#### 12th Standard Maths English Medium Differentials and Partial Derivatives Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:40:00 Hrs
Total Marks : 60

Multiple Choice Questions

15 x 1 = 15
1. 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

2. 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

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. 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

5. 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

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

(a)

1.3968

(b)

1.3898

(c)

1.3893

(d)

none

7. 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

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

(a)

0

(b)

2

(c)

1

(d)

9. lf u = (x-y)4+(y-z)4 +(z-x)4 then $\sum { \frac { \partial u }{ \partial x } }$ =

(a)

4

(b)

1

(c)

0

(d)

-4

10. The cube root of 127 is ............

(a)

5.026

(b)

5.26

(c)

5.028

(d)

5.075

11. 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)

$\frac{22}{14}$

12. If u = yx then $\frac { \partial u }{ \partial y }$ = ............

(a)

xyx-1

(b)

yxy-1

(c)

0

(d)

1

13. If u = sin-1 $\left( \frac { { x }^{ 4 }+{ y }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right)$ and f= sin u then f is a homogeneous function of degree ..................

(a)

0

(b)

1

(c)

2

(d)

4

14. If u = y sin x then $\frac { { \partial }^{ 2 }u }{ \partial x\partial y }$ = ..........

(a)

cos x

(b)

cos y

(c)

sin x

(d)

0

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. Find differential dy for each of the following function
y = ex2-5x+7 cos (x2 - 1)

19. Evaluate $\begin{matrix} lim \\ (x,y)\rightarrow (1,2) \end{matrix}$g(x, y), if the limit exists, where g$(x,y)=\frac { { 3x }^{ 2 }-xy }{ { x }^{ 2 }+{ y }^{ 2 }+3 }$

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

22. If u=x2y+y2z+z2x, then prove that $\cfrac { \partial u }{ \partial x } +\cfrac { \partial u }{ \partial y } +\cfrac { \partial u }{ \partial z } =\left( x+y+z \right) ^{ 2 }$

23. If $u={ e }^{ \frac { x }{ y } }sin\left( \cfrac { x }{ y } \right) +{ e }^{ \frac { y }{ x } }cos\left( \cfrac { y }{ x } \right)$ ,then prove that $x\cfrac { \vartheta u }{ \vartheta x } +y\cfrac { \vartheta u }{ \vartheta y } =0$

24. Find a linear approximation to f(x)=3xe2x-10 at x=5

25. Calculate df for $f=\sqrt { 2x+5 }$ when x = 22 and dx = 3.

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. Use the linear approximation to find approximate values of
$\sqrt [ 3 ]{ 26 }$

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

30. A circular plate expands uniformly under the influence of heat. If it’s radius increases from 10.5 cm to 10.75 cm, then find an approximate change in the area and the approximate percentage change in the area.

31. A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x2 − 0.05y2 and C(x, y) = 8x + 6y + 2000 respectively
Find the profit function P(x, y)

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

33. If u=sin3xcos4y, then prove that $\cfrac { { \partial }^{ 2 }u }{ \partial x\partial y } =\cfrac { { \partial }^{ 2 }u }{ \eth y\partial x }$

34. Find the approximate value of $\left( \cfrac { 17 }{ 81 } \right) ^{ \frac { 1 }{ 4 } }$ using linear approximation.

35. If w=x+2y+z2 and x=cos t,y=sint,z=t then find $\cfrac { dw }{ dt }$

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. If w(x, y) = xy + sin (xy), then prove that $\frac { { \partial }^{ 2 }w }{ \partial y\partial x } =\frac { { \partial }^{ 2 }w }{ \partial x\partial y }$

38. Let U(x, y) = ex sin y, where x = st2, y = s2 t, s, t ∈ R. Find $\frac { \partial U }{ \partial s } ,\frac { \partial U }{ \partial t }$ and evaluate them at s = t = 1.

39. If v(x,y) = log $\left( \frac { { x }^{ 2 }+{ y }^{ 2 } }{ x+y } \right)$, prove that $x\frac { \partial v }{ \partial x } +y\frac { \partial u }{ \partial y }$ = 1

40. If w(x,y, z) = log $\left( \frac { { 5x }^{ 3 }{ y }^{ 4 }+7{ y }^{ 2 }{ xz }^{ 4 }-{ 75y }^{ 3 }{ z }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right)$ find $x\frac { \partial w }{ \partial x } +y\frac { \partial w }{ \partial y } +z\frac { \partial w }{ \partial z }$

41. If u = tan -1 $\left( \frac { { x }^{ 3 }+{ y }^{ 3 } }{ x-y } \right)$
Prove that $x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y }$ sin 2u.

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