#### Applications of Differentiation Model Question Paper

11th Standard

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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 $K{∂P\over ∂ K}+L{∂P\over ∂L}=P$