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Differentials and Partial Derivatives Model Question Paper

12th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 35
    5 x 1 = 5
  1. A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is

    (a)

    0.2%

    (b)

    0.4%

    (c)

    0.04%

    (d)

    0.08%

  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. 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

  4. If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) 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

  5. If \(f(x)=\frac{x}{x+1}\), then its differential is given by

    (a)

    \(\frac { -1 }{ ({ x+1) }^{ 2 } } dx\)

    (b)

    \(\frac { 1 }{ ({ x+1) }^{ 2 } } dx\)

    (c)

    \(\frac { 1 }{ x+1 } dx\)

    (d)

    \(\frac {- 1 }{ x+1 } dx\)

  6. 4 x 2 = 8
  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. A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find approximations for the following:
    (i) change in the volume
    (ii) change in the surface area

  9. Find df for f(x) = x2 + 3x and evaluate it for
    x = 2 and dx = 0.1

  10. 4 x 3 = 12
  11. Use linear approximation to find an approximate value of \(\sqrt { 9.2 } \) without using a calculator.

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

  13. 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

  14. Show that F(x,y) = \(\frac { { x }^{ 2 }+5xy-10{ y }^{ 2 } }{ 3x+7y } \) is a homogeneous function of degree 1.

  15. 2 x 5 = 10
  16. Let (x, y) = e-2y cos(2x) for all (x, y) ∈ R2. Prove that u is a harmonic function in R2.

  17. Verify the above theorem for F(x, y) = x2 - 2y2 + 2xy and x(t) = cos t, y(t) = sin t, t ∈ [0, 2\(\pi\)]

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