# 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

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