    (b)

    Rs 500

    (c)

    Rs 200

    (d)

    Rs 400

  14. If ‘a’ is the annual payment, ‘n’ is the number of periods and ‘i’ is compound interest for Rs 1 then future amount of the annuity is

    (a)

    A = \(\frac{a}{i}(1+i)(1+i)^n-1]\)

    (b)

    A = \(\frac{a}{i}[(1+i)^n-1]\)

    (c)

    P = \(\frac{a}{i}\)

    (d)

    P = \(\frac{a}{i}(1+i)[1-(1+i)^{-n}]\)

  15. Harmonic mean is better than other means if the data are for

    (a)

    Speed or rates.

    (b)

    Heights or lengths.

    (c)

    Binary values like 0 and 1.

    (d)

    Ratios or proportions.

  16. The probability of drawing a spade from a pack of card is

    (a)

    1/52

    (b)

    1/13

    (c)

    4/13

    (d)

    1/4

  17. Correlation co-efficient lies between

    (a)

    0 to ∞

    (b)

    -1 to +1

    (c)

    -1 to 0

    (d)

    -1 to ∞

  18. If r(X,Y) = 0 the variables X and Y are said to be

    (a)

    Positive correlation

    (b)

    Negative correlation

    (c)

    No correlation

    (d)

    Perfect positive correlation

  19. The critical path of the following network is

    (a)

    1 – 2 – 4 – 5

    (b)

    1– 3– 5

    (c)

    1 – 2 – 3 – 5

    (d)

    1 – 2 – 3 – 4 – 5

  20. Maximize: z=3x1+4x2 subject to 2x1+x2≤40, 2x1+5x2≤180, x1,x2≥0 in the LPP, which one of the following is feasible corner point?

    (a)

    x1=18, x2=24

    (b)

    x1=15, x2=30

    (c)

    x1=2.5, x2=35

    (d)

    x1=20, x2=19

  21. II. Answer any seven questions. Q. No 30 is compulsory :
    7 x 2 =14
  22. The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.6 & 0.9 \\ 0.20 & 0.80 \end{bmatrix}\) .Test whether the system is viable as per Hawkins-Simon conditions.

  23. Resolve into partial fractions for the following 
    \(\frac { 3x+7 }{ { x }^{ 2 }-3x+2 } \)

  24. Show that perpendicular distances of the line x-y+5=0 from origin and from the point P(2, 2) are equal.

  25. Find the principal value of \(\cos^{-1}\left(\frac{-1}{\sqrt2}\right)\)

  26. Evaluate: \(\underset { x\rightarrow 1 }{ lim } \frac { { x }^{ 3 }-1 }{ x-1 } \)

  27. If f(x,y) = 3x2 + 4y3 + 6xy - x2y3 + 6. Find fx(1, -1)

  28. How much will be required to buy 125 of Rs 25 shares at a discount of Rs 7

  29. A die is thrown. Find the probability of getting
    (i) a prime number
    (ii) a number greater than or equal to 3

  30. Calculate the correlation coefficient from the following data
    N=9, ΣX=45, ΣY=108, ΣX2=285, ΣY2=1356, ΣXY=597

  31. A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods X and Y. To produce one unit of X, 2 unit of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Yare priced at HOO and H20 per unit respectively, how should the producer use his resources to maximize the total revenue? Formulate the LPP for the above.

  32. III. Answer any seven questions. Q. No 40 is compulsory :

    7 x 3 = 21
  33. If \({ A }^{ -1 }=\left[ \begin{matrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{matrix} \right] \) then, find A.

  34. Find the number of arrangements that can be made out of the letters of the word "ASSASSINATION".

  35. Find the slope of the lines which make an angle of 45° with the line 3x - y + 5 = 0.

  36. Prove that \(\frac { sin(-\theta )tan({ 90 }^{ o }-\theta )sec\left( { 180 }^{ o }-\theta \right) }{ sin(180+\theta )cot(360-\theta )cosec({ 90 }^{ o }-\theta ) } =1\)

  37. Differentiate the following with respect to x \(\frac { { e }^{ x } }{ 1+{ e }^{ x } } \)

  38. Verify Euler’s theorem for the function \(u=\frac{1}{\sqrt{x^2+y^2}}\)

  39. A man bought 6% stock of Rs.12,000 at 92 and sold it when the rise to 96.Find his gain.

  40. Compared to the previous year the overhead expenses went up by 32% in 1995, they increased by 40% in the next year and by 50% in the following year. Calculate the average rate of increase in overhead expenses over the three years.

  41. If two lines of regression are 3X-2Y+1=0, 2X-Y-2=0, find \(\bar {X}\)and \(\bar{Y}\).

  42. Solve the following LPP graphically.
    Maximize \(Z=6{ x }_{ 1 }+5{ x }_{ 2 }\) Subject to the constraints \(3{ x }_{ 1 }+5{ x }_{ 2 }\le 15,\quad 5{ x }_{ 1 }+2{ x }_{ 2 }\le 10\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

  43. IV. Answer all the questions :

    7 x 5 = 35
    1. Solve by matrix inversion method: 3x - y + 2z = 13 ; 2x + Y - z = 3 ; x + 3y - 5z = - 8.

    2. By the principle of mathematical induction, prove the following.
      13+23+33+.......+n3=\(\frac { { n }^{ 2 }(n+1)^{ 2 } }{ 4 } \) for all \(n\in N\).

    1. How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2, 3?

    2. Show that the given lines 3x-4y-13=0, 8x-11y=33 and2x-3y-7=0 are concurrent and find the concurrent point.

    1. If tan x=\(\frac { -4 }{ 3 } \) and x is in II quadrant,find \(sin\frac { x }{ 2 } ,cos\frac { x }{ 2 } \) and \(tan\frac { x }{ 2 } \)

    2. Find the value of tan \(\left( {{\pi}\over{8}}\right).\)

    1. if y = 2 sin x + 3 cos x, then show that y2 + y = 0

    2.  The demand for a commodity x is q= 5-2p1+P2 -\({ p }_{ 1 }^{ 2 }{ p }_{ 2 }\) .Find the partial elasticities \(\frac { Eq }{ { EP }_{ 1 } } \) and \(\frac { Eq }{ { EP }_{ 2 } } \) when p1=3 and p2=7

    1. For the total revenue function R = - 90 + 6x2 - x3 find when R is increasing and when it is decreasing. Also, discuss the behaviour of marginal revenue.

    2. Kamal sold Rs.9000 worth 7% stock at 80 and invested the proceeds in 15% stock at 120. Find the change in his income?

    1. Calculate the Mean deviation about median and its relative measure for the following data.

      X 15 25 35 45 55 65 75 85
      frequency 12 11 10 15 22 13 18 19
    2. Find the equation of the regression line of Y on X, if the observations ( Xt, Yi) are the following (1,4) (2,8) (3,2) ( 4,12) ( 5, 10) ( 6, 14) ( 7, 16) ( 8, 6) (9, 18)

    1. A company is producing three products P1, P2 and P3, with profit contribution of Rs.20, Rs.25 and Rs.15 per unit respectively. The resource requirements per unit of each of the products and total availability are given below.​​​​​​​

      Product P1 P2 P3 Total availability
      Man hours/unit 6 3 12 200
      Machine hours/unit 2 5 4 350
      Material/unit 1kg 2kg 1kg 100kg

      Formulate the above as a linear programming model.

    2. Every gram of wheat provides 0.1 g of proteins and 0.25 g of carbohydrates. The corresponding values of rice are 0.05 g and 0.5 g respectively. Wheat cost Rs.4 per kg and rice cost Rs.6 per kg. The minimum daily requirements of proteins and carbohydrate for an average child are 50 g and 200 g respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrate at minimum cost. Frame an LPP and solve it graphically. 

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