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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