# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/23) | Q#8

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Question

i.
By differentiating , show that if  θ then .

ii.       Hence show that

Giving the values of a and b.

iii.       Find the exact value of

Solution

i.

We are given that;

We are required to show that;

Since   provided that ;

Therefore;

If  and  are functions of , and if , then;

If , then;

Rule for differentiation of  is:

Rule for differentiation of  is;

Since    provided that  and , therefore;

ii.

We are required to show that;

We have found in (i) that;

Second derivative is the derivative of the derivative. If we have derivative
of the curve
as , then expression for the second derivative of the curve  is;

If  and  are functions of , and if , then;

If , then;

Let  and ; then,

Rule for differentiation of  is;

Rule for differentiation of  is;

We have trigonometric identity;

iii.

We are required to find the exact value of;

Rule for integration of  is:

We have trigonometric identity;

Rule for integration of  is:

Rule for differentiation of  is;

Therefore,

provided that