Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | Oct-Nov | (P2-9709/02) | Q#8

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Question

     i.
 

                  a.   Prove the identity

                 b.   Hence prove that

   ii.       By differentiating , show that if  then .

 

  iii.       Using the results of parts (i) and (ii), find the exact value of

 

Solution

     i.
 

       a.
 

              We are required to show that;

We know that;

  provided that

 

b.
 

We are required to prove that;

We have demonstrated in i(a) that;

Therefore;

We have the trigonometric identity;

We have algebraic formula;

   ii.
 

We are given that;

We are required to show that;

We know that;

  provided that

Therefore;

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is:

Therefore;

If  and  are functions of , and if , then;

If , then;

Rule for differentiation of  is:

Rule for differentiation of  is;

We know that;

  provided that

  iii.
 

From i(b) we know that;

Therefore;

Rule for integration of  is:

Rule for integration of  is:

 

We
have demonstrated in (ii) that;

Therefore reverse process of derivative, that is integral, of will yield .

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