Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2007 | June | Q#6

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Question

a.   The polynomial f(x) is given by  .

                                               i.    Use the Factor Theorem to show that (x-1) is a factor of f(x).

                                              ii.    Express f(x) in the form  , where p and q are integers.

                                            iii.    Hence show that the equation f(x)=0 has exactly one real root and state its value.

b.   The curve with equation  is sketched below.

The curve cuts the x-axis at the point A (1, 0) and the point B (2, 11) lies on the curve.

                                               i.    Find

                                              ii.    Hence find the area of the shaded region bounded by the curve and the line AB .

Solution

a.
 

            i.
 

Factor theorem states that if  is a factor of   then;

For the given case  is factor of .

Here  and . Hence;

Hence,  is factor of .

 

          ii.
 

If  is a polynomial of degree  then  will have exactly  factors, some of which may repeat.

We are given a polynomial of degree ,

Therefore, it will have 03 factors some of which may repeat.

From (a:i) we already have 01 factor  of .

We may divide the given polynomial with any of the already known linear factor(s) to get a quadratic  factor and then factorize the obtained quadratic factor to find the 2nd and 3rd linear  factors.

For the given case;

Already known factor from (a:i) is .

We may divide the given polynomial by this factor .

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Therefore, we get the quadratic factor  for given polynomial.

Hence,   can be written as product of linear and quadratic factors as follows;

 

         iii.
 

From (a:ii), we know that  can be written as; 

It is evident from this and (a:i) that (x-1) is a factor of f(x).

Therefore, one of the roots of f(x) can be found from;

Remaining roots of f(x) can be found from;

For a quadratic equation , the expression for solution is;

Where  is called discriminant.

If , the equation will have two roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

For the given equation;

Since , the equation will have no real roots. 

Hence, f(x) has only one real root.

 

b.
 

                           i.
 

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

 

                          ii.
 

It is evident from the diagram that;

To find the area of region under the curve , we need to integrate the curve from point  to   along x-axis.

We are given equation of the curve; however, we do not have equation of the line AB.

Let’s find equation of the line AB. We are given coordinates of two points on the line  and .

Two-Point form of the equation of the line is;

Therefore;

Therefore;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

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