Past Papers’ Solutions  Assessment & Qualification Alliance (AQA)  AS & A level  Mathematics 6360  Pure Core 1 (6360MPC1)  Year 2005  January  Q#4
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Question
a. The function f is defined for all values of by .
i. Find the remainder when is divided by .
ii. Given that and , write down two linear factors of .
iii. Hence express as the product of three linear factors.
b. The curve with equation is sketched below.
i. The curve intersects the yaxis at the point . Find the ycoordinate of .
ii. The curve crosses the xaxis when , when and also at the point . Use the results from part (a) to find the xcoordinate of .
c.
i. Find
ii. Hence find the area of the shaded region bounded by the curve and the xaxis.
Solution
a.
i.
Remainder theorem states that if is divided by then;
For the given case is divided by .
We can write the divisor in standard form as;
Here and . Hence;
ii.
Factor theorem states that if is a factor of then;
For the given case;
Given 
Factor 
Factor 




Or


iii.
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:ii) we already have 02 factors and .
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 2^{nd} and 3^{rd} linear factors.
For the given case;
Already known factors from (a:ii) are and .
We may divide the given polynomial by any of these two factors. We choose .
Therefore, we get the quadratic factor for given polynomial. Now we factorize this quadratic factor.
Hence, can be written as product of three linear factors as follows;
b.
i.
Equation of the given curve is;
We are given that point is the yintercept of the given curve.
The point at which curve (or line) intercepts yaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Therefore, at ;
ii.
From (a:iii) we have;
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
Since for each root/factor , that means curve intersects xaxis at each factor.
Therefore, xintercepts of the given curve can be found from its factors;
c.
i.
Rule for integration of is:
Rule for integration of is:
Rule for integration of is:
ii.
To find the area of region under the curve , we need to integrate the curve from point to along xaxis.
It is evident from the diagram that shaded region lies under the curve and extends from to .
Therefore;
Rule for integration of is:
From (c:i) we have
Hence;
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