# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#1

Question

i.
Express  in the form , where a and b are constants.

ii.   Hence, or otherwise, find the set of values of  for which .

Solution

i.

We have the expression;

We use method of “completing square” to obtain the desired form.

Next we complete the square for the terms which involve .

We have the algebraic formula;

For the given case we can compare the given terms with the formula as below;

Therefore we can deduce that;

Hence we can write;

To complete the square we can add and subtract the deduced value of ;

ii.

To find the set of values of x for which;

We solve the following equation to find critical values of ;

Now we have two options;

Hence the critical points on the curve for the given condition are 1 & -7.

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph.
If
(‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the  graph.

We recognize that  is an upwards opening parabola.

Therefore conditions for  are;