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1.
Find
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∫
4
⎛ 4
⎞
⎜⎝ 2 x − x + 3⎟⎠ d x
√
giving each term in its simplest form.
(4)
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2
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Q1
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2.
Express 93x + 1 in the form 3 y, giving y in the form ax + b, where a and b are constants.
(2)
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Q2
(Total 2 marks)
*P46715A0328*
3
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3.
(a) Simplify
giving your answer in the form a 2 , where a is an integer.
(2)
(b) Hence, or otherwise, simplify
12 3
50 − 18
giving your answer in the form b c , where b and c are integers and b
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50 − 18
(3)
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Q3
(Total 5 marks)
*P46715A0528*
5
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4.
y
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A (–2, 4)
x
O
B (3, –8)
Figure 1
On separate diagrams, sketch the curve with equation
(a) y = 3f(x),
(2)
(b) y = f(x) – 4
(3)
On each diagram, show clearly the coordinates of the maximum and the minimum points
and the coordinates of the point where the curve crosses the y-axis.
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Figure 1 shows a sketch of part of the curve with equation y = f(x). The curve has a
maximum point A at (–2, 4) and a minimum point B at (3, –8) and passes through the
origin O.
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Q4
(Total 5 marks)
*P46715A0728*
7
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5.
Solve the simultaneous equations
=0
y2 + 5x2 + 2x = 0
(6)
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y + 4x + 1
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Q5
(Total 6 marks)
*P46715A0928*
9
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6.
A sequence a1, a2, a3, … is defined by
an + 1 = 5 – kan ,
n.1
where k is a constant.
(a) Write down expressions for a2 and a3 in terms of k.
(2)
Find
3
(b)
∑ (1 + a ) in terms of k, giving your answer in its simplest form,
r
r =1
(c)
100
∑ (a
r =1
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a1 = 4,
r +1
+ kar )
(3)
(1)
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Q6
(Total 6 marks)
*P46715A01128*
11
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7.
Given that
1
find
2 x3 − 7
, x0
3√ x
dy
. Give each term in your answer in its simplified form.
dx
(6)
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y = 3x 2 + 6 x 3 +
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Q7
(Total 6 marks)
*P46715A01328*
13
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8.
The straight line with equation y = 3x – 7 does not cross or touch the curve with
equation y = 2px2 – 6px + 4p, where p is a constant.
(4)
(b) Hence find the set of possible values of p.
(4)
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(a) Show that 4p2 – 20p + 9 < 0
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Q8
(Total 8 marks)
*P46715A01528*
15
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9.
(a) Show that, immediately after his 12th birthday, the total of these gifts was £225
(1)
(b) Find the amount that John received from his uncle as a birthday gift on his
18th birthday.
(2)
(c) Find the total of these birthday gifts that John had received from his uncle up to and
including his 21st birthday.
(3)
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On John’s 10th birthday he received the first of an annual birthday gift of money from his
uncle. This first gift was £60 and on each subsequent birthday the gift was £15 more than
the year before. The amounts of these gifts form an arithmetic sequence.
When John had received n of these birthday gifts, the total money that he had received
from these gifts was £3375
(d) Show that n2 + 7n = 25 × 18
(e) Find the value of n, when he had received £3375 in total, and so determine John’s age
at this time.
(2)
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(3)
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Q9
(Total 11 marks)
*P46715A01928*
19
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10.
y
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l1
Not to scale
Q
P
R
O
l2
x
Figure 2
The points P (0, 2) and Q (3, 7) lie on the line l1, as shown in Figure 2.
The line l2 is perpendicular to l1, passes through Q and crosses the x-axis at the point R, as
shown in Figure 2.
(a) an equation for l2, giving your answer in the form a x + b y + c = 0, where a, b and c
are integers,
(5)
(b) the exact coordinates of R,
(2)
(c) the exact area of the quadrilateral ORQP, where O is the origin.
(5)
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Find
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Q10
(Total 12 marks)
*P46715A02328*
23
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11. The curve C has equation y = 2x3 + kx2 + 5x + 6, where k is a constant.
dy
dx
(2)
The point P, where x = –2, lies on C.
The tangent to C at the point P is parallel to the line with equation 2y – 17 x – 1 = 0
Find
(b) the value of k,
(4)
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(a) Find
(c) the value of the y coordinate of P,
(2)
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(d) the equation of the tangent to C at P, giving your answer in the form ax + by + c = 0,
where a, b and c are integers.
(2)
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(Total 10 marks)
TOTAL FOR PAPER: 75 MARKS
END
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