## 2017_1 TMA Solutions - MTH382, Mathematical Methods IV

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### 2017_1 TMA Solutions - MTH382, Mathematical Methods IV

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

Posts: 1931
Joined: Thu Aug 10, 2017 2:00 am
Contact:

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

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
Posts: 1931
Joined: Thu Aug 10, 2017 2:00 am
Contact:

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

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
Posts: 1931
Joined: Thu Aug 10, 2017 2:00 am
Contact:

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

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 ââ‚¬¦ââ‚¬¦ââ‚¬¦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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