Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q# 9

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Question

The function  is defined by  for .

     i.       Express  in the form .

   ii.       State the range of .

  iii.        Find the set of values of  for which .

The function g is defined by  for .

  iv.       Find the value of the constant  for which the equation  has two equal roots.

Solution


i.
 

We have the expression;

We use method of “completing square” to obtain the desired form. We take out factor
‘2’ from the terms which involve
;

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.
 

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 given curve  , is a parabola opening upwards.

Vertex form of a quadratic equation is;

The given curve , as demonstrated in (i) can be written in vertex form as;

Coordinates of the vertex are .Since this is a parabola opening upwards the vertex is the minimum point on the graph.
Here y-coordinate of vertex represents least value of
 and x-coordinate of vertex represents corresponding value of .

For the given case, vertex is . Therefore, least value of  is -11 and corresponding value of  is 3.

Hence range of ;


iii.
 

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.

Since, as demonstrated in (ii) it is an upwards opening parabola.

Therefore conditions for  are;


iv.
 

We have the functions;

We have;

Standard form of quadratic equation is;

Expression for discriminant of a quadratic equation is;

If   ;        quadratic equation has two real roots.

If   ;        Quadratic equation has no real roots.

If   ;  Quadratic equation has one real root/two equal roots.

For the given case;

We are given that the equation  has two equal roots, therefore;

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