#### 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 $\sqrt [ 3 ]{ 1.02 } +\sqrt { 1.02 }$