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Applications of Differentiation Model Question Paper

11th Standard

    Reg.No. :
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Business Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. The elasticity of demand for the demand function x = \(\frac { 1 }{ p } \) os

    (a)

    0

    (b)

    1

    (c)

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

    (d)

    \(\infty \)

  2. Relationship among MR, AR and ηd is

    (a)

    \({ n }_{ d }=\frac { AR }{ AR-MR } \)

    (b)

    n4 =  AR - MR

    (c)

    MR = AR = n4

    (d)

    \(AR=\frac { MR }{ { n }_{ 4 } } \)

  3. if u = 4x2 + 4xy + y2 + 32 + 16 , then \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is equal to

    (a)

    8x + 4y + 4

    (b)

    4

    (c)

    2y + 32

    (d)

    0

  4. If u = x3 + 3xy2 + y3 then  \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \)

    (a)

    3

    (b)

    6y

    (c)

    6x

    (d)

    2

  5. if q = 1000 + 8p1 - p2 then, \(\frac { \partial q }{ \partial { p }_{ 1 } } \) is

    (a)

    -1

    (b)

    8

    (c)

    1000

    (d)

    1000 - p2

  6. 5 x 2 = 10
  7. If y=x-1/x, prove that y is a strictly increasing function for all real vaules of x(x\(\neq\)0).

  8. If y=1+1/x, show that y is a strictly decreasing function for all real values of x(x\(\neq\)0).

  9. Prove that 75-12x+6x2-x3 always decreases as x increases.

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

  11. If f(x,y) = 3x2 + 4y3 + 6xy - x2y3 + 6. Find fxy(2,1)

  12. 5 x 3 = 15
  13. Find the stationary points and stationary values of the function f(x) = x3 - 3x2 - 9x + 5.

  14. Show that the function x3 + 3x2 + 3x + 7 is an increasing function for all real values of x.

  15. Separate the intervals in which the function x3 + 8x2 + 5x - 2 is increasing or decreasing.

  16. Find the maximum and minimum values of the function x2 + 16/x

  17. For the production function P= 5(L)0.7(K)0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)0·3 = 0.6968; (3.33)0·7 = 2.2322]

  18. 4 x 5 = 20
  19. A firm has revenue function R = 8x and production cost function \(C = 150000 + 60\left(x^2\over 900\right)\) Find the total profit function and the number of units to be sold to get the maximum profit.

  20. Verify Euler's theorem for the function \(u=\sqrt{x^2+y^2}\)

  21. If \(u= e^{x/y} sin\left(x\over y\right)+e^{y/x}cos\left(y\over x\right)\) show that \(x{∂u\over ∂x}+y{∂u\over ∂y}=0\) using Euler's theorem.

  22. For the production function P = C(L)α(K)β where C is a positive constant and if α + β = 1, show that \(K{∂P\over ∂ K}+L{∂P\over ∂L}=P\)

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