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#### Differentials and Partial Derivatives Model Question Paper 1

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?

(a)

$\frac{1}{31}$

(b)

$\frac15$

(c)

5

(d)

31

2. If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is

(a)

0.4 cu.cm

(b)

0.45 cu.cm

(c)

2 cu.cm

(d)

4.8 cu.cm

3. If u = log $\sqrt { { x }^{ 2 }+{ y }^{ 2 } }$, then $\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } }$ is

(a)

$\sqrt { { x }^{ 2 }+{ y }^{ 2 } }$

(b)

0

(c)

u

(d)

2u

4. If f(x, y, z) = sin (xy) + sin (yz) + sin (zx) then fxx is

(a)

-y sin (xy) + z2 cos (xz)

(b)

y sin (xy) - z2 cos (xz)

(c)

y sin (xy) + z2 cos (xz)

(d)

-y sin (xy) - z2 cos (xz)

5. If f(x,y) = 2x2 - 3xy + 5y + 7 then f(0, 0) and f(1, 1) is

(a)

7,11

(b)

11,7

(c)

0,7

(d)

1,0

6. 5 x 2 = 10
7. Find a linear approximation for the following functions at the indicated points.
f(x) = x3 - 5x + 12, x0 = 2

8. Find a linear approximation for the following functions at the indicated points.
g(x) = $g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4$

9. The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = 2ㅠ$\sqrt { \frac { 1 }{ g } }$, where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of 1

10. Use differentials to find $\sqrt{25.2}$

11. IF u(x, y) = x2 + 3xy + y2, x, y, ∈ R, find tha linear appraoximation for u at (2, 1)

12. 5 x 3 = 15
13. The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm.find the following in calculating the area of the circular plate:
Relative error

14. The trunk of a tree has diameter 30 cm. During the following year, the circumference grew 6cm.
What is the percentage increase in area of the tree's cross-section?

15. A coat of paint of thickness 0.2 cm is applied to the faces of a cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of paint is used to paint this cube. Also calculate the exact amount of paint used to paint this cube.

16. Find the approximate value of f (3.02) where f(x) = 3x2 + 5x +3.

17. If w = xy + z where x = cos t; y = sin t; z = t find $\frac{dw}{dt}$

18. 4 x 5 = 20
19. Let (x, y) = e-2y cos(2x) for all (x, y) ∈ R2. Prove that u is a harmonic function in R2.

20. Let g(x, y) = $\frac { { e }^{ y }sinx }{ x }$, for x ≠ 0 and g(0, 0) = 1. Show that g is continuous at (0,0).

21. Find $\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } ,\frac { { \partial }^{ 2 }f }{ { \partial y }^{ 2 } }$  at x = 2, y = 3 if f(x,y) = 2x2 + 3y2 - 2xy

22. If V = log r and r2 = x2 +y2 + z2, then prove that $\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } }$