If v (x, y) = log (e^x + e^y), then \(\frac { { \partial }v }{ \partial x } +\frac { \partial v }{ \partial y } \) is equal to

(a)

e^x + e^y

(b)

\(\frac{1}{e^x + e^y}\)

(c)

2

(d)

1

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

(a)

xye^xy

(b)

(1 +xy)e^xy

(c)

(1 +y)e^xy

(d)

(1 + x)e^xy

If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to

(a)

-4

(b)

-3

(c)

-7

(d)

13

If the radius of the sphere is measured as 9 cm 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

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

If u = log (x³ + y³ + z³ - 3xyz) then \(\frac { { \partial }u }{ \partial { x } } +\frac { { \partial }u }{ { \partial y } }+ \frac { { \partial }u }{ \partial z } \) = _____________

(a)

\(\frac { 3 }{ x+y+z } \)

(b)

x+y+z

(c)

\(\frac { -9 }{ { (x+y+z) }^{ 2 } } \)

(d)

\(\frac { -9 }{ { (x+y+z) }^{ 2 } } \)

If u = y^x then \(\frac { \partial u }{ \partial y } \) = ............

(a)

xy^x-1

(b)

yx^y-1

(c)

0

(d)

1

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

If u = \((\frac{y}{x})\) then x \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } \) = .....................

(a)

0

(b)

1

(c)

2u

(d)

u

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