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%

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

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

(a)

\(e^{x^{2}+y^{2}}\)

(b)

2xu

(c)

x²u

(d)

y²u

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 w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

(a)

x^y log x

(b)

y log x

(c)

yx^y-1

(d)

x log y

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 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

The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is

(a)

12 x_o+dx

(b)

12x_o dx

(c)

6x_o dx

(d)

6x_o+ dx

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)

0.03x m³

(c)

0.03x² m³

(d)

0.03x³ m³

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)

6e^2t + 5 sin t - 4 cos t sin t

(b)

6e^2t- 5 sin t + 4 cos t sin t

(c)

3e^2t+ 5 sin t + 4 cos t sin t

(d)

3e^2t - 5 sin t + 4 cos t sin t

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)

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

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

Linear approximation for g(x) = cos x at \(x=\frac{\pi}{2}\) is

(a)

\(x+\frac{\pi}{2}\)

(b)

\(-x +\frac{\pi}{2}\)

(c)

\(x - \frac{\pi}{2}\)

(d)

\(-x - \frac{\pi}{2}\)

If w (x, y, z) = x² (y - 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

If f(x,y, z) = xy +yz +zx, then f_x - f_z is equal to

(a)

z - x

(b)

y - z

(c)

x - z

(d)

y - x