2017_1 TMA Solutions - MTH382, Mathematical Methods IV

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2017_1 TMA Solutions - MTH382, Mathematical Methods IV

Postby admin » Mon Sep 25, 2017 12:51 am

1 solve \[\frac{d^{2} A}{d t^{2}}-4\frac{d A}{d t}-5A=0\]

\[A=C_{1} e^{5t}+C_{2} e^{-t}\]


2 Let \[A=x^{2}yzi-2xz^{3}j-xz^{2}\] and \, find \[\frac{\partial^{2}}{\partial x\partial y}(A\times B)\] at (1,0,-2)

\[-4i-8j\]


3 If \[A=\sin u i+\cos u j+ u k\],\ and \[C=2i+3j-k\], evaluate \[\frac{d}{d u}(A\times (B\times C))\] at u=0

\[7i+6j-6k\]


4 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot A)\]

\[100t^{3}+2t+6t^{5}\]


5 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\times B)\]

\[(t^{3}\sin t-3t^{2}\cos t)i-(t^{3}\cos t-3t^{2}\sin t)j +(5t^{2}\sin t-11t\cos t-\sin t)k\]


6 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot B)\]

\[(5t^{2}-1)\cos t +11 t \sin t\]


7 Determine the unit tangent at the point where t=2 on the curve \[x=t^{2}+1\],\[y=4t-3\] and \[z=2t^2-6t\].

\[\frac{2}{3}i+\frac{2}{3}j+\frac{1}{3}k\]


8 A particle moves along the curve \[x=2t^{2}\],\[y=t^{2}-4t\] and \[z=3t-5\],where t is the time. Find the components of the velocity at t=1 in the direction \[i-3j+2k\]

\[8\frac{\sqrt(14)}{7} \]


9 A particle moves along a curve whose parameter equations are \[x=e^{-t}\],\[y=2\cos 3t\],\[z=2\sin 3t\]. Find the magnitude of the acceleration at t=0

\[\sqrt(325)\]


10 Given that \[A=\sin ti+\cos tj+tk\], evaluate \[\left|\frac{d^{2} A}{d t^{2}}\right|\]

1


11 solve \[\frac{d^{2} A}{d t^{2}}-4\frac{d A}{d t}-5A=0\]

\[A=C_{1} e^{5t}+C_{2} e^{-t}\]


12 Let \[A=x^{2}yzi-2xz^{3}j-xz^{2}\] and \, find \[\frac{\partial^{2}}{\partial x\partial y}(A\times B)\] at (1,0,-2)

\[-4i-8j\ ]


13 If \[A=\sin u i+\cos u j+ u k\],\ and \[C=2i+3j-k\], evaluate \[\frac{d}{d u}(A\times (B\times C))\] at u=0

\[7i+6j-6k\]


14 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot A)\]

\[100t^{3}+2t+6t^{5}\]


15 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\times B)\]

\[(t^{3}\sin t-3t^{2}\cos t)i-(t^{3}\cos t-3t^{2}\sin t)j +(5t^{2}\sin t-11t\cos t-\sin t)k\]


16 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot B)\]

\[(5t^{2}-1)\cos t +11 t \sin t\]


17 Determine the unit tangent at the point where t=2 on the curve \[x=t^{2}+1\],\[y=4t-3\] and \[z=2t^2-6t\].

\[\frac{2}{3}i+\frac{2}{3}j+\frac{1}{3}k\]


18 A particle moves along the curve \[x=2t^{2}\],\[y=t^{2}-4t\] and \[z=3t-5\],where t is the time. Find the components of the velocity at t=1 in the direction \[i-3j+2k\]

\[8\frac{\sqrt(14)}{7}\]


19 A particle moves along a curve whose parameter equations are \[x=e^{-t}\],\[y=2\cos 3t\],\[z=2\sin 3t\]. Find the magnitude of the acceleration at t=0

\[\sqrt(325)\]


20 Given that \[A=\sin ti+\cos tj+tk\], evaluate \[\left|\frac{d^{2} A}{d t^{2}}\right|\]

1



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Posts: 1900
Joined: Thu Aug 10, 2017 2:00 am
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Re: 2017_1 TMA Solutions - MTH382, Mathematical Methods IV

Postby admin » Mon Sep 25, 2017 12:53 am

1 solve \[\frac{d^{2} A}{d t^{2}}-4\frac{d A}{d t}-5A=0\]

\[A=C_{1} e^{5t}+C_{2} e^{-t}\]


2 Let \[A=x^{2}yzi-2xz^{3}j-xz^{2}\] and \, find \[\frac{\partial^{2}}{\partial x\partial y}(A\times B)\] at (1,0,-2)

\[-4i-8j\]


3 If \[A=\sin u i+\cos u j+ u k\],\ and \[C=2i+3j-k\], evaluate \[\frac{d}{d u}(A\times (B\times C))\] at u=0

\[7i+6j-6k\]


4 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot A)\]

\[100t^{3}+2t+6t^{5}\]


5 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\times B)\]

\[(t^{3}\sin t-3t^{2}\cos t)i-(t^{3}\cos t-3t^{2}\sin t)j +(5t^{2}\sin t-11t\cos t-\sin t)k\]


6 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot B)\]

\[(5t^{2}-1)\cos t +11 t \sin t\]


7 Determine the unit tangent at the point where t=2 on the curve \[x=t^{2}+1\],\[y=4t-3\] and \[z=2t^2-6t\].

\[\frac{2}{3}i+\frac{2}{3}j+\frac{1}{3}k\]


8 A particle moves along the curve \[x=2t^{2}\],\[y=t^{2}-4t\] and \[z=3t-5\],where t is the time. Find the components of the velocity at t=1 in the direction \[i-3j+2k\]

\[8\frac{\sqrt(14)}{7} \]


9 A particle moves along a curve whose parameter equations are \[x=e^{-t}\],\[y=2\cos 3t\],\[z=2\sin 3t\]. Find the magnitude of the acceleration at t=0

\[\sqrt(325)\]


10 Given that \[A=\sin ti+\cos tj+tk\], evaluate \[\left|\frac{d^{2} A}{d t^{2}}\right|\]

1
admin
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Posts: 1900
Joined: Thu Aug 10, 2017 2:00 am
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Re: 2017_1 TMA Solutions - MTH382, Mathematical Methods IV

Postby admin » Mon Sep 25, 2017 12:55 am

1 If a and b are non-collinear vectors and \[A=(x+y)a+(2x+y+1)b\]

x=2,y=1


2 The following forces act on a particle P:\[F_{1}=2i+3j-5k\], \[F_{2}=-5i+j+3k\],\[F_{3}=i-2j+4k\],\[F_{4}=4i-3j-2k\], Find the magnitude of the resultant

\[2i-j\]


3 Given the scalar defined by \[\phi(x,y,z)=3x^{2}z-xy^{2}+5\],find \[\phi\] at the points (-1,-2,-3)

19


4 Find a unit vector parallel to the resultant vector \[A_{1}=2i+4j-5k\],\[A_{2}=1+2j+3k\]

\[\frac{3}{7}i+\frac{6}{7}j-\frac{2}{7}k\]


5 If \[A_{1}=3i-j-4k\], \[A_{2}=-2i+4j-3k\],\[A_{3}=i+2j-k\], find \[\left|3A_{1}-2A_{3}+4A_{3}\right|\]

\[\sqrt (398)\]


6 A car travels 3km due north, then 5km northeast. Determine the resultant displacement

7.43


7 Let a and b be vectors, then \[a \times b= ab\sin \theta\] is the __________

vector


8 Given that \[A_{1}=2i-j+k\],\[A_{2}=i+3j-2k\],\[A_{3}=3i+2j+5k\] and \[A_{4}=3i+2j+5k\],Find scalars a, b, c such that \[A_{4}=a A_{1}
+b A_{2}+c A_{3}\]

a=-2,b=1,c=-3


9 Given that \[A_{1}=3i-2j+k\],\[A_{2}=2i-4j-3k\],\[A_{3}=-i+2j+2k\], find the magnitudes of \[2A_{1}-3 A_{2}-5 A_{3}\]

\[\sqrt 30\ ]


10 Find the magnitude of vector \[A=3i-2j+2k\]

3


11 solve \[\frac{d^{2} A}{d t^{2}}-4\frac{d A}{d t}-5A=0\]

\[A=C_{1} e^{5t}+C_{2} e^{-t}\]


12 Let \[A=x^{2}yzi-2xz^{3}j-xz^{2}\] and \, find \[\frac{\partial^{2}}{\partial x\partial y}(A\times B)\] at (1,0,-2)

\[-4i-8j\ ]


13 If \[A=\sin u i+\cos u j+ u k\],\ and \[C=2i+3j-k\], evaluate \[\frac{d}{d u}(A\times (B\times C))\] at u=0

\[7i+6j-6k\]


14 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot A)\]

\[100t^{3}+2t+6t^{5}\]


15 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\times B)\]

\[(t^{3}\sin t-3t^{2}\cos t)i-(t^{3}\cos t-3t^{2}\sin t)j +(5t^{2}\sin t-11t\cos t-\sin t)k\]


16 If \[A= 5t^{2}+tj-t^{3}k\] and \. evaluate \[\frac{d}{dt}(A\cdot B)\]

\[(5t^{2}-1)\cos t +11 t \sin t\]


17 Determine the unit tangent at the point where t=2 on the curve \[x=t^{2}+1\],\[y=4t-3\] and \[z=2t^2-6t\].

\[\frac{2}{3}i+\frac{2}{3}j+\frac{1}{3}k\]


18 A particle moves along the curve \[x=2t^{2}\],\[y=t^{2}-4t\] and \[z=3t-5\],where t is the time. Find the components of the velocity at t=1 in the direction \[i-3j+2k\]

\[8\frac{\sqrt(14)}{7}\]


19 A particle moves along a curve whose parameter equations are \[x=e^{-t}\],\[y=2\cos 3t\],\[z=2\sin 3t\]. Find the magnitude of the acceleration at t=0

\[\sqrt(325)\]


20 Given that \[A=\sin ti+\cos tj+tk\], evaluate \[\left|\frac{d^{2} A}{d t^{2}}\right|\]

1
admin
Site Admin
Posts: 1900
Joined: Thu Aug 10, 2017 2:00 am
Contact:

Re: 2017_1 TMA Solutions - MTH382, Mathematical Methods IV

Postby admin » Mon Sep 25, 2017 12:58 am

1 If a and b are non-collinear vectors and \[A=(x+y)a+(2x+y+1)b\]

x=2,y=1


2 The following forces act on a particle P:\[F_{1}=2i+3j-5k\], \[F_{2}=-5i+j+3k\],\[F_{3}=i-2j+4k\],\[F_{4}=4i-3j-2k\], Find the magnitude of the resultant

\[2i-j\]


3 Given the scalar defined by \[\phi(x,y,z)=3x^{2}z-xy^{2}+5\],find \[\phi\] at the points (-1,-2,-3)

19


4 Find a unit vector parallel to the resultant vector \[A_{1}=2i+4j-5k\],\[A_{2}=1+2j+3k\]

\[\frac{3}{7}i+\frac{6}{7}j-\frac{2}{7}k\]


5 If \[A_{1}=3i-j-4k\], \[A_{2}=-2i+4j-3k\],\[A_{3}=i+2j-k\], find \[\left|3A_{1}-2A_{3}+4A_{3}\right|\]

\[\sqrt (398)\]


6 A car travels 3km due north, then 5km northeast. Determine the resultant displacement

7.43


7 Let a and b be vectors, then \[a \times b= ab\sin \theta\] is the ………product

vector


Given that \[A_{1}=2i-j+k\],\[A_{2}=i+3j-2k\],\[A_{3}=3i+2j+5k\] and \[A_{4}=3i+2j+5k\],Find scalars a, b, c such that \[A_{4}=a A_{1}
+b A_{2}+c A_{3}\]

a=-2,b=1,c=-3


9 Given that \[A_{1}=3i-2j+k\],\[A_{2}=2i-4j-3k\],\[A_{3}=-i+2j+2k\], find the magnitudes of \[2A_{1}-3 A_{2}-5 A_{3}\]

\[\sqrt 30\ ]


10 Find the magnitude of vector \[A=3i-2j+2k\]

3


11 If a and b are non-collinear vectors and \[A=(x+y)a+(2x+y+1)b\]

x=2,y=1


12 The following forces act on a particle P:\[F_{1}=2i+3j-5k\], \[F_{2}=-5i+j+3k\],\[F_{3}=i-2j+4k\],\[F_{4}=4i-3j-2k\], Find the magnitude of the resultant

\[2i-j\]


13 Given the scalar defined by \[\phi(x,y,z)=3x^{2}z-xy^{2}+5\],find \[\phi\] at the points (-1,-2,-3)

19


14 Find a unit vector parallel to the resultant vector \[A_{1}=2i+4j-5k\],\[A_{2}=1+2j+3k\]

\[\frac{3}{7}i+\frac{6}{7}j-\frac{2}{7}k\]


15 If \[A_{1}=3i-j-4k\], \[A_{2}=-2i+4j-3k\],\[A_{3}=i+2j-k\], find \[\left|3A_{1}-2A_{3}+4A_{3}\right|\]

\[\sqrt (398)\]


16 A car travels 3km due north, then 5km northeast. Determine the resultant displacement

7.43


17 Let a and b be vectors, then \[a \times b= ab\sin \theta\] is the __________

vector


18 Given that \[A_{1}=2i-j+k\],\[A_{2}=i+3j-2k\],\[A_{3}=3i+2j+5k\] and \[A_{4}=3i+2j+5k\],Find scalars a, b, c such that \[A_{4}=a A_{1}
+b A_{2}+c A_{3}\]

a=-2,b=1,c=-3


19 Given that \[A_{1}=3i-2j+k\],\[A_{2}=2i-4j-3k\],\[A_{3}=-i+2j+2k\], find the magnitudes of \[2A_{1}-3 A_{2}-5 A_{3}\]

\[\sqrt 30\ ]


20 Find the magnitude of vector \[A=3i-2j+2k\]

3


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