3. (a) A projectile fired at an angle? has its horizontal (x) and vertical (y) distance as a function of time, t, given as:
x =utcos(?)
y =utsin(?) ?^{gt}_{2}^{2}
where g is the acceleration due to gravity, and u is the initial velocity of the projectile.
At what time does the projectile reaches the highest point in its trajectory? Express your answer in terms of u, g, and?
(b) Similarly, determine the maximum height.
(c) At what time after launch does the projectile hit the ground?
^{(d) }Evaluate^{dy} and^{d} ^{2}^{y}
dx dx^{2}
[3 marks] [3 marks] [3 marks]
[6 marks]
(e) Use that u = 100 m/s and g = 10 m/s^{2}. Make a sketch of the trajectory of the projectile for a launch at an angle with the horizon of?= 30° and in the
same sketch, add the trajectory for a launch angle?= 70°
(f) Show that the horizontal distance travelled by the projectile can be written as:
x =^{2}_{g}^{u}^{2}sin ?cos ?
For which value of? does the projectile travels furthest?
4. | (a) | Let z= 1+ j | and 1? j be two complex numbers. Determine the following: | ||||||
(i) z+ w | (ii) z? w | (iii) zw | (iv) | z | |||||
w | |||||||||
(b) Express the complex number?4 in modulus-argument form. Hence find all solutions z to the equation
z^{4}+4=0and mark them on an Argand diagram.
(c)
[5 marks]
[5 marks]
[9 marks]
[9 marks]
Page 3 of 5
y
P( t)
R
?t
x
Fig 3
5 (a)
(b)
(c)
(d)
Consider a simple harmonic oscillator whose motion is illustrated in the diagram above. We can describe the motion in complex form as
P(t)= Re^{j}^{?}^{t}.
Determine P(t)= Re^{j}^{?}^{t} in rectangular form for R= 2 and
(i)?t= | ? | (ii)?t= | ? | (iii)?t=? |
4 | 2 |
Solve the following linear system of equations: | ||||||
10x | + | y | ? 5z | = | 18 | |
?20x +3y + | 20z | =14 | ||||
5x | + | 3 y | + | 5z | = | 9 |
Determine AB if possible for the following matrices:
1 | 1 | |||||||||||||||||||||||
(a) | B= | (1 | 2 | 3 | 4) | (b) A= | B=(5 | 6) | ||||||||||||||||
A= | ^{,} | 2 | , | |||||||||||||||||||||
^{7} | 3 | |||||||||||||||||||||||
(c) | A=(1 | 3 | (d) | A = | 3 | B=(1 | 2) | |||||||||||||||||
2) , B= | 4 | 4 | ^{,} | |||||||||||||||||||||
?1 | 0 | 0 | ||||||||||||||||||||||
0 | ?1 | 0 | . DetermineA | 2013 | . | |||||||||||||||||||
LetA= | ||||||||||||||||||||||||
0 | 0 | ?1 | ||||||||||||||||||||||
1 | 1 | 0 | 0 | |||||||||||||||||||||
LetA=(1 | 2 | 3 | ), | B | = | 2 | 0 | 2 | 0 | . Evaluate ADB. | ||||||||||||||
and D= | ||||||||||||||||||||||||
3 | 0 | 0 | 3 | |||||||||||||||||||||
[7 marks]
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5 marks
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- (a) Evaluate the following:
- _{?}_{2}^{4} x^{3}dx
- _{?}_{4}^{2} x^{3}dx
6marks
(b) Let f( x)=_{?}_{0}^{3} e^{x}^{2} dx Determine the derivative f?( x) .
4 marks
- (i) Show that
_{dx}^{d} sin(x)ln(x)= cos( x) ln( x)+^{sin}_{x}^{(}^{x}^{)}
4 marks
- Find
? | cos( x) ln( x)+ | sin( x)_{dx} | |
x | |||
3 marks