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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

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