# Subject: Maths

## Topic: C3

## C3 EXAM PRACTICE QUESTIONS

### To improve your C3 Exam performance, attempt our C3 Exam questions, then view our C3 Exam Solutions and work on your weaknesses.

**Question**

For the curve with equation

Find the values of and

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**Question**

Differentiate with respect to ,

(a)

(b)

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**Question**

(a) Prove the identity

(b) Solve the equation

for

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**Question**

(a) Differentiate with respect to

(b) Use your answer from part (a) to differentiate with respect to

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**Question**

(i) Using the chain rule and an appropriate substitution differentiate with respect to

(ii) Hence show that

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**Question**

Differentiate with respect to ,

i.

ii.

simplify your answer where possible.

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**Question**

Show that

(a)

(b)

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**Question**

(a) Differentiate with respect to and simplify where possible.

(b) show that

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**Question**

show that

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**Question**

A curve has equation

(a) Using the product rule for differentiation find ,

and hence the coordinates of the turning points of

(b) Determine the nature of these turning points.

(c) Given that the equation of the tangent to at the point where

can be expressed in the form find and

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**Question**

The curve shown in the figure has equation

(a) Clearly from the figure the curve has three turning points. Find them all.

(b) Using your answer from part (a) find .

(c) Hence determine the nature of the turning points of

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**Question**

A curve has equation

(a) show that in the interval there is a root to the equation

(b) show that can be written in the form

(c) Hence using an appropriate iterative formula for and an appropriate value for

find the values of and to decimal places.

(d) Find, to decimal places, the coordinates of the turning points of

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**Question**

(a) Express in terms of

(b) Prove the identity

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**Question**

(a) Solve, for the equation

(b) Show that

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**Question**

(a) Show that

(b) Solve, for the equation

giving your answer/s to one decimal place.

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**Question**

(a) By writing show that

(b) Show that

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**Question**

Given that can be expressed in the form

where and

(a) Find the values of and to decimal places.

(b) Hence solve

for

(c) Write down the maximum value of and the smallest possible value of for which this occurs.

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**Question**

(a) By expressing in the form where and

solve for

(b) i. Write down the maximum value of

ii. Find the smallest negative value of for which this maximum occurs.

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**Question**

The figure shows a sketch of the curve with equation

(a) Find .

(b) Hence find the coordinates of the turning points of

(c) Show that

and hence determine the nature of the turning points.

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**Question**

Differentiate the following functions with respect to and simplify where possible.

(a)

(b)

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**Question**

(a) Show that there lies a root of in the interval

(b) Show that is equivalent to

(c) Use the iterative formula

to calculate the values of and to five decimal places.

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**Question**

The figure shows the curve with equation

which intersects the -axis at point and has two turning points.

(a) Find the coordinates of the turning points of the curve.

Let point have -coordinate To find an approximation to the iterative formula

is used.

(b) Taking find to five decimal places the values of and

(c) Show that, correct to five decimal places,

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**Question**

The function is given by

(a) Show that

The function is given by

(b) Show that

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Express as a single fraction in its simplest from.

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**Question**

(a) Simplify

(b) Given that

find in terms of

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**Question**

(a) Differentiate with respect to

(b) Find for

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**Question**

Differentiate with respect to

(a)

(b)

simplifying where possible.

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**Question**

Differentiate with respect to

(a)

(b)

simplifying where possible.

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**Question**

(a) Find at the point on the curve with equation

(b) Differentiate with respect to

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