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12th Standard Maths English Medium Differentials and Partial Derivatives Reduced Syllabus Important Questions With Answer Key 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  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. The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is

    (a)

    0.3xdx m3

    (b)

    0.03 xm3

    (c)

    0.03.x2 m3

    (d)

    0.03x3m3

  3. 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

  4. 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)

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

    (d)

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

  5. If w (x, y, z) = x2 (v - z) + y2 (z - x) + z2(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

    (a)

    xy + yz + zx

    (b)

    x(y + z)

    (c)

    y(z + x)

    (d)

    0

  6. If (x,y, z) = xy +yz +zx, then fx - fz is equal to

    (a)

    z - x

    (b)

    y - z

    (c)

    x - z

    (d)

    y - x

  7. If y = x4 - 10 and if x changes from 2 to 1.99, the approximate change in y is

    (a)

    -32

    (b)

    -0.32

    (c)

    - 10

    (d)

    10

  8. If the radius of the sphere is measured as 9 em with an error of 0.03 cm, the approximate error in calculating its volume is

    (a)

    9.72 cm3

    (b)

    0.972 cm3

    (c)

    0.972π cm3

    (d)

    9.72π cm3

  9. If loge4 = 1.3868, then loge4.01 =

    (a)

    1.3968

    (b)

    1.3898

    (c)

    1.3893

    (d)

    none

  10. 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

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

    (a)

    0

    (b)

    2

    (c)

    1

    (d)

  12. If f (x, y) = x3 + y3 - 3xythen \(\frac { { \partial }f }{ \partial { x } } \) at x = 2,

    (a)

    -15

    (b)

    15

    (c)

    -9

    (d)

    16

  13. 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

  14. If x = r cos θ, y = r sin, then \(\frac { \partial r }{ \partial x } \) = ....................

    (a)

    sec θ

    (b)

    sin θ

    (c)

    cos θ

    (d)

    cosec θ

  15. If is a homogeneous function of x and y of degree n, then \(x\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +y\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = .............. \(\frac { { \partial }u }{ \partial { x } } \)

    (a)

    n

    (b)

    0

    (c)

    1

    (d)

    n - 1

    1. 2 Marks


    10 x 2 = 20
  16. Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.

  17. 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:
    change in the volume

  18. 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

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

  20. A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10cm.

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

  22. If \(u=log\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \) then prove that \(\left( \cfrac { \vartheta u }{ \vartheta x } \right) +\left( \cfrac { \vartheta u }{ \vartheta y } \right) =\cfrac { 1 }{ { x }^{ 2 }+{ y }^{ 2 } } \)

  23. If u=x2+y2+z2-3xyz, then prove that \(x\cfrac { \partial u }{ \partial x } +y\cfrac { \partial u }{ \partial y } +z\cfrac { \partial u }{ \partial z } =3u\)

  24. If u=x2+3xy2+y2, then prove that \(\cfrac { { \partial }^{ 2 }u }{ \partial x\partial y } =\cfrac { { \partial }^{ 2 }u }{ \partial y\partial x } \)

  25. If w=xyexy find \(\cfrac { { \partial }^{ 2 }u }{ \partial x\partial y } \)

    1. 3 Marks


    10 x 3 = 30
  26. Find the linear approximation for f(x) = \(\sqrt { 1+x } ,x\ge -1\) at x0 = 3. Use the linear approximation to estimate f(3.2) 

  27. A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.

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

  29. 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:
    Absolute error

  30. 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:
    Percentage error

  31. 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?

  32. An egg of a particular bird is very nearly spherical. If the radius to the inside of the shell is 5 mm and radius to the outside of the shell is 5.3 mm, find the volume of the shell approximately.

  33. Find the linear approximation to \(g(z)=\sqrt [ 4 ]{ zat } z=2\)

  34. If u=(x-y)(y-z)(z-x), then prove that ux+uy+uz=0

  35. Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 0,0 \right) }{ lim } \cfrac { { x }^{ 2 }-xy }{ \sqrt { x } -\sqrt { y } } \)

    1. 5 Marks


    7 x 5 = 35
  36. Let f(x, y) = sin(xy2) + ex3+5y for all ∈ R2. Calculate \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } } \)and \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)

  37. For each of the following functions find the fx, fy, and show that fxy =fyx
    f(x,y) = tan-1 \(\left( \frac { x }{ y } \right) \)

  38. If \(u=log\sqrt { { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } } \) , then prove that \(\cfrac { { \partial }^{ 2 }u }{ { \partial x }^{ 2 } } +\cfrac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } +\cfrac { { \partial }^{ 2 }u }{ { \partial z }^{ 2 } } =\cfrac { 1 }{ { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } } \)

  39. If w=u2ev where \( u=\cfrac { x }{ y } \) and v=logx. Find \(\cfrac { \partial w }{ \partial x } \) and \(\cfrac { \partial w }{ \partial y } \) 

  40. Find \(\cfrac { \partial w }{ \partial u } ,\cfrac { \partial w }{ \partial v } \) if w=sin-1(x,y) where x=u+v,y=u-v

  41. If \(w=x^{ 2 }sin\left( \cfrac { x }{ y } \right) +{ y }^{ 2 }cos\left( \cfrac { x }{ y } \right) +xytan\left( \cfrac { x }{ y } \right) \),then prove that \(x\cfrac { \partial w }{ \partial x } +y\cfrac { \partial w }{ \partial y } =2w\)

  42. Find the approximate value of \(\sqrt [ 3 ]{ 1.02 } +\sqrt { 1.02 } \)

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