Engineering Mathematics Test 4

Question 1
1 / -0

Which of the following vector function is solenoidal?

A
(x + y)î + (2y – 3z)ĵ + (3x – 4z)k̂

B
(x - y)î + (2y – z)ĵ + (3x – 4z)k̂

C
(2x - y)î + (y – 3z)ĵ + (3x + 4z)k̂

D
(2x + y)î + (2y – 3z)ĵ + (3x – 4z)k̂

Question 2
1 / -0

If \(\vec F = \left( {{x^2}y + 3z} \right)\hat i + \left( {x{z^3} - 2y} \right)\hat j + {x^2}z\hat k.\) Then the value of grad(div \(\vec F\)) at the point (1, 2, 3) is

A
î - 2ĵ

B
2î + 3ĵ

C
6î + 2ĵ

D
6î + ĵ - k̂

Question 3
1 / -0

Consider a function f(x, y, z) given by f(x, y, z) = (4x³y – 3y²x² + 4z) ln y. The value of f_y \((\frac{\partial f}{\partial y})\)at point x = 2, y = 1, z = 2 is

A
28-28

B
284-281

C
287-286

D
289-282

Question 4
1 / -0

If \(u = \ln \left( {\frac{{{x^3} + {y^3}}}{{x + y}}} \right)\), then find the value of \(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}\) ________?

A
2-2

B
24-21

C
27-26

D
29-22

Question 5
1 / -0

Let \(\vec u = - x\hat i + y\hat j + z\hat k\) and \(\vec v = y{z^2}\hat i + x{z^2}\hat j + 2xyz\hat k\). If \(\vec u\;and\;\vec v\) are irrotational vectors satisfying the condition \(\vec \nabla \cdot \left( {\vec u \times \vec v} \right) + f\left( {x,y,z} \right) + \vec u \cdot \vec v = 0,\) then f(x, y, z) is equal to

A
xy²z

B
–2xyz²

C
xy² + z

D
-2xy² + z²

Question 6
1 / -0

If u = x log xy where \({x^3} + {y^3} + 3xy = 1,\) find du/dx

A
\(1 - \log xy + \frac{x}{y}\frac{{\left( {{x^2} + y} \right)}}{{\left( {x + {y^2}} \right)}}\)

B
\(1 + \log xy + \frac{x}{y}\frac{{\left( {{x^2} + y} \right)}}{{\left( {x + {y^2}} \right)}}\)

C
\(1 - \log xy - \frac{x}{y}\left( {\frac{{{x^2} + y}}{{x + {y^2}}}} \right)\)

D
\(1 + \log xy - \frac{x}{y}\frac{{\left( {{x^2} + y} \right)}}{{\left( {x + {y^2}} \right)}}\)

Question 7
1 / -0

The directional derivative of ϕ = x²yz + 4 xz²at (1, -2, -1) in the direction 2î - ĵ - 2k [upto 2 decimals]

A
12.30-12.36

B
12.304-12.361

C
12.307-12.366

D
12.309-12.362

Question 8
1 / -0

The vector \(\vec V = \left( {x + y + az} \right)i + \left( {bx + 2y - z} \right)j + + \left( { - x + cy + 2z} \right)k\) is irrotational. Where a, b and c are constants. Find the divergence of the vector \(\vec V\).

A
5-5

B
54-51

C
57-56

D
59-52

Question 9

1 / -0

Match the following:

A.	If \(u\; = \;\frac{{{x^2}y}}{{x + y}}\) then \(x\frac{{{\partial ^2}u}}{{\partial {x^2}}} + y\frac{{{\partial ^{2u}}}}{{\partial x\partial y}}\)	1.	\(\frac{{ - 3}}{{16}}u\)
B.	If \(u\; = \;\frac{{\sqrt x - \sqrt y }}{{{x^{\frac{1}{4}}} + {y^{\frac{1}{4}}}}}\) then \({x^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} + 2xy\frac{{{\partial ^2}u}}{{\partial x\partial y}} + {y^2}\frac{{{\partial ^2}u}}{{\partial {y^2}}}\)	2.	\(\frac{{\partial u}}{{\partial x}}\)
C.	If \(u\; = \;{x^{\frac{1}{2}}} + {y^{\frac{1}{2}}}\) then \({x^2}\frac{{{\partial ^2}u}}{{\partial {x^2}}} + 2xy\frac{{{\partial ^2}u}}{{\partial x\partial y}} + {y^2}\frac{{{\partial ^2}u}}{{\partial {y^2}}}\;\)	3.	0
D.	If \(u\; = \;f\left( {\frac{y}{x}} \right)\) then \(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}\)	4.	\(- \frac{1}{4}u\)

A – 1, B – 2, C – 3, D – 4

A – 2, B – 1, C – 4, D – 3

A – 2, B – 1, C – 3, D – 4

A – 1, B – 2, C – 4, D – 3

Question 10
1 / -0

The potential function for the vector field \(\vec F = \left( {4xy + {y^2}z} \right)\hat i + \left( {2{x^2} + 2xyz + {z^2}} \right)\hat j + \left( {x{y^2} + 2yz} \right)\hat k\) is given by

A
ϕ (x, y, z) = 2x²y + xy²z + z²y + c

B
ϕ (x, y, z) = 2x²y + xyz + z² + c

C
ϕ (x, y, z) = x²y + xy² + z²y + c

D
ϕ (x, y, z) = x²y + xy²z + z² + c

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Engineering Mat...

Question 1

(x + y)î + (2y – 3z)ĵ + (3x – 4z)k̂

(x - y)î + (2y – z)ĵ + (3x – 4z)k̂

(2x - y)î + (y – 3z)ĵ + (3x + 4z)k̂

(2x + y)î + (2y – 3z)ĵ + (3x – 4z)k̂

Question 2

î - 2ĵ

2î + 3ĵ

6î + 2ĵ

6î + ĵ - k̂

Question 3

28-28

284-281

287-286

289-282

Question 4

2-2

24-21

27-26

29-22

Question 5

xy2z

–2xyz2

xy2 + z

-2xy2 + z2

Question 6

\(1 - \log xy + \frac{x}{y}\frac{{\left( {{x^2} + y} \right)}}{{\left( {x + {y^2}} \right)}}\)

\(1 + \log xy + \frac{x}{y}\frac{{\left( {{x^2} + y} \right)}}{{\left( {x + {y^2}} \right)}}\)

\(1 - \log xy - \frac{x}{y}\left( {\frac{{{x^2} + y}}{{x + {y^2}}}} \right)\)

\(1 + \log xy - \frac{x}{y}\frac{{\left( {{x^2} + y} \right)}}{{\left( {x + {y^2}} \right)}}\)

Question 7

12.30-12.36

12.304-12.361

12.307-12.366

12.309-12.362

Question 8

5-5

54-51

57-56

59-52

Question 9

A – 1, B – 2, C – 3, D – 4

A – 2, B – 1, C – 4, D – 3

A – 2, B – 1, C – 3, D – 4

A – 1, B – 2, C – 4, D – 3

Question 10

ϕ (x, y, z) = 2x2y + xy2z + z2y + c

ϕ (x, y, z) = 2x2y + xyz + z2 + c

ϕ (x, y, z) = x2y + xy2 + z2y + c

ϕ (x, y, z) = x2y + xy2z + z2 + c

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xy²z

–2xyz²

xy² + z

-2xy² + z²

ϕ (x, y, z) = 2x²y + xy²z + z²y + c

ϕ (x, y, z) = 2x²y + xyz + z² + c

ϕ (x, y, z) = x²y + xy² + z²y + c

ϕ (x, y, z) = x²y + xy²z + z² + c