Principal Math’s Mixed Questions to be done 2020

during the COVID-19 holiday break

MBABAZI RODNEY NIWAGABA
S6 PCM/ICT

SECTION A (100 MARKS)

1. a) Prove the formula cos3 A  4 cos3 A  3 cos A. (Formulae for Cos2A and sin2A) may be
assumed). Substitute x  12  cos in the equation 8 x 3  12x 2  1  0 and with aid of the above
formula; solve the resulting equation in  giving values between 00 and 1800 . Hence find the three
roots of the cubic equation in x giving the two correct to two decimal places.
1 1
b) Given that sin  sin  a and cos  cos  b , show that cos2 (   )  (a 2  b 2 )
2 4
c) The parametric equation of a curve are x  3(2  sin 2 and y  3(1  cos 2 )

The tangent and normal at the point P where   meets the Y axis at L and X axis at M
4
9
respectively. Show that the area of the triangle PLM is given by (  2) 2
4

2. Find the equation of the line through P(1,0,1) and Q(3,2,1) in form of r  a  b
3. A curve has parametric equations x  cos 2 , y  cos .
dy 1
a) Show that  .
dx 4 y
b) State the domain and range of the curve
c) By eliminating  , find the Cartesian equation of the curve
d) Sketch the curve

2
4. i) Estimate the value of  x 2 dx using four trapeziums of equal width.
1
ii) Find the true value of the integral.
iii) Show that the error in your estimate is less than 0.5%
x
b) If z  and x and y have possible errors of x and y f show that the maximum possible
y
x y
error in z, z is given by z  x 2 .
y y
5. a) Show that the circle on the line joining the points ( x1, y1 ) and (x2 , y2 ) as diameter has the
equation ( x  x1 )(x  x2 )  ( y  y1 )( y  y2 )  0
c
b) Prove that the normal to the rectangular parabola xy=c2 at a point A (ct , ) meets the hyperbola
t
c
again at a point B ( 3 ,ct )
3

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 Principal Math’s Mixed Questions to be done 2020
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f ( )
6. i) If f ( )  sec  tan  , show that  sec .
f ( )
ii) If t  tan( 2 ), show that
dt 1
 (1  t 2 )
d 2
1 t 1 t2
iii) show that f ( )  b and sec( ) 
1 t 1 t2

iv) Find the integral  sec dx in two ways firstly by making use of f ( ) and secondly by

making use of t  tan( 2 ) and show that your answers are equivalent.
7. If  ,  and  are roots of the equation ax3  bx2  cx  d  0, prove that
b c d
i)  +  + = ii)      = iii)  =
a d a
b)  ,  and  are roots of the equation 12x  28x  3x  18  0, state values of
3 2

 +  + ,      , 
c) If the equation has two equal roots, solve the equation completely

8. a) Prove that when a polynomial f (x ) is divided by ( x  a) the remainder is f (a ) , when f (x ) is

divided by ( x  a )( x  b) is written in the form A( x  a)  B( x  b) prove that provided ( a  b)
f (b) f (b)
A and B 
ba ab
b) When a polynomial ax3  bx2  cx  d is divide by x 2  1, the remainder is kx where k is a
1 1
constant. When it is divided by x 2  4, is prove that 𝑎 = −𝑏 = 4 𝑐 = 𝑑 = − 3 𝑘
9. a) Sketch the curve y 2  x 2 (2  x)
b) A uniform solid of revolution is formed by rotating about the x-axis the area enclosed by the
curve and the positive x-axis.
i) Find the volume of this solid
ii) Find also the coordinates of the centre of mass given that the solid has uniform density

10. A rectangular metal sheet measuring 8cm by 5cm is to be converted into an open box. Equal squares of size
3
x are removed from each corner, and edges then turned up to make an open box of volume Vcm . Show
that V  40x  26x  4 x . Hence find the maximum possible volume and the corresponding value of x
2 3

11. Find the equation of the line through the intersection of the lines 3x  4 y  6  0 and
5 x  y  y  13  0
Which
(i) passes through the point (2,4),
(ii) Makes an angle of 600 with the x-axis.
12. Show that (cos  j sin )  (cos  j sin )1
Show that the points which represent the roots of the equation ( z  1)  z  0 lie on a line which is
n n

parallel to the imaginary axis

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 Principal Math’s Mixed Questions to be done 2020
during the COVID-19 holiday break
SECTION B

NUMERICAL METHODS
x2 x4 x6
13. The valves of cos x can be found from the infinite series cos x  1     .....
2! 4! 6!
i) Form a flow chart diagram that can be used to read in a value of x and print out the
corresponding of cos x correct to degree of accuracy  .
ii) Perform a dry run considering 5 iterations giving your answer to 4 significant figures
dy
14. Evaluate y (0.5) given that  y  x 2 and y (0)  1 using a simple step- by -step method with
dx
a step of 0.1
Given that the solution to the given equation y  3e x  (2  2 x  x 2 ), show that the step-by step
method gives you an under estimate of the value of the value of y (0.5) which is less than a 4%
error
15. i) On the same axes with the same scale draw accurately the graph of y  ln x and
y  ( x  3)( x  2) for 1  x  4
ii) From your graph, estimate to one decimal place the two roots of the equation
ln x  ( x  3)( x  2)  0
iii) Use Newton’s method twice to calculate a better approximation for the larger of the roots
iv) Form a flow chart diagram that can be used to read in a value of x and print out the
corresponding root using the small of the two roots.
v) Perform three integrations using the smaller of roots an give your answer to 3 significant
figures

DIFFERENTIAL EQUATIONS AND APPLICATION

16. The number of bacteria in a culture is increasing at a rate which is proportional to the number
present. The initial count of bacteria was 500 and two hours later was 800.
(a) How many bacteria will be expected to be present after another hour?
(b) How many hours after the initial count will it take for the count to reach 1500?

dy 1 1 3
17. i) Solve the differential equation  x 3  2  , given that x  1 when y  
dx x x 4
dy 6
ii) Solve the differential equation  3 y  e 2 x given that that x  0 when y 
dx 5

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 Principal Math’s Mixed Questions to be done 2020
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18. The temperature of a cooling liquid at any time t is decreasing at a rate proportional to (T-To),
where To is the constant temperature of the surroundings.
i) Form a differential equation and use it to show that (T-To) is proportional to e  t where 
is a positive constant
1
ii) Initially the temperature T=1200C and To = 200C. Given that    and t is measured in
2
minutes, show that the temperature of the liquid will differ from the surroundings
temperature by less than 10C after 4 ln10 minutes.

di
19. The current, I in an RL circuit satisfies the equation L  Ri  E , where R is the resistance, L
dt
the inductance and E is the e.m.f. In a certain RL circuit with no source of e.m.f. the initial current is
I0. Find an expression for I in terms of L, R, and I0.

If the circuit is provided with an e.m.f. source of 4 sin t volts and R=100Ω, L=4 henries, determine
the expression for the current flowing at any time given that initially there is no current.

20. (a) The rate at which the number M of teachers decreases is found to be proportional to the
number present at any time t.
i) Write down the differential equation describing the process
ii) Given that initially there were 2000 mathematics teachers and the number reduced to
half in ten years, find the number of teachers after 15 years.

(b) Show that equation x(3xy  4 y 3  6)dx  ( x3  6 x 2 y 2  1)dy  0 is exact. Hence solve the
equation
d 2 y 2dy
(c) Solve the initial value problem   3 y  0, y (0)  0, y(0)  3
dx2 dx
(d) Prove that when y  at 2  2bt  c, t  ax 2  2bx  c where a,b,c are constant
d3y
3
 24a 2 (ax  b)
dx

MECHANICS
21. a) An aeroplane P is heading due west with a speed of relative to the air of 192kmh-1. A wind speed of
80kmh-1 is blowing from the south to north.
i) Calculate the magnitude and velocity of P relative to the earth.
ii) In what direction should the pilot of head in order to travel due west
b) C is a point 40 km south-east of a point A, and B is a point 100Km east of A. in still air, and aircraft Q is
travelling at 300km-1 due west towards A. When Q is at B, another aircraft R having airspeed of 500Kmh-1
sets of from C to intercept Q as quickly as possible.
i) Find by scale drawing the direction in which R must head.
ii) Find by trigonometry calculation the direction in which R must head.

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 Principal Math’s Mixed Questions to be done 2020
during the COVID-19 holiday break
22. A uniform wooden block of mass M in the form of a cuboid is suspended by four equal vertical weightless
strings attached to the four corners of its horizontal top face.

A bullet of mass m, moving with velocity v, strikes the centre of a vertical face at right angles and becomes
embedded in the block.

The block, with the embedded bullet, after swinging, first comes to rest at a height h above its original
position. Show that the original velocity of the bullet is given by;
mM
v (2 gh) .
m

Given that M = 2kg and m = 2grams show that approximately 99.9% of the original KE is lost in the
impact.

23. (i) A body of mass 100kg is being dragged along a rough horizontal surface by a force of 50 N
making a constant angle of tan 1 34 with the surface.

The frictional force between the body and the surface is 25N. Find the increase in the kinetic
energy of the body when it moves a distance of 20m along the surface. Hence find the velocity at
the end of 20m if the initial velocity was 0.5m s 1

(ii) A body of mass 50kg rests on a horizontal table, the coefficient of friction between body and
table being 0.2. A force T, at a constant angle of tan 1 34 to the table is required to drag the body
along at a constant speed.

Express the normal reaction between the table and body in terms of T. Hence find T.

24. A particle is projected at an angle of  to the horizontal. Air resistance produces a retardation
which is equal to k times the velocity of the particle. The particle returns to the horizontal plane
through the point of projection after a time t and hits the plane at an angle  to the horizontal. If the
initial Velocity is V, Show that the range is
V cos
(1  e  kt )
k
gt  Vsi
and that t an  (1  e  kt )
V cose  kt

25. The force F acting on a particle P of mass m is given as a function of the time t by the equation
 cost 
F   
 sin t 
0 
The particle starts at rest at the point (c,0) with an initial velocity  
V0 
Find the equation, in terms of t, the path of the particle.

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 Principal Math’s Mixed Questions to be done 2020
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26. P is a point on a horizontal plane. A particle is projected from P with a speed of 50m/s at an angle
3
of tan 1 to the horizontal.
4
b) Find the position of the particle and the magnitude and direction of its velocity after 2 seconds.
c) Find the time at which the particle reaches the highest point of its flight and the height of this
point
d) Calculate the distance from P of the point R at which the particle hits the plane
e) Q is a point on PR such that PQ = 150m. Find the height of the particle when it passes over Q

PROBABILITY AND STATISTCS

27. The probability density function of a random variable X is
ka( x 2  2)  a  x0

f ( x )  ka( x  a ) 0 xa
0 else where.


(i) Sketch the graph of f(x).

(ii) Determine the distribution function F(x) and the value of k. Hence evaluate E(X) in terms of a only.

28. The random variable X has a probability density function f (x ) defined as

k (1  x) 2 0  x 1
f ( x)  
0 elsewhere.

(iii) Show that k = 3

1
(iv) Given that evaluate E(X) = calculate the variance of x.
4
(v) Determine the (cumulative) distribution function F(x) of X. Hence show that the median of this
1
distribution is 1  2 3

29. The probability density function of a random variable X is

ae bx 0  x  10
f ( x)  
0 else where.

(vi) Given that E(X)= 300, find a and b

(vii) Determine the median and the variance of X.

GENERAL QUESTIONS (MIXED)

30. One side of the hill has the form of an inclined plane The line of the greatest slope AB being due north A
man walks from A to B the distance being d Km at B, the man walks for a distance r Km in a straight lone
at an acute angle  to AB reaching a point C. The vertical heights of B and C above A are h Km and k Km
respectively
d ( k  h)
a) Prove that cos 
hr
hr sind
b) The bearing of C from A is  east or west of north. Prove that tan  
k (d 2  h 2 ) 2
1

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