1
A convergent geometric progression G has first term a and common ratio r. The sum to infinity of the even-numbered terms of G, i.e. the second, fourth, sixth, … terms is 78. The sum to infinity of every third-term of G, i.e. the third, sixth, ninth, … terms is 24. Find the values of a and r. [5]
2
A curve C is described by the equation
( x − 2) ⎛ 5 ⎞ ⎜ ⎟ ⎝ 2 ⎠
2
2
( y − 3) − b2
2
= 1,
where b is a positive constant. It is given that the line y = (i)
8 1 x − is one of the asymptotes of C. 5 5
Show that the value of b is 4.
[2]
(ii) Sketch the curve of C, showing clearly the equation of the asymptotes and the coordinates of the vertices. [3] Hence, state the set of values of m for which the line with equation [1] y = m( x − 2) + 3 does not intersect C. 3
[Do not use a calculator for this question.] (i) Prove, using a non-graphical method, that x 2 − x + 3 is always positive for all real x. [1] 2x − 3 1 (ii) Hence, solve the inequality 2 [4] ≥ . x − x+3 x (iii) Using your result in part (ii), find the set of values of x for which
2 x2 − 3 1 ≥ 2. 4 2 x − x +3 x
4
[3]
A curve has parametric equations
x = t + 2 , y = t 2 + 3t + 5 . (i)
(
)
Show that the gradient of the line passing through P p + 2, p2 + 3 p + 5 and (2,4)
p2 + 3 p + 1 . [1] p (ii) It is given that the line in part (i) is also the tangent to the curve at P. Find the possible values of p. Hence, find the equations of the tangents. [5] can be expressed as
2
5
By using the substitution x =
u , where u is a function of t, show that the differential t
equation
t
dx = 4 − x − 9 x 2t 2 , for t > 0, dt
(1)
can be expressed as
du = 4 − 9u 2 . dt Find the general solution of the differential equation (1), giving your answer in the form [6] x = f (t ) .
6
The graphs of y 2 = f ( x) and y = f '( x) are shown below.
y 2 = f ( x) ( 5, 2 ) O
3
y = f '( x)
O
(i) Determine the nature of all stationary points of the graph of y = f ( x) .
[2]
(ii) State the set of values of x where f is strictly decreasing. [1] (iii) Sketch the graph of y = f ( x) , indicating clearly the asymptotes, coordinates of the stationary points and the intersections with the axes.
[4]
[Turn over 3
7
A man’s running speed on the beach is 4 times as fast as his swimming speed in the sea.
The distances AB, BC and BD are 100 m, h m and 300 m respectively, as shown in the diagram above. The man swims in a straight line from A to C, and then runs from C to D on the beach. (i) Show that the time taken for him to get from point A to point D through C, is ⎛ 1002 + h 2 ( 300 − h ) ⎞ given by ⎜ + ⎟ s, where v m/s represents his constant ⎜ ⎟ v 4v ⎝ ⎠ swimming speed in the sea. [2] (ii) Hence, by differentiation, find the value of h that will allow the man to reach D in the shortest amount of time. [4]
8
(a)
(i)
Find the derivative of 1 − x 2 .
(ii)
Hence, find
(i)
2 ⎛ dy ⎞ d y Given that y + y = 2 x − x , show that 6 y ⎜ ⎟ + 2 ( 3 y 2 + 1) = 4. ⎝ dx ⎠ dx
[2]
(ii)
Find the series expansion of y, up to and including the term in x2.
[3]
∫ sin
−1
[2]
x dx .
[3] 2
(b)
3
2
The diagram below shows the graph of C, given by y 3 + y = 2 x 2 − x for x ≥ 0 and the line x = 0.6.
(iii) Find an approximation for the area of the shaded region. 4
[2]
9
(i)
⎛ 2 ⎞ By expressing ln ⎜1 − ⎟ as ln A − ln B , where A and B are expressions of k, ⎝ k ⎠ m +1
prove that
∑ ln ⎛⎜⎝1 − k2 ⎞⎟⎠ = ln m(m2+ 1) .
[4]
k =3
17
(ii)
Using the result in part (i), calculate the exact value of
∑ ln ⎛⎜⎝1 − k2 ⎞⎟⎠.
[2]
k =7
m
(iii) Using the result in part (i), find
∑ ln ⎛⎜⎝1 − k 2+ 1 ⎞⎟⎠ in terms of m.
[3]
k =3
10
2
Sketch the graph of ( x − 93) + y 2 = 152 , showing the coordinates of the points with the maximum and minimum y –values. [2] The shape of a circular tube can be modelled by rotating the region bounded by 2
( x − 93)
+ y 2 = 152 completely about the y-axis. 15
Show the volume of the tube is given by 372π
∫
152 − y 2 dy .
[3]
−15
By using the substitution y = 15sin θ , find the exact volume of the tube.
11
(a)
[5]
To take part in the National Stamp Collection Campaign, John started to buy stamps every week. The number of stamps he bought in the first week is a and subsequently the number of stamps he bought in a week is d more than that of the previous week. It is given that he bought 10 stamps in the 3rd week and the total number of stamps he bought in the 7 th and 9 th weeks is 50. Find a and d. [2] Given that the total number of stamps he bought from nth week to (n + 19)th week (inclusive) is 2450, find the value of n. [3]
(b)
At the beginning of January 2015, Gilbert took a home loan of $400 000 from a bank that charges 0.2% monthly interest on the outstanding amount at the end of every month. Gilbert makes payment of $x on the 1st of every month, starting from February 2015. Taking January 2015 as the first month, show that the outstanding amount Gilbert still owes the bank on the 2nd of n th month is
400 000 (1.002 )
n −1
− 500 x ⎡(1.002 ) ⎣
n −1
− 1⎤ . ⎦
[3]
Gilbert intends to pay the loan fully after the payment on 1st of January 2030. Find the value of x. [2] [Turn over 5
12
The functions f and g are defined by
f :x g: x
x , x + a2 2
x ∈ , x > 0,
− ln ( x + a ) , x ∈ , x > −a ,
where a is a constant greater than 2. (i)
Sketch, on separate diagrams, the graphs of y = f ( x ) and y = g ( x ) .
[6]
[You should state the equation of any asymptotes, coordinates of the turning points and the coordinates of any points of intersection with the axes.] (ii) Explain why g-1 exists. [1] (iii) Giving your reasons, determine which of the composite functions fg −1 or g −1f exists. [3] Hence, define in a similar form, the composite function that exists and find its exact range. [5]
End of paper
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