Updated MCQ for CV, ME PDF

+-------------+-------------+-------------+-------------+-------------+ | 80. | **The unit | | | | | | normal | | | | | | vector to | | | | |...

+-------------+-------------+-------------+-------------+-------------+ | 80. | **The unit | | | | | | normal | | | | | | vector to | | | | | | the | | | | | | surface** | | | | | | [**x**^**2* | | | | | | *^**+y**^** | | | | | | 2**^ **−**  | | | | | | **2z** **+* | | | | | | * **3** **= | | | | | | ** **0**]{. | | | | | | math | | | | | |.inline} | | | | | | **at** | | | | | | [**(1,** ** | | | | | | 2,**  **−** | | | | | |  **1)**]{.m | | | | | | ath | | | | | |.inline} | | | | | | **is** | | | | +=============+=============+=============+=============+=============+ | | A) | \ | B) | [\$\\frac{2 | | | | [\$\$\\frac | | \\widehat{i | | | | {2\\widehat | | } | | | | {i} | | + | | | | + | | 4\\widehat{ | | | | 4\\widehat{ | | j}}{\\sqrt{ | | | | j} | | 20}}\$]{.ma | | | | + | | th | | | | 2\\widehat{ | |.inline} | | | | k}}{\\sqrt{ | | | | | | 24}}\\text{ | | | | | | \\ | | | | | | \\ \\ \\ \\ | | | | | | \\ \\ \\ \\ | | | | | | \\ \\ \\ | | | | | | }\$\$]{.mat | | | | | | h | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | [\$\\frac{2 | | | | [\$\$\\frac | | \\widehat{i | | | | {2\\widehat | | } | | | | {i} | | + | | | | + | | 4\\widehat{ | | | | 2\\widehat{ | | j} | | | | j}}{\\sqrt{ | | - | | | | 20}}\$\$]{. | | 2\\widehat{ | | | | math | | k}}{\\sqrt{ | | | |.display}\ | | 24}}\$]{.ma | | | | | | th | | | | | |.inline} | +-------------+-------------+-------------+-------------+-------------+ | 1. | **The value | | | | | | of** | | | | | | [**curl(gra | | | | | | df)**]{.mat | | | | | | h | | | | | |.inline}**, | | | | | | where** | | | | | | [**f** **=* | | | | | | * **2x**^** | | | | | | 2**^ **−**  | | | | | | **3y**^**2* | | | | | | *^ **+** ** | | | | | | 4z**^**2**^ | | | | | | ]{.math | | | | | |.inline} | | | | | | **is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | [4*x* − 6*y | B) | \ | | | | * + 8*z*]{. | | [\$\$\\over | | | | math | | rightarrow{ | | | |.inline} | | 0}\$\$]{.ma | | | | | | th | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | 3 | | | | [4*xî* − 6 | | | | | | *yĵ* + 8*z | | | | | | k̂*]{.math | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 2. | **A vector | | | | | | field which | | | | | | has a | | | | | | vanishing | | | | | | divergence | | | | | | is called | | | | | | as** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Solenoidal | B) | irrotationa | | | | field | | l | | | | | | field | +-------------+-------------+-------------+-------------+-------------+ | | C) | rotational | D) | scalar | | | | field | | field | +-------------+-------------+-------------+-------------+-------------+ | 3. | **In | | | | | | spherical | | | | | | coordinate | | | | | | system** | | | | | | [**(r,** ** | | | | | | θ,** **ϕ)** | | | | | | ]{.math | | | | | |.inline}**, | | | | | | **[ | | | | | | **θ** **=** | | | | | |  **a**]{.ma | | | | | | th | | | | | |.inline}**, | | | | | | where** | | | | | | [**a**]{.ma | | | | | | th | | | | | |.inline} | | | | | | **is a | | | | | | constant | | | | | | represents* | | | | | | * | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | A plane | B) | A cylinder | | | | | | with radius | | | | | | *a* | +-------------+-------------+-------------+-------------+-------------+ | | C) | A cone | D) | A Sphere | +-------------+-------------+-------------+-------------+-------------+ | 4. | **A unit | | | | | | tangent | | | | | | vector to | | | | | | the | | | | | | surface** | | | | | | [**x** **=* | | | | | | * **t,** ** | | | | | | y=e**^**t** | | | | | | ^**,** **z* | | | | | | * **=**  ** | | | | | | −** **3t**^ | | | | | | **2**^ | | | | | | ]{.math | | | | | |.inline}**a | | | | | | t** | | | | | | [**t** **=* | | | | | | * **0**   ] | | | | | | {.math | | | | | |.inline}**i | | | | | | s** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [*î* + *ĵ | | [\$\$\\frac | | | | *]{.math | | {1}{\\sqrt{ | | | |.display}\ | | 2}}\\wideha | | | | | | t{i} | | | | | | + | | | | | | \\frac{1}{\ | | | | | | \sqrt{2}}\\ | | | | | | widehat{j}\ | | | | | | $\$]{.math | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | None of | | | | [\$\$\\frac | | these | | | | {1}{\\sqrt{ | | | | | | 3}}\\wideha | | | | | | t{i} | | | | | | + | | | | | | \\frac{1}{\ | | | | | | \sqrt{3}}\\ | | | | | | widehat{j}\ | | | | | | $\$]{.math | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 5. | **If** | | | | | | [**ϕ=x**^** | | | | | | 2**^ **+**  | | | | | | **y** **−** | | | | | |  **z** **−* | | | | | | * **1,**  ] | | | | | | {.math | | | | | |.inline}**t | | | | | | hen** | | | | | | [\| **▽** * | | | | | | *ϕ**\| | | | | | | ]{.math | | | | | |.inline}**a | | | | | | t** | | | | | | [(**1,** ** | | | | | | 0,** **0**) | | | | | | ]{.math | | | | | |.inline}**i | | | | | | s** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | [\$\\sqrt{5 | B) | \ | | | | }\\ | | [\$\$\\sqrt | | | | \$]{.math | | {2}\$\$]{.m | | | |.inline} | | ath | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | 1 | | | | [\$\$\\sqrt | | | | | | {6}\$\$]{.m | | | | | | ath | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 6. | **If** | | | | | | [\$\\overri | | | | | | ghtarrow{\\ | | | | | | mathbf{r}}\ | | | | | | \mathbf{= | | | | | | x}\\widehat | | | | | | {\\mathbf{i | | | | | | }}\\mathbf{ | | | | | | + | | | | | | y}\\widehat | | | | | | {\\mathbf{j | | | | | | }}\\mathbf{ | | | | | | + | | | | | | z}\\widehat | | | | | | {\\mathbf{k | | | | | | }}\$]{.math | | | | | |.inline}**, | | | | | | then | | | | | | *div***[\$\ | | | | | | \overrighta | | | | | | rrow{\\math | | | | | | bf{r}}\\mat | | | | | | hbf{\\ | | | | | | }\$]{.math | | | | | |.inline}**= | | | | | | \_\_\_\_\_\ | | | | | | _\_\_\_\_\_ | | | | | | and** | | | | | | [\$\\mathbf | | | | | | {\\text{cur | | | | | | l}}\\overri | | | | | | ghtarrow{\\ | | | | | | mathbf{r}}\ | | | | | | \mathbf{=}\ | | | | | | $]{.math | | | | | |.inline}**\ | | | | | | _\_\_\_\_\_ | | | | | | \_\_\_** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | [*î* + *ĵ | B) | 3 and | | | | * + *k̂*]{. | | [\$\\overri | | | | math | | ghtarrow{0} | | | |.inline} | | \$]{.math | | | | and 0 | |.inline} | +-------------+-------------+-------------+-------------+-------------+ | | C) | 3 and 0 | D) | None of | | | | | | these | +-------------+-------------+-------------+-------------+-------------+ | 7. | **In | | | | | | cylindrical | | | | | | coordinate | | | | | | system** | | | | | | [**(r,** ** | | | | | | θ,** **z)** | | | | | | ]{.math | | | | | |.inline}**, | | | | | | if**[ **r** | | | | | |  **=** **a* | | | | | | *]{.math | | | | | |.inline}**, | | | | | | a constant | | | | | | then it | | | | | | represents* | | | | | | * | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | A plane | B) | A circle | +-------------+-------------+-------------+-------------+-------------+ | | C) | A cone | D) | A cylinder | | | | | | with radius | | | | | | *a* | +-------------+-------------+-------------+-------------+-------------+ | 8. | **If** | | | | | | [\$\\overse | | | | | | t{\\mathbf{ | | | | | | \\rightarro | | | | | | w}}{\\mathb | | | | | | f{F}}\\math | | | | | | bf{= | | | | | | 7}\\widehat | | | | | | {\\mathbf{i | | | | | | }}\\mathbf{ | | | | | | + | | | | | | 2}\\widehat | | | | | | {\\mathbf{j | | | | | | }}\\mathbf{ | | | | | | -}\\widehat | | | | | | {\\mathbf{k | | | | | | }}\\mathbf{ | | | | | | \\ | | | | | | }\$]{.math | | | | | |.inline}**t | | | | | | hen** | | | | | | [\$\\mathbf | | | | | | {\\nabla}\\ | | | | | | mathbf{.}\\ | | | | | | overset{\\m | | | | | | athbf{\\rig | | | | | | htarrow}}{\ | | | | | | \mathbf{F}} | | | | | | \\mathbf{= | | | | | | \\\_\\\_\\\ | | | | | | _\\\_\\\_\\ | | | | | | \_}\$]{.mat | | | | | | h | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | 6 | B) | 2 | +-------------+-------------+-------------+-------------+-------------+ | | C) | 4 | D) | 0 | +-------------+-------------+-------------+-------------+-------------+ | 9. | **If** | | | | | | [\$\\overri | | | | | | ghtarrow{\\ | | | | | | mathbf{r}}\ | | | | | | \mathbf{= | | | | | | 2}\\mathbf{ | | | | | | x}\\widehat | | | | | | {\\mathbf{i | | | | | | }}\\mathbf{ | | | | | | + | | | | | | 2}\\mathbf{ | | | | | | y}\\widehat | | | | | | {\\mathbf{j | | | | | | }}\\mathbf{ | | | | | | + | | | | | | 2}\\mathbf{ | | | | | | z}\\widehat | | | | | | {\\mathbf{k | | | | | | }}\$]{.math | | | | | |.inline} | | | | | | **then** | | | | | | [\$\\mathbf | | | | | | {\\nabla}\\ | | | | | | mathbf{.}\\ | | | | | | overset{\\m | | | | | | athbf{\\rig | | | | | | htarrow}}{\ | | | | | | \mathbf{r}} | | | | | | \\mathbf{= | | | | | | \\\_\\\_\\\ | | | | | | _\\\_\\\_\\ | | | | | | \_}\$]{.mat | | | | | | h | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | 1 | B) | 6 | +-------------+-------------+-------------+-------------+-------------+ | | C) | 0 | D) | 2 | +-------------+-------------+-------------+-------------+-------------+ | 10. | **If** | | | | | | [\$\\overse | | | | | | t{\\mathbf{ | | | | | | \\rightarro | | | | | | w}}{\\mathb | | | | | | f{F}}\\math | | | | | | bf{= | | | | | | yz}\\wideha | | | | | | t{\\mathbf{ | | | | | | i}}\\mathbf | | | | | | {+}\\mathbf | | | | | | {y}\^{\\mat | | | | | | hbf{2}}\\ma | | | | | | thbf{x}\\wi | | | | | | dehat{\\mat | | | | | | hbf{j}}\\ma | | | | | | thbf{+ | | | | | | z}\\widehat | | | | | | {\\mathbf{k | | | | | | }}\$]{.math | | | | | |.inline}**t | | | | | | hen | | | | | | the value | | | | | | of** | | | | | | [\$\\mathbf | | | | | | {\\nabla}\\ | | | | | | mathbf{\\bu | | | | | | llet}\\math | | | | | | bf{(}\\math | | | | | | bf{\\nabla} | | | | | | \\mathbf{\\ | | | | | | times}\\ove | | | | | | rset{\\math | | | | | | bf{\\righta | | | | | | rrow}}{\\ma | | | | | | thbf{F}}\\m | | | | | | athbf{) | | | | | | = | | | | | | \\\_\\\_\\\ | | | | | | _\\\_\\\_\\ | | | | | | \_}\$]{.mat | | | | | | h | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | 0 | B) | 1 | +-------------+-------------+-------------+-------------+-------------+ | | C) | 2 | D) | 3 | +-------------+-------------+-------------+-------------+-------------+ | 11. | **If**[ | | | | | | ]{.math | | | | | |.inline} | | | | | | [**φ** **=* | | | | | | * **xyz**]{ | | | | | |.math | | | | | |.inline}**, | | | | | | then | | | | | | at (2,2,2) | | | | | | the value | | | | | | of** | | | | | | [**∇φ** **= | | | | | | ** **\_\_\_ | | | | | | \_\_\_**]{. | | | | | | math | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [2*î* + *j | | [2*î* − 2* | | | | ̂* + 2*k̂*] | | ĵ* + 2*k̂* | | | | {.math | | ]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [4*î* + 4* | | [4*î* + 4* | | | | ĵ* + 4*k̂* | | ĵ* − 4*k̂* | | | | ]{.math | | ]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 12. | **If** | | | | | | [\$\\overse | | | | | | t{\\mathbf{ | | | | | | \\rightarro | | | | | | w}}{\\mathb | | | | | | f{F}}\\math | | | | | | bf{=}\\wide | | | | | | hat{\\mathb | | | | | | f{i}}\\math | | | | | | bf{+}\\wide | | | | | | hat{\\mathb | | | | | | f{j}}\\math | | | | | | bf{+ | | | | | | 3}\\widehat | | | | | | {\\mathbf{k | | | | | | }}\$]{.math | | | | | |.inline}**t | | | | | | hen** | | | | | | [\$\\mathbf | | | | | | {\\nabla.}\ | | | | | | \overset{\\ | | | | | | mathbf{\\ri | | | | | | ghtarrow}}{ | | | | | | \\mathbf{F} | | | | | | }\\mathbf{= | | | | | | \\\_\\\_\\\ | | | | | | _\\\_\\\_\\ | | | | | | \_}\$]{.mat | | | | | | h | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | 3 | B) | 0 | +-------------+-------------+-------------+-------------+-------------+ | | C) | 6 | D) | 1 | +-------------+-------------+-------------+-------------+-------------+ | 13. | **If** | | | | | | [\$\\overri | | | | | | ghtarrow{\\ | | | | | | mathbf{r}}\ | | | | | | \mathbf{= | | | | | | x}\\widehat | | | | | | {\\mathbf{i | | | | | | }}\\mathbf{ | | | | | | + | | | | | | y}\\widehat | | | | | | {\\mathbf{j | | | | | | }}\\mathbf{ | | | | | | + | | | | | | z}\\widehat | | | | | | {\\mathbf{k | | | | | | }}\$]{.math | | | | | |.inline} | | | | | | **then** | | | | | | [\$\\mathbf | | | | | | {\\nabla | | | | | | \\bullet}\\ | | | | | | overset{\\m | | | | | | athbf{\\rig | | | | | | htarrow}}{\ | | | | | | \mathbf{r}} | | | | | | \\mathbf{= | | | | | | \\\_\\\_\\\ | | | | | | _\\\_\\\_\\ | | | | | | \_}\$]{.mat | | | | | | h | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | 0 | B) | 2 | +-------------+-------------+-------------+-------------+-------------+ | | C) | 1 | D) | 3 | +-------------+-------------+-------------+-------------+-------------+ | 14. | **If**[ | | | | | | ]{.math | | | | | |.inline} | | | | | | [**φ** **=* | | | | | | * **x** **+ | | | | | | ** **y** ** | | | | | | +** **z**]{ | | | | | |.math | | | | | |.inline}**, | | | | | | then** | | | | | | [**∇φ** **= | | | | | | ** **\_\_\_ | | | | | | \_\_\_**]{. | | | | | | math | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [*î* + *ĵ | | [2*î* − *j | | | | * + *k̂*]{. | | ̂* + *k̂*]{ | | | | math | |.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [*î* − *ĵ | | [*î* − *ĵ | | | | * + *k̂*]{. | | * + 2*k̂*]{ | | | | math | |.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 15. | **If** | | | | | | [\$\\overse | | | | | | t{\\mathbf{ | | | | | | \\rightarro | | | | | | w}}{\\mathb | | | | | | f{F}}\\math | | | | | | bf{= | | | | | | 7}\\widehat | | | | | | {\\mathbf{i | | | | | | }}\\mathbf{ | | | | | | + | | | | | | 5}\\widehat | | | | | | {\\mathbf{j | | | | | | }}\\mathbf{ | | | | | | + | | | | | | 2}\\widehat | | | | | | {\\mathbf{k | | | | | | }}\$]{.math | | | | | |.inline}**t | | | | | | hen | | | | | | the value | | | | | | of** | | | | | | [\$\\mathbf | | | | | | {(\\nabla | | | | | | \\times}\\o | | | | | | verset{\\ma | | | | | | thbf{\\righ | | | | | | tarrow}}{\\ | | | | | | mathbf{F}}\ | | | | | | \mathbf{) | | | | | | = | | | | | | \\\_\\\_\\\ | | | | | | _\\\_\\\_\\ | | | | | | \_}\$]{.mat | | | | | | h | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [\$\$\\over | | [2*î* − 7* | | | | set{\\right | | ĵ* − 5*k̂* | | | | arrow}{0}\$ | | ]{.math | | | | \$]{.math | |.display}\ | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [2*î* − 7* | | [2*î* + 7* | | | | ĵ* + 5*k̂* | | ĵ* + 5*k̂* | | | | ]{.math | | ]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 16. | [\$\\mathbf | | | | | | {\\int}\\ov | | | | | | errightarro | | | | | | w{\\mathbf{ | | | | | | F}}\\mathbf | | | | | | {\\bullet | | | | | | d}\\overrig | | | | | | htarrow{\\m | | | | | | athbf{r}}\$ | | | | | | ]{.math | | | | | |.inline} | | | | | | **is | | | | | | independent | | | | | | of the path | | | | | | of | | | | | | integration | | | | | | if** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | [*F⃗*]{.mat | B) | [*F⃗* =  ▽  | | | | h | | *ϕ*,   *ϕ*] | | | |.inline} is | | {.math | | | | conservativ | |.inline} is | | | | e | | a scalar | | | | | | function | +-------------+-------------+-------------+-------------+-------------+ | | C) | [∮~*C*~*F⃗* | D) | (A), (B), | | | |    *dr⃗* =  | | (C) are | | | | 0]{.math | | true | | | |.inline} | | | +-------------+-------------+-------------+-------------+-------------+ | 17. | **Using the | | | | | | Green\'s | | | | | | theorem, | | | | | | the value | | | | | | of the line | | | | | | integral** | | | | | | [∫~**C**~** | | | | | | y**^**2**^* | | | | | | *dx+x**^**2 | | | | | | **^dy | | | | | | ]{.math | | | | | |.inline}**, | | | | | | where C is | | | | | | the | | | | | | boundary of | | | | | | the | | | | | | square** | | | | | | [ **−** **1 | | | | | | ** **≤** ** | | | | | | x** **≤** * | | | | | | *1,**  **−* | | | | | | * **1** **≤ | | | | | | ** **y** ** | | | | | | ≤** **1**]{ | | | | | |.math | | | | | |.inline} | | | | | | **is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | 0 | B) | \ | | | | | | [2(*x* + *y | | | | | | *)]{.math | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | 4 | D) | 4/3 | +-------------+-------------+-------------+-------------+-------------+ | 18. | **Suppose** | | | | | | [**S**]{.ma | | | | | | th | | | | | |.inline} | | | | | | **is an | | | | | | open, | | | | | | two-sided | | | | | | surface | | | | | | bounded by | | | | | | a simple | | | | | | closed | | | | | | curve C | | | | | | and** | | | | | | [\$\\overse | | | | | | t{\\mathbf{ | | | | | | \\rightarro | | | | | | w}}{\\mathb | | | | | | f{\\text{F\ | | | | | | \ | | | | | | }}}\$]{.mat | | | | | | h | | | | | |.inline}**h | | | | | | as | | | | | | continuous | | | | | | derivatives | | | | | |. | | | | | | Then by | | | | | | Stokes' | | | | | | theorem | | | | | | which of | | | | | | the | | | | | | following | | | | | | is true?** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | [\$\\int\_{ | B) | \ | | | | C}\^{}\\ove | | [\$\$\\int\ | | | | rset{\\righ | | _{C}\^{}\\o | | | | tarrow}{F} | | verset{\\ri | | | | \\bullet | | ghtarrow}{F | | | | d\\overset{ | | } | | | | \\rightarro | | \\bullet | | | | w}{r} | | d\\overset{ | | | | = | | \\rightarro | | | | \\iint\_{S} | | w}{r} | | | | \^{}\\left( | | = | | | | \\nabla | | \\iint\_{S} | | | | \\times | | \^{}\\left( | | | | \\overset{\ | | \\nabla | | | | \rightarrow | | \\bullet | | | | }{F} | | \\overset{\ | | | | \\right) | | \rightarrow | | | | \\bullet | | }{F} | | | | \\widehat{n | | \\right) | | | | }\\text{dS} | | \\bullet | | | | \$]{.math | | \\widehat{n | | | |.inline} | | }\\text{dS} | | | | | | \$\$]{.math | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | [\$\\int\_{ | D) | \ | | | | C}\^{}\\ove | | [\$\$\\int\ | | | | rset{\\righ | | _{C}\^{}\\o | | | | tarrow}{F} | | verset{\\ri | | | | \\bullet | | ghtarrow}{F | | | | d\\overset{ | | } | | | | \\rightarro | | \\times | | | | w}{r} | | d\\overset{ | | | | = | | \\rightarro | | | | \\iint\_{S} | | w}{r} | | | | \^{}\\left( | | = | | | | \\nabla | | \\iint\_{S} | | | | \\times | | \^{}\\left( | | | | \\overset{\ | | \\nabla | | | | \rightarrow | | \\times | | | | }{F} | | \\overset{\ | | | | \\right)\\t | | \rightarrow | | | | ext{dS}\$]{ | | }{F} | | | |.math | | \\right) | | | |.inline} | | \\bullet | | | | | | \\widehat{n | | | | | | }\\text{dS} | | | | | | \$\$]{.math | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 19. | **Gauss | | | | | | divergence | | | | | | theorem | | | | | | relates** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Line and | B) | Surface and | | | | surface | | volume | | | | integral | | integral | +-------------+-------------+-------------+-------------+-------------+ | | C) | Line and | D) | Line | | | | volume | | integral, | | | | integral | | surface | | | | | | integral | | | | | | and volume | | | | | | integral | +-------------+-------------+-------------+-------------+-------------+ | 20. | **Let** | | | | | | [**ϕ**]{.ma | | | | | | th | | | | | |.inline} | | | | | | **be a | | | | | | scalar | | | | | | potential | | | | | | of the | | | | | | conservativ | | | | | | e | | | | | | vector | | | | | | field** | | | | | | [\$\\overse | | | | | | t{\\mathbf{ | | | | | | \\rightarro | | | | | | w}}{\\mathb | | | | | | f{F}}\$]{.m | | | | | | ath | | | | | |.inline}**. | | | | | | Then the | | | | | | work done | | | | | | in moving a | | | | | | particle | | | | | | along a | | | | | | curve from | | | | | | point** | | | | | | [**A**]{.ma | | | | | | th | | | | | |.inline} | | | | | | **to is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [ *ϕ*(*B*)  | | [*ϕ*(*B*)]{ | | | | − *ϕ*(*A*)] | |.math | | | | {.math | |.display}\ | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | 0 | D) | \ | | | | | | [*ϕ*(*A*)]{ | | | | | |.math | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 21. | **Using | | | | | | Green's | | | | | | theorem in | | | | | | the plane, | | | | | | the value | | | | | | of the | | | | | | integral** | | | | | | [∫~**C**~(* | | | | | | *3x**^**2** | | | | | | ^ **−** **8 | | | | | | y**^**2**^) | | | | | | **dx+**(**4 | | | | | | y** **−** * | | | | | | *6**xy)dy]{ | | | | | |.math | | | | | |.inline} | | | | | | **where** | | | | | | [**C**]{.ma | | | | | | th | | | | | |.inline} | | | | | | **is the | | | | | | boundary of | | | | | | the region | | | | | | bounded | | | | | | by** | | | | | | [**x** **=* | | | | | | * **0,** ** | | | | | | x** **=** * | | | | | | *1,** **y** | | | | | |  **=** **0, | | | | | | ** **y** ** | | | | | | =** **1**]{ | | | | | |.math | | | | | |.inline} | | | | | | **is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | 10 | B) | 5 | +-------------+-------------+-------------+-------------+-------------+ | | C) | 25 | D) | 15 | +-------------+-------------+-------------+-------------+-------------+ | 22. | **Stoke's | | | | | | theorem | | | | | | relates** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | line and | B) | surface and | | | | surface | | volume | | | | integral | | integral | +-------------+-------------+-------------+-------------+-------------+ | | C) | line and | D) | line | | | | volume | | integral, | | | | integral | | surface | | | | | | integral | | | | | | and volume | | | | | | integral | +-------------+-------------+-------------+-------------+-------------+ | 23. | **Using | | | | | | Gauss | | | | | | divergence | | | | | | theorem, | | | | | | relation | | | | | | between | | | | | | surface and | | | | | | volume | | | | | | integral | | | | | | is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [∬~*S*~*F⃗* | | [∬~*S*~*F⃗* | | | |    *n̂*ds = | |    *n̂*ds = | | | |  ∭~*V*~ ▽   | |  ∭~*V*~ ▽   | | | | × *F⃗* | |  *F⃗* | | | | dV]{.math | | dV]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | [∫~*C*~*F⃗* | | | | [∬~*S*~*F⃗* | |    *dr⃗* =  | | | |    *n̂*ds = | | ∬~*S*~( ▽   | | | |  ∭~*V*~ ▽ * | | × *F⃗*)   * | | | | F⃗* | | ds⃗*]{.math | | | | dV]{.math | |.inline} | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 24. | **The Gauss | | | | | | divergence | | | | | | theorem | | | | | | uses which | | | | | | of the | | | | | | following | | | | | | operations? | | | | | | ** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Gradient | B) | Divergence | +-------------+-------------+-------------+-------------+-------------+ | | C) | Curl | D) | Laplacian | +-------------+-------------+-------------+-------------+-------------+ | 25. | **Which of | | | | | | the | | | | | | following | | | | | | theorem | | | | | | converts | | | | | | line | | | | | | integral to | | | | | | surface | | | | | | integral?** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Gauss | B) | Green\'s | | | | divergence | | theorem | | | | and | | only | | | | Stoke\'s | | | | | | theorem | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | Stoke\'s | D) | Stoke\'s | | | | theorem | | and | | | | only | | Green\'s | | | | | | theorem | +-------------+-------------+-------------+-------------+-------------+ | 26. | **By the | | | | | | Green\'s | | | | | | theorem, | | | | | | value of** | | | | | | [∮~**C**~** | | | | | | Pdx** **+** | | | | | |  **Qdy** ** | | | | | | =** **\_\_\ | | | | | | _\_\_\_**]{ | | | | | |.math | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [\$\$\\iint | | [\$\$\\iint | | | | \_{R}\^{}{( | | \_{R}\^{}{( | | | | \\frac{\\pa | | \\frac{\\pa | | | | rtial | | rtial | | | | Q}{\\partia | | Q}{\\partia | | | | l | | l | | | | x}} - | | x}} + | | | | \\frac{\\pa | | \\frac{\\pa | | | | rtial | | rtial | | | | P}{\\partia | | P}{\\partia | | | | l | | l | | | | y})dxdy\$\$ | | y})dxdy\$\$ | | | | ]{.math | | ]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [\$\$\\iint | | [\$\$\\iint | | | | \_{R}\^{}{( | | \_{R}\^{}{( | | | | \\frac{\\pa | | \\frac{\\pa | | | | rtial | | rtial | | | | P}{\\partia | | P}{\\partia | | | | l | | l | | | | x}} - | | x}} - | | | | \\frac{\\pa | | \\frac{\\pa | | | | rtial | | rtial | | | | Q}{\\partia | | P}{\\partia | | | | l | | l | | | | y})dxdy\$\$ | | y})dxdy\$\$ | | | | ]{.math | | ]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 27. | **The Gauss | | | | | | divergence | | | | | | theorem | | | | | | converts | | | | | | \_\_\_\_\_\ | | | | | | _\_\_\_\_** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Line to | B) | Line to | | | | surface | | volume | | | | integral | | integral | +-------------+-------------+-------------+-------------+-------------+ | | C) | surface to | D) | Surface to | | | | line | | volume | | | | integral | | integral | +-------------+-------------+-------------+-------------+-------------+ | 28. | **By | | | | | | Stoke\'s | | | | | | theorem, | | | | | | relation | | | | | | between the | | | | | | line and | | | | | | the surface | | | | | | integral is | | | | | | \_\_\_\_\_\ | | | | | | _\_\_\_\_\_ | | | | | | \_\_\_** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [\$\$\\oint | | [\$\$\\oint | | | | \_{C}\^{}\\ | | \_{C}\^{}\\ | | | | overset{\\r | | overset{\\r | | | | ightarrow}{ | | ightarrow}{ | | | | F} | | F} | | | | \\bullet | | \\bullet | | | | \\overset{\ | | \\overset{\ | | | | \rightarrow | | \rightarrow | | | | }{\\text{dr | | }{\\text{dr | | | | }} | | }} | | | | = | | = | | | | \\iint\_{S} | | \\iint\_{S} | | | | \^{}\\overs | | \^{}\\overs | | | | et{\\righta | | et{}{\\nabl | | | | rrow}{F} | | a} | | | | \\bullet | | \\bullet | | | | \\overset{\ | | \\overset{\ | | | | \land}{n}\\ | | \land}{n}\\ | | | | text{dS}\$\ | | text{dS}\$\ | | | | $]{.math | | $]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [\$\$\\oint | | [\$\$\\oint | | | | \_{C}\^{}\\ | | \_{C}\^{}\\ | | | | overset{\\r | | overset{\\r | | | | ightarrow}{ | | ightarrow}{ | | | | F} | | F} | | | | \\bullet | | \\bullet | | | | \\overset{\ | | \\overset{\ | | | | \rightarrow | | \rightarrow | | | | }{\\text{dr | | }{\\text{dr | | | | }} | | }} | | | | = | | = | | | | \\iint\_{S} | | \\iint\_{S} | | | | \^{}{(\\nab | | \^{}{(\\nab | | | | la\\varphi} | | la | | | | ) | | \\times | | | | \\bullet | | \\overset{\ | | | | \\overset{\ | | \rightarrow | | | | \land}{n}\\ | | }{F}}) | | | | text{dS}\$\ | | \\bullet | | | | $]{.math | | \\overset{\ | | | |.display}\ | | \land}{n}\\ | | | | | | text{dS}\$\ | | | | | | $]{.math | | | | | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 29. | **By the | | | | | | Gauss | | | | | | divergence | | | | | | theorem, | | | | | | relation | | | | | | between the | | | | | | surface and | | | | | | the volume | | | | | | integral | | | | | | is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [\$\$\\oint | | [\$\$\\iint | | | | \_{C}\^{}\\ | | \_{S}\^{}{\ | | | | overset{\\r | | \overset{\\ | | | | ightarrow}{ | | rightarrow} | | | | F} | | {F} | | | | \\bullet | | \\bullet}\\ | | | | \\overset{\ | | overset{\\l | | | | \rightarrow | | and}{n}dS | | | | }{\\text{dr | | = | | | | }} | | \\iiint\_{V | | | | = | | }\^{}{\\nab | | | | \\iint\_{S} | | la | | | | \^{}\\overs | | \\times | | | | et{}{\\nabl | | \\overset{\ | | | | a} | | \rightarrow | | | | \\bullet | | }{F}}\\text | | | | \\overset{\ | | {dV}\$\$]{. | | | | \land}{n}\\ | | math | | | | text{dS}\$\ | |.display}\ | | | | $]{.math | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [\$\$\\oint | | [\$\$\\iint | | | | \_{C}\^{}\\ | | \_{S}\^{}{\ | | | | overset{\\r | | \overset{\\ | | | | ightarrow}{ | | rightarrow} | | | | F} | | {F} | | | | \\bullet | | \\bullet}\\ | | | | \\overset{\ | | overset{\\l | | | | \rightarrow | | and}{n}dS | | | | }{\\text{dr | | = | | | | }} | | \\iiint\_{V | | | | = | | }\^{}{\\nab | | | | \\iint\_{S} | | la | | | | \^{}{(\\nab | | \\bullet | | | | la | | \\overset{\ | | | | \\times | | \rightarrow | | | | \\overset{\ | | }{F}}\\text | | | | \rightarrow | | {dV}\$\$]{. | | | | }{F}}) | | math | | | | \\bullet | |.display}\ | | | | \\overset{\ | | | | | | \land}{n}\\ | | | | | | text{dS}\$\ | | | | | | $]{.math | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 30. | **The | | | | | | function** | | | | | | [**f(z)=log | | | | | | ** **(z** * | | | | | | *+** **1)** | | | | | | ]{.math | | | | | |.inline} | | | | | | **is | | | | | | \_\_\_\_\_\ | | | | | | _** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Analytic at | B) | Not | | | | z=-1 | | analytic at | | | | | | z=-1 | +-------------+-------------+-------------+-------------+-------------+ | | C) | Differentia | D) | differentia | | | | ble | | ble | | | | at z=-1 | | everywhere | | | | | | except at | | | | | | the origin | +-------------+-------------+-------------+-------------+-------------+ | 31. | **Stoke\'s | | | | | | theorem | | | | | | uses which | | | | | | of the | | | | | | following | | | | | | operations? | | | | | | ** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Gradient | B) | Divergence | +-------------+-------------+-------------+-------------+-------------+ | | C) | Curl | D) | Laplacian | +-------------+-------------+-------------+-------------+-------------+ | 32. | **If** | | | | | | [**f(z)=u(r | | | | | | ,θ)+**iv**( | | | | | | r,θ)**]{.ma | | | | | | th | | | | | |.inline} | | | | | | **is | | | | | | analytic | | | | | | then** | | | | | | [\$\\frac{\ | | | | | | \mathbf{\\p | | | | | | artial | | | | | | u}}{\\mathb | | | | | | f{\\partial | | | | | | r}}\\mathbf | | | | | | {=}\$]{.mat | | | | | | h | | | | | |.inline} | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | [\$\\frac{1 | | | | [\$\$\\frac | | }{r}\\frac{ | | | | {\\partial | | \\partial | | | | v}{\\partia | | v}{\\partia | | | | l\\theta}\$ | | l\\theta}\$ | | | | \$]{.math | | ]{.math | | | |.display}\ | |.inline} | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | [\$\\frac{1 | | | | [\$\$- | | }{r}\\frac{ | | | | r\\frac{\\p | | \\partial | | | | artial | | v}{\\partia | | | | v}{\\partia | | l | | | | l\\theta}\$ | | r}\$]{.math | | | | \$]{.math | |.inline} | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 33. | **A | | | | | | function** | | | | | | [**u(x,y)** | | | | | |  ]{.math | | | | | |.inline}**i | | | | | | s | | | | | | said to be | | | | | | harmonic | | | | | | if** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [*u*~xx~ −  | | [*u*~xy~ +  | | | | *u*~yy~ = 0 | | *u*~yx~ = 0 | | | | ]{.math | | ]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [*u*~xx~ +  | | [*u*~xx~ =  | | | | *u*~yy~ = 0 | | *u*~yy~]{.m | | | | ]{.math | | ath | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 34. | **If** | | | | | | [**f(z)=x+* | | | | | | *ay**+i(**b | | | | | | x**+**cy**) | | | | | | **]{.math | | | | | |.inline} | | | | | | **is | | | | | | analytic, | | | | | | then the | | | | | | values of** | | | | | | [**a,b,c**] | | | | | | {.math | | | | | |.inline} | | | | | | **are** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | [*a* = 1]{. | B) | [*c* = 1]{. | | | | math | | math | | | |.inline} | |.inline} | | | | and | | and | | | | [*c* =  − * | | [*a* =  − * | | | | b*]{.math | | b*]{.math | | | |.inline} | |.inline} | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | [*b* = 1]{. | | | | [*a* = *b*  | | math | | | | = *c* = 1]{ | |.inline} | | | |.math | | and | | | |.display}\ | | [*a* =  − * | | | | | | c*]{.math | | | | | |.inline} | +-------------+-------------+-------------+-------------+-------------+ | 35. | **The | | | | | | bilinear | | | | | | transformat | | | | | | ion | | | | | | that maps | | | | | | the | | | | | | points** | | | | | | [**0,i,** * | | | | | | *∞** ]{.mat | | | | | | h | | | | | |.inline}**r | | | | | | espectively | | | | | | into** | | | | | | [**0,** **1 | | | | | | ,** **∞**]{ | | | | | |.math | | | | | |.inline} | | | | | | **is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [\$\$w = | | [*w* =  − * | | | | \\frac{1}{z | | z*]{.math | | | | }\$\$]{.mat | |.display}\ | | | | h | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [*w* = *iz* | | [*w* =  − * | | | | ]{.math | | iz*]{.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 36. | **If** | | | | | | [**f(z)** * | | | | | | *=** **u(x, | | | | | | ** **y)** * | | | | | | *+** **iv(x | | | | | | ,** **y)**  | | | | | | ]{.math | | | | | |.inline}**i | | | | | | s | | | | | | analytic | | | | | | then the | | | | | | family of | | | | | | curves** | | | | | | [**u(x,** * | | | | | | *y)=c**~**1 | | | | | | **~]{.math | | | | | |.inline} | | | | | | **and** | | | | | | [**v(x,** * | | | | | | *y)=c**~**2 | | | | | | **~ ]{.math | | | | | |.inline}**a | | | | | | re** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | Parallel to | B) | Coincide | | | | each other | | each other | +-------------+-------------+-------------+-------------+-------------+ | | C) | Intersect | D) | Orthogonal | | | | at any | | trajectorie | | | | angle | | s | | | | | | of each | | | | | | other | +-------------+-------------+-------------+-------------+-------------+ | 37. | **If** | | | | | | [**e**^ax^* | | | | | | *cos** **y* | | | | | | *]{.math | | | | | |.inline} | | | | | | **is | | | | | | harmonic, | | | | | | then the | | | | | | value of** | | | | | | [**a**]{.ma | | | | | | th | | | | | |.inline} | | | | | | **is** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [ ± *i*]{.m | | {.math | | | | ath | |.display}\ | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [ ± 1]{.mat | | {.math | | | | h | |.display}\ | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 38. | **If** | | | | | | [\$\\mathbf | | | | | | {w | | | | | | =}\\frac{\\ | | | | | | mathbf{az | | | | | | + | | | | | | b}}{\\mathb | | | | | | f{cz | | | | | | + | | | | | | d}}\$]{.mat | | | | | | h | | | | | |.inline} | | | | | | **is a | | | | | | bilinear | | | | | | transformat | | | | | | ion | | | | | | then** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | [ad − *bc*  | | | | [ab − dc ≠  | | = 0]{.math | | | | 0]{.math | |.inline} | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | [ab − *dc*  | | | | [ad − bc ≠  | | = 0]{.math | | | | 0]{.math | |.inline} | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ | 39. | **Cauchy -- | | | | | | Reimann | | | | | | equations | | | | | | are** | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | [*u*~*x*~ = | B) | [*u*~*x*~ = | | | |  *v*~*x*~]{ | |  *v*~*y*~ ] | | | |.math | | {.math | | | |.inline}and | |.inline} | | | | [*u*~*y*~ = | | and | | | |  *v*~*y*~]{ | | [*u*~*y*~ = | | | |.math | |   −  | | | |.inline} | | v~*x*~]{.ma | | | | | | th | | | | | |.inline} | +-------------+-------------+-------------+-------------+-------------+ | | C) | [*u*~*x*~ = | D) | [*u*~*x*~ = | | | |  *v*~*y*~]{ | |   − *v*~*y* | | | |.math | | ~]{.math | | | |.inline}and | |.inline}and | | | | [*u*~*y*~ = | | [*u*~*y*~ = | | | |  *v*~*x*~]{ | |   − *v*~*x* | | | |.math | | ~]{.math | | | |.inline} | |.inline} | +-------------+-------------+-------------+-------------+-------------+ | 40. | ### If the | | | | | | function f( | | | | | | z) satisfie | | | | | | s Cauchy-Ri | | | | | | emann equat | | | | | | ions, then | | | | | | the first d | | | | | | erivative o | | | | | | f f(*z*) is | | | | | | \_\_\_\_\_ | | | | | | \_\_\_\_\_ | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [*u*~*x*~ − | | [*u*~*x*~ + | | | |  *iv*~*x*~] | |  *iv*~*x*~] | | | | {.math | | {.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | | C) | \ | D) | \ | | | | [*iu*~*x*~  | | [*iu*~*x*~  | | | | − *v*~*x*~] | | + *v*~*x*~] | | | | {.math | | {.math | | | |.display}\ | |.display}\ | +-------------+-------------+-------------+-------------+-------------+ | 41. | ### Cauchy- | | | | | | Riemann equ | | | | | | ations in C | | | | | | artesian fo | | | | | | rm are \_\_ | | | | | | \_\_\_\_\_\ | | | | | | _\_\_\_\_ | | | | +-------------+-------------+-------------+-------------+-------------+ | | A) | \ | B) | \ | | | | [*u*~*x*~ = | | [*u*~*x*~ = |