" /> -->

Differentials and Partial Derivatives Model Question Paper

12th Standard EM

    Reg.No. :
  •  
  •  
  •  
  •  
  •  
  •  

Maths

Time : 01:00:00 Hrs
Total Marks : 40
    5 x 1 = 5
  1. If w (x, y) = xy, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

    (a)

    xy log x

    (b)

    y log x

    (c)

    yxy-1

    (d)

    x log y

  2. If f (x, y) = exy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to

    (a)

    xyexy

    (b)

    (1 +xy)exy

    (c)

    (1 +y)exy

    (d)

    (1 + x)exy

  3. The change in the surface area S = 6x2 of a cube when the edge length varies from xo to xo+ dx is

    (a)

    12 xo+dx

    (b)

    12xo dx

    (c)

    6xo dx

    (d)

    6xo+ dx

  4. 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

  5. If u = xy + yx then ux + uy at x = y = 1 is

    (a)

    0

    (b)

    2

    (c)

    1

    (d)

  6. 5 x 2 = 10
  7. Find a linear approximation for the following functions at the indicated points.
    g(x) = \(g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4\)

  8. Find a linear approximation for the following functions at the indicated points.
    \({ h }({ x })=\frac { x }{ 1+x } =\frac { 1 }{ 2 } \)

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

  11. If of f(x, y) = x2 + y3 + 2xy2 find fxx, fyy, fxy and fyx.

  12. 5 x 3 = 15
  13. f(x,y) = \(\frac { xy }{ { x }^{ 2 }+{ y }^{ 2 } } \), (x,y) ≠ (0,0) and f (0,0) = 0 Show that f is not continuous at f, -(0,0) and continuous at all other points of R2

  14. Consider g(x,y) = \(\frac { 2{ x }^{ 2 }y }{ { x }^{ 2 }+{ y }^{ 2 } } \), if (x,y) ≠ (0,0) and g(0,0) = 0 Show that g is continuous on R2

  15. Let U(x, y, z) = x2 − xy + 3 sin z, x, y, z ∈ R Find the linear approximation for U at (2,−1,0).

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

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

  18. 2 x 5 = 10
  19. Let (x, y) = e-2y cos(2x) for all (x, y) ∈ R2. Prove that u is a harmonic function in R2.

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

*****************************************

TN 12th Standard EM Maths free Online practice tests

Reviews & Comments about 12th Maths - Differentials and Partial Derivatives Model Question Paper

Write your Comment