3.            (a) A projectile fired at an angle α has its horizontal (x) and vertical (y) distance as a function of time, t, given as:

 

x   = utcos(α)

 

y   = utsin(α)gt22

 

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 α

 

(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/s2. Make a sketch of the trajectory of the projectile for a launch at an angle with the horizon of α= 30° and in the

 

same sketch, add the trajectory for a launch angle α= 70°

 

(f)             Show that the horizontal distance travelled by the projectile can be written as:

x   = 2gu2sin α cos α

 

For which value of α 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

 

z4+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

 

ωt

 

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 jωt.

 

Determine P (t)= Re jωt  in rectangular form for R = 2 and

 

(i) ωt=

π

(ii) ωt=

π

(iii) ωt=π

 

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

 

. Determine A

2013

.

 

 

 

 

 

 

 

 

Let A=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

 

0

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

1

0

 

0

 

 

 

 

 

Let A=(1

2

 

3

) ,

B

=

 

2

 

 

 

0

2

 

0

 

. Evaluate ADB.

 

 

 

 

 

and D =

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

 

 

0

0

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

[7 marks]

 

 

 

 

 

 

 

 

 

10 marks

 

 

 

 

 

 

 

 

 

8 marks

 

 

 

 

2 marks

 

 

 

 

 

 

5 marks


 

 

 

 

 

 

 

 

 

Page 4 of 5


 

 

  1. (a) Evaluate the following:

     

    1. 24 x3dx

    2. 42 x3dx

       

      6                                                                                                                                             marks

       

      (b) Let f ( x)=03 ex2 dx Determine the derivative f ( x) .

       

      4 marks

       

 

  1. (i) Show that

     

    dxd sin (x)ln (x)= cos ( x) ln ( x)+sinx(x)

    4 marks

     

 

  1. Find

     

 

cos ( x) ln ( x)+

sin( x)dx

 

 

x

 

 

 

 

 

 

3 marks

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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