3
Sec 4 EOY Math Exam Paper 2
Answer all the Questions.
1
Differentiate the following with respect to x
4
1

(a)  3x 2   ,
x

(b) ln
e
4x
1  x 1  x  .
[3]
f( x) dx  e 4 x cos 4 x  c where c is a constant, find an expression for f( x) .
[3]
2
If
3
Given that
4
[2]
5
1
1
5
 f ( x) dx  10 , evaluate
 f ( x)  x dx .
Without using calculator, list the following in order of size, starting with the smallest,
2320 ,
5
[4]
3240 ,
5160
[3]
Differentiate tan3 8x with respect to x.
[2]
Hence find
6
(i)
 tan
2
8 x sec2 8 x dx ,
[2]
(ii)
 sec
4
8x dx .
[3]
(a) The coefficient of x3 in the expansion of  2  ax 1  3x  is 405.
Find the value of a.
6
1
(b) Find the coefficient of
x
7
3
[4]
9
1 

in the expansion of  2 x 2   .
2x 

[4]
(a) The function f is defined by f : x  8  ( x  3)2 for the domain 0  x  4 .
(i) Find the range of f.
[3]
(ii) Determine whether function f has an inverse, stating your reason.
[1]
(iii) Write down the largest possible domain that gives the same range as in part (i).
[1]
(b) The function g is defined by g : x  8  ( x  3)2 for the domain x  k.
Write down the smallest value of k for which g 1 exists.
[1]
4
8
A liquid is heated for N minutes to M C and then allowed to cool. The diagram shows how the
temperature of the liquid varies during the heating and the cooling processes. The temperature, T ,
in C, of the liquid is given by T  e x2 for 0  x  N and T  e223x for x  N where x is the
time in minutes after heating started.
Calculate
9
(i) the initial temperature of the liquid,
[1]
(ii) the value of N,
[2]
(iii) the value of M.
[2]
(a) Find all angles between 0 and 360 for which
(i) sin( x  30)  2 cos x ,
(ii) sin y cos y 
[4]
1
.
4
[4]
(b) Solve cos3z  cos 2z  cos z  0 for 0  z   .
10 ABCD is a parallelogram. The point E on BD is such that BE 

and AD  4q .
[5]

1
BD . Given that AB  4p
4
(a) Express the following in terms of p and/or q,

(i) CD ,

(ii) BD ,

(iii) AE .
(b) AE produced meets BC at the point X such that 3AX  4 AE . Given that BX  hBC ,
BX
find the value of h. Hence find the numerical value of
.
XC
[1]
[1]
[1]
[3]
5
11 (a) The diagram shows part of the graph of y  x 2 and the tangent to the curve at x  2 .
(i) Find the equation of the tangent at x  2 .
[2]
(ii) Find the area bounded by the curve, the tangent and the x-axis.
[4]
y
x
(b) The curve y  f( x ) is a strictly increasing function passing through the origin.
2
8
Given that the point (2, 8) lies on the curve, find the value of  y dx   x dy .
0
[2]
0
12 A particle P moves in a straight line so that at time t seconds after leaving a fixed point O its
velocity, v ms1 , is given by v  5  8e2t .
(i) Find an expression for the acceleration of the particle.
[2]
(ii) Find an expression for the displacement of the particle from O.
[3]
(iii) Find the distance of the particle from O at the instant at which the particle is
instantaneously at rest.
[3]
(iv) Particle Q travels at a constant speed of 5 ms1 .
Explain briefly whether particle P can travel faster than particle Q.
[2]
6
13 (a) Show that
1  sec 2
 cot  for all values of  .
tan 2
[4]
(b) The diagram shows part of the graph of y  cos x  sin x .
1
The graph intersects the x  axis at Q and the line y 
at R.
2
P is a minimum point on the curve. Find the coordinates of P, Q and R.
You may leave your answers in surd form.
[9]
y
R
Q x
P
14 A man lives on an island, denoted by A, located at 2 km from a point B on the mainland. Everyday he
takes a ferry from A to C, then walks from C to D. The distance between point B and point D is 3 km.
The ferry travels at a constant rate of 2 kmh 1 and the man walks at a constant rate of 4 kmh 1 .
A
B
C
D
Given that BC = x km, show that
dT
x
1


dx 2 x 2  4 4
where T is the total time, in hours, taken for the man to travel from A to D.
[3]
How far should point C be from point B so that he can reach point D in the shortest time
possible?
[4]
Show clearly why your answer will give the shortest time possible.
[2]
End of Paper