(i) Integrate the following functions by using the given substitution

- [substitution: ]
- [substitution: ]

(iii) If ,
prove that and hence
show that *y* increases as *x* increases in the domain .

(iv) By giving *x* an appropriate value in
prove that

(a) (b)

[ans: (i) (a) (b) ]

**Question 2**

(i) Use the substitution to show that

(ii) Factorise the polynomial ,
given that *P(x)* has a repeated root.

(iii) The polynomial *Qx)* has the form *Qx)
= P(x) (x+a)* with *P(x)* as in (ii) and where the constant *a*
is chosen so that *Q(x)* __>__ 0 for all real *x*. Find possible
values of *a*.

(iv) A particle undergoes a SHM about the origin
O. Its displacement *x* from O is given by
where *t* is the time (in seconds).

(a) Express the acceleration as a function of displacement

(b) write down the amplitude

(c) find *x* for which the speed is maximum
and this speed.

**Question 3**

(i) A meeting room contains a round table with 10 chairs. These chairs are equally spaced and indistinguishable.

(a) A committee of 10 persons included 3 teenagers are to seat. How many sitting arrangements can be done if the teenagers are to sit together ?

(b) Elections are held for the position of chairperson and secretary in a 2nd committee of 10 people seated around the table. What is the probability that the 2 elected people are sitting directly opposite each other ?

(ii) A scientist found that the amount Q(t) of
a substance present in a mineral at time *t* is satisfied the equation

(a) verify that *Q(t)* =
satisfied this equation for any constant A>0.

(b) If Q(0) = 10 mg, find the maximum value of Q(t) and the time at which this occurs.

(c) Describe what happen to Q(t) when t increases indefinitely.

**Question 4**

(i) If prove that .

(ii) A sequence (known as the Fibonacci sequence)
is defined as (*n *>
2). Prove by mathematical induction that

(iii) If .
Express in terms of *y*.
Hence solve the equation .

**Question 5**

(i) In a certain GP series, the sum of the first
4 terms is -5 and the sum of the first 8 terms is .
Show that and that .
Hence by division, show that
and that *r =* __+__1/2. Show that there are two possible sequences
and find them.

(ii) By considering a right angled triangle in
which the hypotenuse is 1 unit and the other sides is *x* units. Show
that for 0<x<1, .
Furthermore, prove by substitution that
for (a) *x* = 0 (b) 1<*x<*0.

(iii) Hence verify the result for the derivative of obtained in (ii).

**Question 6**

(i) Prove that

(ii) *P *and *Q* are points on a circle
and the tangents to the circle at P and Q meets at S. R is a pont on the
circle so that PR // QS. Prove that PQ = QR.

(iii) A circle has equation

(a) Find the centre and radius of the circle.

(b) The line *x + 2y = 0* meets the circle
at A and B. Find the coordinates of A and B and hence its length.

**Question 7**

(i) A wall 27 m high stands on a horizontal plane.
From a point O in this plane, a ball is projected in a plane at right angles
to the wall at an angle
with the horizontal, where *tan*
= 3/4. The ball just clears the top of the wall. If O is 96 m from the
wall, find the time taken to reach the wall and show that the speed of
projection is 40 m/s.

(a) Find the greatest height to which the ball will rise beyond the horizontal plane.

(b) Find the range.

(ii) The chord PQ in the parabola subtends a right angle at the vertex of the parabola. The normals to P and Q meets at R.

(a) Prove that *pq* = -4 where *p *and
*q* are parameters of the points P and Q.

(b) Show that as P and Q take varied positions on the parabola, the locus of R is .