Download Maths practice questions and more Quizzes Mathematics in PDF only on Docsity!
IYGB GCE
Core Mathematics C
Advanced
Practice Paper Z
Difficulty Rating: 3.9600/1.
Time: 2 hours
Candidates may use any calculator allowed by the Regulations of the Joint Council for Qualifications.
Information for Candidates
This practice paper follows the Edexcel Syllabus. The standard booklet “Mathematical Formulae and Statistical Tables” may be used. Full marks may be obtained for answers to ALL questions. The marks for the parts of questions are shown in round brackets, e.g. (2). There are 8 questions in this question paper. The total mark for this paper is 75.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. Non exact answers should be given to an appropriate degree of accuracy. The examiner may refuse to mark any parts of questions if deemed not to be legible.
Question 1 (+)*
3 x = ln y^2 + 9 2.
Show clearly that
dy y dx y
Question 2 (+)*
Prove the validity of the following trigonometric identity
sin 2 cos 2 sec sin cos
Question 3 (****)
The equation of a curve C is
1 2ln
x y x
, x ∈ , x > 0.
The curve has a single turning point at P.
a) Show that the coordinates of P are (^) ( e, 12 e).
b) Find the exact value of
2 2
d y dx
at P and hence determine its nature.
(^6 )
( 5 )
(^6 )
( 4 )
Question 5 (****)
A scientist investigating the growth of a certain species of mushroom observes that this mushroom grows to a height of 41 mm in 5 hours.
He decides to model the height, H mm, t hours after the mushroom started forming, by the equation
1 H = k 1 −e −^12 t
, t ≥ 0
where k is a positive constant.
a) Show that k = 120 , correct to three significant figures.
The equation
1 H = 120 1^ −e −^12 t
, t ≥ 0 ,
is to be used for the rest of this question.
b) Determine the value of t when H = 90 , giving the answer in the form a ln 2, where a is an integer.
c) Show clearly that
dH H dt
d) Hence find the value of H when the height is the mushroom is growing at the rate of 7.5 mm per hour.
e) State the maximum height of the mushroom according to this model.
(^3 )
(^2 )
(^2 )
(^4 )
(^1 )
Question 6 (****)
The figure above shows the graph of
f ( x ) = a + cos bx , 0 ≤ x ≤ 2 π,
where a and b are non zero constants.
The stationary points (^) ( 0, 4) and (^) ( 2 π , 2)are the endpoints of the graph.
a) State the range of f (^) ( x (^) )and hence find the value of a and b.
b) Find an expression for f −^1 ( x ), the inverse function of f (^) ( x (^) ).
c) State the domain and range of f −^1 ( x ).
d) Determine the gradient at the point on f (^) ( x (^) )with coordinates (^) ( 43 π ,^52 ).
e) State the gradient at the point on f −^1 ( x )with coordinates (^) ( (^5) 2 3^ ,^4 π (^) ).
f (^) ( x (^) ) = a +cos bx
O 2 π
y
x
( 0, 4)
( 2 )
(^1 )
(^2 )
(^3 )
(^5 )
