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#### Differentials and Partial Derivatives Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 40
4 x 1 = 4
1. If g(x, y) = 3x2 - 5y + 2y, x(t) = et and y(t) = cos t, then $\frac{dg}{dt}$ is equal to

(a)

6e2t+5 sin t - 4 cos t sin t

(b)

6e2t- 5 sin t + 4 cos t sin t

(c)

3e2t+ 5 sin t + 4 cos t sin t

(d)

3e2t - 5 sin t + 4 cos t sin t

2. If u(x, y) = x2+ 3xy + y - 2019, then $\frac { \partial u }{ \partial x }$(4, -5) is equal to

(a)

-4

(b)

-3

(c)

-7

(d)

13

3. 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)

4. If y = sin x and x changes from $\frac{\pi}{2}$ to ㅠ the approximate change in y is ..............

(a)

0

(b)

1

(c)

$\frac{\pi}{2}$

(d)

$\frac{22}{14}$

5. 1 x 2 = 2
6. If u = log $\left( \frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } \right)$ then
(1) u is a homogeneous function
(2) $x\frac { { \partial }u }{ \partial { x } } +y\frac { { \partial }u }{ { \partial y } }$ = 0
(3) $\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy }$ is a homogeneous function
(4) $\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy }$ is a homogeneous function of  degree 0.

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

9. Find differential dy for each of the following function
y = (3 + sin(2x)) 2/3

10. Let g(x, y) = $\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } }$ for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that $\begin{matrix} lim \\ (x,y)\rightarrow (1,2) \end{matrix}$ g(x, y) = 0 along every line y = mx, m ∈ R

11. If f (x, y) = 2x3 - 11x2y + 3y3, prove that $x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } =3f$

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

13. 3 x 3 = 9
14. The trunk of a tree has diameter 30 cm. During the following year, the circumference grew 6cm.
Approximately, how much did the tree's diameter grow?

15. prove that g(x, y) = x log$\left( \frac { y }{ x } \right)$ is homogeneous; what is the degree? Verify Euler's Theorem for g.

16. Using differentials find the approximate value of tan 46° if it is given that 10 = 0.01745 radians

17. 3 x 5 = 15
18. Assuming log10e = 0.4343, find an approximate value of log10 1003

19. For each of the following functions find the gxy, gxx, gyy and gyx.
g(x, y) = log (5x + 3y)

20. 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