# 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      