Summer Packet IB Math SL Sample Questions 1. (a)

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Summer Packet IB Math SL Sample Questions
1.
Consider the infinite geometric sequence 3000, – 1800, 1080, – 648, … .
(a)
Find the common ratio.
(2)
(b)
Find the 10th term.
(2)
(c)
Find the exact sum of the infinite sequence.
(2)
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(Total 6 marks)
2.
Consider the arithmetic sequence 2, 5, 8, 11, .....
(a)
Find u101.
(3)
(b)
Find the value of n so that un = 152.
(3)
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(Total 6 marks)
3.
The first term of an infinite geometric sequence is 18, while the third term is 8. There are two
possible sequences. Find the sum of each sequence.
Working:
Answers:
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(Total 6 marks)
4.
$1000 is invested at the beginning of each year for 10 years.
The rate of interest is fixed at 7.5% per annum. Interest is compounded annually.
Calculate, giving your answers to the nearest dollar
(a)
how much the first $1000 is worth at the end of the ten years;
(b)
the total value of the investments at the end of the ten years.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
5.
(a)
Write down the first three terms of the sequence un = 3n, for n ≥1.
(1)
(b)
Find
20
(i)
∑ 3n ;
n =1
100
(ii)
∑ 3n
n = 21
.
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(5)
(Total 6 marks)
6.
The diagrams below show the first four squares in a sequence of squares which are
1
subdivided in half. The area of the shaded square A is 4 .
A
A
B
Diagram 1
Diagram 2
A
A
B
B
C
Diagram 3
(a)
(i)
Find the area of square B and of square C.
C
Diagram 4
(ii)
Show that the areas of squares A, B and C are in geometric progression.
(iii)
Write down the common ratio of the progression.
(5)
(b)
(i)
Find the total area shaded in diagram 2.
(ii)
Find the total area shaded in the 8th diagram of this sequence.
Give your answer correct to six significant figures.
(4)
(c)
The dividing and shading process illustrated is continued indefinitely.
Find the total area shaded.
(2)
(Total 11 marks)
7.
(a)
Consider the geometric sequence −3, 6, −12, 24, ….
(i)
Write down the common ratio.
(ii)
Find the 15th term.
Consider the sequence x − 3, x +1, 2x + 8, ….
(3)
(b)
When x = 5, the sequence is geometric.
(i)
Write down the first three terms.
(ii)
Find the common ratio.
(2)
(c)
Find the other value of x for which the sequence is geometric.
(4)
(d)
For this value of x, find
(i)
the common ratio;
(ii)
the sum of the infinite sequence.
(3)
(Total 12 marks)
2
8.
⎛ P ⎞
⎟
log10 ⎜⎜
QR 3 ⎟⎠
⎝
Let log10P = x , log10Q = y and log10R = z. Express
in terms of x , y and z.
Working:
Answer:
....................................................................
(Total 4 marks)
9.
Solve the equation log27 x = 1 – log27 (x – 0.4).
Working:
Answer:
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(Total 6 marks)
10.
(a)
Given that log3 x – log3 (x – 5) = log3 A, express A in terms of x.
(b)
Hence or otherwise, solve the equation log3 x – log3 (x – 5) = 1.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
11.
Given that p = loga 5, q = loga 2, express the following in terms of p and/or q.
(a)
loga 10
(b)
loga 8
(c)
loga 2.5
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(Total 6 marks)
12.
(
Given that 3 +
(a)
p;
(b)
q.
7
)
3
= p + q 7 where p and q are integers, find
Working:
Answers:
(a) .................................................
(b) .................................................
(Total 6 marks)
9
13.
⎛ 2 1⎞
⎜ 3x – ⎟ .
x⎠
Consider the expansion of ⎝
(a)
How many terms are there in this expansion?
(b)
Find the constant term in this expansion.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
8
14.
⎛2
⎞
⎜ x − 3⎟ .
⎠
Find the term in x3 in the expansion of ⎝ 3
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(Total 5 marks)
15.
Each year for the past five years the population of a certain country has increased at a steady
rate of 2.7% per annum. The present population is 15.2 million.
(a)
What was the population one year ago?
(b)
What was the population five years ago?
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
16.
Initially a tank contains 10 000 litres of liquid. At the time t = 0 minutes a tap is opened, and
liquid then flows out of the tank. The volume of liquid, V litres, which remains in the tank
after t minutes is given by
V = 10 000 (0.933t).
(a)
Find the value of V after 5 minutes.
(1)
(b)
Find how long, to the nearest second, it takes for half of the initial amount of liquid to
flow out of the tank.
(3)
(c)
The tank is regarded as effectively empty when 95% of the liquid has flowed out.
Show that it takes almost three-quarters of an hour for this to happen.
(3)
(d)
(i)
Find the value of 10 000 – V when t = 0.001 minutes.
(ii)
Hence or otherwise, estimate the initial flow rate of the liquid.
Give your answer in litres per minute, correct to two significant figures.
(3)
(Total 10 marks)
17.
The population of a city at the end of 1972 was 250 000. The population increases by 1.3%
per year.
(a)
Write down the population at the end of 1973.
(b)
Find the population at the end of 2002.
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(Total 6 marks)
18.
(a)
Let y = –16x2 + 160x –256. Given that y has a maximum value, find
(i)
the value of x giving the maximum value of y;
(ii)
this maximum value of y.
The triangle XYZ has XZ = 6, YZ = x, XY = z as shown below. The perimeter of triangle
XYZ is 16.
(4)
(b)
(i)
Express z in terms of x.
(ii)
Using the cosine rule, express z2 in terms of x and cos Z.
(iii)
5 x −16
Hence, show that cos Z = 3x .
(7)
Let the area of triangle XYZ be A.
(c)
Show that A2 = 9x2 sin2 Z.
(2)
(d)
Hence, show that A2 = –16x2 + 160x – 256.
(4)
(e)
(i)
Hence, write down the maximum area for triangle XYZ.
(ii)
What type of triangle is the triangle with maximum area?
(3)
(Total 20 marks)
19.
The following diagram shows part of the graph of a quadratic function, with equation in the
form y = (x − p)(x − q), where p, q ∈ .
(a)
Write down
(i)
the value of p and of q;
(ii)
the equation of the axis of symmetry of the curve.
(3)
(b)
Find the equation of the function in the form y = (x − h)2 + k, where h, k ∈
.
(3)
(c)
dy
Find dx .
(2)
(d)
Let T be the tangent to the curve at the point (0, 5). Find the equation of T.
(2)
(Total 10 marks)
20.
2
Let f (x) = 4 tan x – 4 sin x,
(a)
−
π
π
≤ x≤ .
3
3
On the grid below, sketch the graph of y = f (x).
(3)
(b)
Solve the equation f (x) = 1.
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(3)
(Total 6 marks)
21.
The diagram below shows two circles which have the same centre O and radii 16 cm and 10
cm respectively. The two arcs AB and CD have the same sector angle θ = 1.5 radians.
A
B
D
C
O
Find the area of the shaded region.
Working:
Answer:
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(Total 6 marks)
22.
The diagram below shows a circle centre O, with radius r. The length of arc ABC is 3π cm
2π
.
and AÔC = 9
(a)
Find the value of r.
(2)
(b)
Find the perimeter of sector OABC.
(2)
(c)
Find the area of sector OABC.
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(2)
(Total 6 marks)
23.
Let f : x a sin3 x.
(a)
(i)
Write down the range of the function f.
(ii)
Consider f (x) =1, 0 ≤ x ≤ 2π. Write down the number of solutions to this
equation. Justify your answer.
(5)
(b)
Find f ′ (x), giving your answer in the form a sinp x cosq x where a, p, q ∈
.
(2)
(c)
1
π
2
3
sin
x
(cos
x
)
Let g (x) =
for 0 ≤ x ≤ 2 . Find the volume generated when the curve
of g is revolved through 2π about the x-axis.
(7)
(Total 14 marks)
24.
(a)
Consider the equation 4x2 + kx + 1 = 0. For what values of k does this equation have
two equal roots?
(3)
Let f be the function f (θ ) = 2 cos 2θ + 4 cos θ + 3, for −360° ≤ θ ≤ 360°.
(b)
Show that this function may be written as f (θ ) = 4 cos2 θ + 4 cos θ + 1.
(1)
(c)
Consider the equation f (θ ) = 0, for −360° ≤ θ ≤ 360°.
(i)
How many distinct values of cos θ satisfy this equation?
(ii)
Find all values of θ which satisfy this equation.
(5)
(d)
Given that f (θ ) = c is satisfied by only three values of θ, find the value of c.
(2)
(Total 11 marks)
25.
Consider the equation 3 cos 2x + sin x = 1
(a)
Write this equation in the form f (x) = 0 , where f (x) = p sin2 x + q sin x + r , and p , q ,
r∈ .
(b)
Factorize f (x).
(c)
Write down the number of solutions of f (x) = 0, for 0 ≤ x ≤ 2π.
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(Total 6 marks)
26.
Solve the equation 2 cos2 x = sin 2x for 0 ≤ x ≤ π, giving your answers in terms of π.
Working:
Answer:
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(Total 6 marks)
27. Part of the graph of y = p + q cos x is shown below. The graph passes through the points (0,
3) and (π, –1).
y
3
2
1
0
π
2π
x
–1
Find the value of
(a)
p;
(b)
q.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
28.
The diagrams show a circular sector of radius 10 cm and angle θ radians which is formed
into a cone of slant height 10 cm. The vertical height h of the cone is equal to the radius r of
its base. Find the angle θ radians.
10cm
h
10cm
r
Working:
Answer:
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(Total 4 marks)
29.
The diagram shows the graph of the function f given by
⎛⎜ π x ⎞⎟
f (x) = A sin ⎝ 2 ⎠ + B,
for 0 ≤ x ≤ 5, where A and B are constants, and x is measured in radians.
y
(1,3)
(5, 3)
2
(0, 1)
x
0
1
2
3
4
5
(3, –1)
The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph.
(a)
Write down the values of f (1) and f (5).
(2)
(b)
Show that the period of f is 4.
(2)
The point (3, –1) is a minimum point of the graph.
(c)
Show that A = 2, and find the value of B.
(5)
(d)
⎛⎜ π x ⎞⎟
Show that f′ (x) = π cos ⎝ 2 ⎠ .
(4)
The line y = k – πx is a tangent line to the graph for 0 ≤ x ≤ 5.
(e)
Find
(i)
the point where this tangent meets the curve;
(ii)
the value of k.
(6)
(f)
Solve the equation f (x) = 2 for 0 ≤ x ≤ 5.
(5)
(Total 24 marks)
30.
⎛⎜ x + π ⎞⎟
9⎠.
Consider y = sin ⎝
(a)
(b)
The graph of y intersects the x-axis at point A. Find the x-coordinate of A, where
0 ≤ x ≤ π.
⎛⎜ x + π ⎞⎟
1
9
⎝
⎠
Solve the equation sin
= – 2 , for 0 ≤ x ≤ 2π.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
31.
The following diagram shows a pentagon ABCDE, with AB = 9.2 cm, BC = 3.2 cm, BD =
7.1 cm, AÊD =110°, AD̂E = 52° and AB̂D = 60°.
(a)
Find AD.
(4)
(b)
Find DE.
(4)
(c)
The area of triangle BCD is 5.68 cm2. Find DB̂C .
(4)
(d)
Find AC.
(4)
(e)
Find the area of quadrilateral ABCD.
(5)
(Total 21 marks)
32.
Consider the function f (x) = cos x + sin x.
(a)
(i)
π
Show that f (– 4 ) = 0.
(ii)
Find in terms of π, the smallest positive value of x which satisfies f (x) = 0.
(3)
The diagram shows the graph of y = ex (cos x + sin x), – 2 ≤ x ≤ 3. The graph has a maximum
turning point at C(a, b) and a point of inflexion at D.
y
6
C(a, b)
4
D
2
–2
(b)
–1
1
2
3
x
dy
Find dx .
(3)
(c)
Find the exact value of a and of b.
(4)
π
(d)
Show that at D, y =
2e 4 .
(5)
(e)
Find the area of the shaded region.
(2)
(Total 17 marks)
33.
The diagram below shows a quadrilateral ABCD. AB = 4, AD = 8, CD =12, B Ĉ D = 25°,
BÂD =θ.
(a)
Use the cosine rule to show that BD =
4 5 − 4 cos θ
.
(2)
Let θ = 40°.
(b)
(i)
Find the value of sin CB̂D .
(ii)
Find the two possible values for the size of CB̂D .
(iii)
Given that CB̂D is an acute angle, find the perimeter of ABCD.
(12)
(c)
Find the area of triangle ABD.
(2)
(Total 16 marks)
34.
⎛1 2 3⎞
⎜
⎟
⎜ 3 1 2⎟
⎜2 0 1⎟
⎠, B =
Let A = ⎝
⎛ 18 ⎞
⎜ ⎟
⎜ 23 ⎟
⎜ 13 ⎟
⎝ ⎠
and X =
⎛ x⎞
⎜ ⎟
⎜ y⎟
⎜z⎟
⎝ ⎠
(a)
Write down the inverse matrix A−1.
(b)
Consider the equation AX = B.
.
(i)
Express X in terms of A−1 and B.
(ii)
Hence, solve for X.
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(Total 6 marks)
35.
2
0⎞
⎛ 1
⎜
⎟
⎜ − 3 1 − 1⎟
⎜ 2 −2 1 ⎟
⎠ has inverse A−1 =
The matrix A = ⎝
(a)
⎛ −1 − 2 − 2⎞
⎜
⎟
1
1 ⎟
⎜ 3
⎜a
6
b ⎟⎠
⎝
.
Write down the value of
(i)
a;
(ii)
b.
Consider the simultaneous equations
x + 2y = 7
–3x + y – z = 10
2x – 2y + z = –12
(b)
Write these equations as a matrix equation.
(c)
Solve the matrix equation.
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(Total 6 marks)
36.
Calculate the acute angle between the lines with equations
⎛ 4⎞
⎛ 4⎞
⎜⎜ ⎟⎟
⎜⎜ ⎟⎟
–1
3
r = ⎝ ⎠ + s ⎝ ⎠ and
⎛ 2⎞
⎛ 1⎞
⎜⎜ ⎟⎟
⎜⎜ ⎟⎟
4
–1
r = ⎝ ⎠ + t⎝ ⎠
Working:
Answer:
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(Total 6 marks)
37.
⎛ 3⎞
⎜⎜ ⎟⎟
4
Find the cosine of the angle between the two vectors ⎝ ⎠ and
⎛ − 2⎞
⎜⎜ ⎟⎟
⎝ 1⎠ .
Working:
Answer:
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(Total 6 marks)
38.
The histogram below represents the ages of 270 people in a village.
(a)
Use the histogram to complete the table below.
Age range
Frequency
Mid-interval
value
0 ≤ age < 20
40
10
20 ≤ age < 40
40 ≤ age < 60
60 ≤ age < 80
80 ≤ age ≤100
(2)
(b)
Hence, calculate an estimate of the mean age.
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(4)
(Total 6 marks)
39.
The 45 students in a class each recorded the number of whole minutes, x, spent doing
experiments on Monday. The results are ∑x = 2230.
(a)
Find the mean number of minutes the students spent doing experiments on Monday.
Two new students joined the class and reported that they spent 37 minutes and 30 minutes
respectively.
(b)
Calculate the new mean including these two students.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
40.
There are 50 boxes in a factory. Their weights, w kg, are divided into 5 classes, as shown in
the following table.
Class
Weight (kg)
Number of boxes
A
9.5 ≤ w <18.5
7
B
18.5 ≤ w < 27.5
12
C
27.5 ≤ w < 36.5
13
D
36.5 ≤ w < 45.5
10
E
45.5 ≤ w < 54.5
8
(a)
Show that the estimated mean weight of the boxes is 32 kg.
(3)
(b)
There are x boxes in the factory marked “Fragile”. They are all in class E. The
estimated mean weight of all the other boxes in the factory is 30 kg. Calculate the
value of x.
(4)
(c)
An additional y boxes, all with a weight in class D, are delivered to the factory. The
total estimated mean weight of all of the boxes in the factory is less than 33 kg. Find
the largest possible value of y.
(5)
(Total 12 marks)
The cumulative frequency graph below shows the heights of 120 girls in a school.
130
120
110
100
Cumulative frequency
41.
90
80
70
60
50
40
30
20
10
0
150 155 160 165 170 175 180 185
Height in centimetres
(a)
Using the graph
(i)
write down the median;
(ii)
(b)
find the interquartile range.
Given that 60% of the girls are taller than a cm, find the value of a.
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(Total 6 marks)
42.
In a suburb of a large city, 100 houses were sold in a three-month period. The following
cumulative frequency table shows the distribution of selling prices (in thousands of
dollars).
Selling price P
($1000)
P ≤ 100
P ≤ 200
P ≤ 300
P ≤ 400
P ≤ 500
Total number
of houses
12
58
87
94
100
(a)
Represent this information on a cumulative frequency curve, using a scale of 1 cm to
represent $50000 on the horizontal axis and 1 cm to represent 5 houses on the vertical
axis.
(4)
(b)
Use your curve to find the interquartile range.
(3)
The information above is represented in the following frequency distribution.
Selling price P
($1000)
0 < P ≤ 100
Number of
houses
12
(c)
100 < P ≤ 200 200 < P ≤ 300 300 < P ≤ 400 400 < P ≤ 500
46
29
a
b
Find the value of a and of b.
(2)
(d)
Use mid-interval values to calculate an estimate for the mean selling price.
(2)
(e)
Houses which sell for more than $350000 are described as De Luxe.
(i)
Use your graph to estimate the number of De Luxe houses sold.
Give your answer to the nearest integer.
(ii)
Two De Luxe houses are selected at random. Find the probability
that both have a selling price of more than $400000.
(4)
(Total 15 marks)
43.
A box contains 100 cards. Each card has a number between one and six written on it.
The following table shows the frequencies for each number.
(a)
Number
1
2
3
4
5
6
Frequency
26
10
20
k
29
11
Calculate the value of k.
(2)
(b)
Find
(i)
the median;
(ii)
the interquartile range.
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(5)
(Total 7 marks
44.
2
1
7
Consider the events A and B, where P(A) = 5 , P(B′) = 4 and P(A ∪ B) = 8 .
(a)
Write down P(B).
(b)
Find P(A ∩ B).
(c)
Find P(A | B).
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(Total 6 marks)
45.
A factory makes switches. The probability that a switch is defective is 0.04.
The factory tests a random sample of 100 switches.
(a)
Find the mean number of defective switches in the sample.
(2)
(b)
Find the probability that there are exactly six defective switches in the sample.
(2)
(c)
Find the probability that there is at least one defective switch in the sample.
(3)
(Total 7 marks)
46.
Let A and B be independent events such that P(A) = 0.3 and P(B) = 0.8.
(a)
Find P(A ∩ B).
(b)
Find P(A ∪ B).
(c)
Are A and B mutually exclusive? Justify your answer.
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(Total 6 marks)
47.
Three students, Kim, Ching Li and Jonathan each have a pack of cards, from which they
select a card at random. Each card has a 0, 3, 4, or 9 printed on it.
(a)
Kim states that the probability distribution for her pack of cards is as follows.
x
0
3
4
9
P(X = x)
0.3
0.45
0.2
0.35
Explain why Kim is incorrect.
(2)
(b)
Ching Li correctly states that the probability distribution for her pack of cards is as
follows.
x
0
3
4
9
P(X = x)
0.4
k
2k
0.3
Find the value of k.
(2)
(c)
Jonathan correctly states that the probability distribution for his pack of cards is given
x +1
by P(X = x) = 20 . One card is drawn at random from his pack.
(i)
Calculate the probability that the number on the card drawn is 0.
(ii)
Calculate the probability that the number on the card drawn is greater than 0.
(4)
(Total 8 marks)
48.
Reaction times of human beings are normally distributed with a mean of 0.76 seconds and a
standard deviation of 0.06 seconds.
(a)
The graph below is that of the standard normal curve. The shaded area represents the
probability that the reaction time of a person chosen at random is between 0.70 and
0.79 seconds.
a
0 b
(i)
Write down the value of a and of b.
(ii)
Calculate the probability that the reaction time of a person chosen at random is
(a)
greater than 0.70 seconds;
(b)
between 0.70 and 0.79 seconds.
(6)
Three percent (3%) of the population have a reaction time less than c seconds.
(b)
(i)
Represent this information on a diagram similar to the one above. Indicate
clearly the area representing 3%.
(ii)
Find c.
(4)
(Total 10 marks)
49.
The heights of certain flowers follow a normal distribution. It is known that 20% of these
flowers have a height less than 3 cm and 10% have a height greater than 8 cm.
Find the value of the mean μ and the standard deviation σ.
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(Total 6 marks)
50.
The weights of a group of children are normally distributed with a mean of 22.5 kg and a
standard deviation of 2.2 kg.
(a)
Write down the probability that a child selected at random has a weight more than
25.8 kg.
(b)
Of the group 95% weigh less than k kilograms. Find the value of k.
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(c)
The diagram below shows a normal curve.
On the diagram, shade the region that represents the following information:
87% of the children weigh less than 25 kg
(Total 6 marks)
51.
A box contains a large number of biscuits. The weights of biscuits are normally distributed
with mean 7 g and standard deviation 0.5 g.
(a)
One biscuit is chosen at random from the box. Find the probability that this biscuit
(i)
weighs less than 8 g;
(ii)
weighs between 6 g and 8 g.
(4)
(b)
Five percent of the biscuits in the box weigh less than d grams.
(i)
Copy and complete the following normal distribution diagram, to represent this
information, by indicating d, and shading the appropriate region.
(ii)
Find the value of d.
(5)
(c)
The weights of biscuits in another box are normally distributed with mean μ and
standard deviation 0.5 g. It is known that 20% of the biscuits in this second box weigh
less than 5 g.
Find the value of μ.
(4)
(Total 13 marks)
52.
The function f is defined as f (x) = ex sin x, where x is in radians. Part of the curve of f is
shown below.
There is a point of inflexion at A, and a local maximum point at B. The curve of f intersects
the x-axis at the point C.
(a)
Write down the x-coordinate of the point C.
(1)
(b)
(i)
Find f ′ (x).
(ii)
Write down the value of f ′ (x) at the point B.
(4)
(c)
Show that f ″(x) = 2ex cos x.
(2)
(d)
(i)
Write down the value of f ″(x) at A, the point of inflexion.
(ii)
Hence, calculate the coordinates of A.
(4)
(e)
Let R be the region enclosed by the curve and the x-axis, between the origin and C.
(i)
Write down an expression for the area of R.
(ii)
Find the area of R.
(4)
(Total 15 marks)
53.
1
Consider f (x) = 3 x3 + 2x2 – 5x. Part of the graph of f is shown below. There is a maximum
point at M, and a point of inflexion at N.
(a)
Find f ′ (x).
(3)
(b)
Find the x-coordinate of M.
(4)
(c)
Find the x-coordinate of N.
(3)
(d)
The line L is the tangent to the curve of f at (3, 12). Find the equation of L in the form
y = ax + b.
(4)
(Total 14 marks)
54.
1
5
2
x
−
5
The function f (x) is defined as f (x) = 3 +
,x ≠ 2.
(a)
Sketch the curve of f for −5 ≤ x ≤ 5, showing the asymptotes.
(3)
(b)
Using your sketch, write down
(i)
the equation of each asymptote;
(ii)
the value of the x-intercept;
(iii)
the value of the y-intercept.
(4)
(c)
The region enclosed by the curve of f, the x-axis, and the lines x = 3 and x = a, is
revolved through 360° about the x-axis. Let V be the volume of the solid formed.
⎛
6
1
∫ ⎜⎜⎝ 9 + 2 x − 5 + (2 x − 5)
2
⎞
⎟
⎟
⎠ dx.
(i)
Find
(ii)
⎛ 28
⎞
π⎜ + 3 ln 3 ⎟
3
⎠ , find the value of a.
Hence, given that V = ⎝
(10)
(Total 17 marks)
55.
The following diagram shows part of the graph of y = cos x for 0 ≤ x ≤ 2π. Regions A and B
are shaded.
(a)
Write down an expression for the area of A.
(1)
(b)
Calculate the area of A.
(1)
(c)
Find the total area of the shaded regions.
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(4)
(Total 6 marks)
56.
Consider the function f (x) = 4x3 + 2x. Find the equation of the normal to the curve of f at the
point where x =1.
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(Total 6 marks)
57.
The following diagram shows the graphs of f (x) = ln (3x – 2) + 1 and g (x) = – 4 cos (0.5x) +
2, for 1 ≤ x ≤ 10.
(a)
Let A be the area of the region enclosed by the curves of f and g.
(i)
Find an expression for A.
(ii)
Calculate the value of A.
(6)
(b)
(i)
Find f ′ (x).
(ii)
Find g′ (x).
(4)
(c)
There are two values of x for which the gradient of f is equal to the gradient of g. Find
both these values of x.
(4)
(Total 14 marks)
58.
3x 2
Let f (x) = 5 x − 1 .
(a)
Write down the equation of the vertical asymptote of y = f (x).
(1)
(b)
ax 2 + bx
2
Find f ′ (x). Give your answer in the form (5 x − 1) where a and b ∈
.
(4)
(Total 5 marks)
59.
2
The diagram below shows the shaded region R enclosed by the graph of y = 2x 1 + x , the
x-axis, and the vertical line x = k.
y
y = 2x 1+x 2
R
k
(a)
x
dy
Find dx .
(3)
(b)
Using the substitution u = 1 + x2 or otherwise, show that
∫ 2x
1+ x2
2
3
2
dx = 3 (1 + x ) 2 + c.
(3)
(c)
Given that the area of R equals 1, find the value of k.
(3)
(Total 9 marks)
60.
A part of the graph of y = 2x – x2 is given in the diagram below.
The shaded region is revolved through 360° about the x-axis.
(a)
Write down an expression for this volume of revolution.
(b)
Calculate this volume.
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(Total 6 marks)
61.
Let f (x) = ex (1 – x2).
(a)
Show that f ′ (x) = ex (1 – 2x – x2).
(3)
Part of the graph of y = f (x), for – 6 ≤ x ≤ 2, is shown below. The x-coordinates of the local
minimum and maximum points are r and s respectively.
(b)
Write down the equation of the horizontal asymptote.
(1)
(c)
Write down the value of r and of s.
(4)
(d)
Let L be the normal to the curve of f at P(0, 1). Show that L has equation x + y = 1.
(4)
(e)
Let R be the region enclosed by the curve y = f (x) and the line L.
(i)
Find an expression for the area of R.
(ii)
Calculate the area of R.
(5)
(Total 17 marks)

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