Download 2014 Summer Practice Problems - Juniors of 2014

2014 Summer Practice Problems – Juniors of 2014-2015 School year (HL 1)
Hello IB HL Students.
Below are IB questions on material that was covered during SL Year 1. However, they are
HL level. This means they will be harder. Take time to read through them and
contemplate what they mean and what they are asking. I have labeled the calculator and
non calculator sections. I will post the answers to these the week before school starts.
Good luck!
NO CALCULATOR (1-18)
1.
Events A and B are such that P(A) = 0.3 and P(B) = 0.4.
(a)
Find the value of P(A  B) when
(i)
A and B are mutually exclusive;
(ii)
A and B are independent.
(4)
(b)
Given that P(A  B) = 0.6, find P(A | B).
(3)
(Total 7 marks)
2.
In a population of rabbits, 1 % are known to have a particular disease. A test is developed for
the disease that gives a positive result for a rabbit that does have the disease in 99 % of cases. It
is also known that the test gives a positive result for a rabbit that does not have the disease in
0.1 % of cases. A rabbit is chosen at random from the population.
(a)
Find the probability that the rabbit tests positive for the disease.
(2)
(b)
Given that the rabbit tests positive for the disease, show that the probability that the rabbit
does not have the disease is less than 10 %.
(3)
(Total 5 marks)
3.
The lengths of the sides of a triangle ABC are x – 2, x and x + 2. The largest angle is 120°.
(a)
Find the value of x.
(6)
(b)
15 3
.
4
Show that the area of the triangle is
(3)
(c)
Find sin A + sin B + sin C giving your answer in the form
p q
r
where p, q, r 
.
(4)
(Total 13 marks)
4.
The obtuse angle B is such that tan B = 
(a)
sin B;
(b)
cos B;
(c)
sin 2B;
5
. Find the values of
12
(1)
(1)
(2)
IB Questionbank Mathematics Higher Level 3rd edition
1
2014 Summer Practice Problems – Juniors of 2014-2015 School year (HL 1)
(d0
cos 2B.
(2)
(Total 6 marks)
5.
(a)
Sketch the curve f(x) = sin 2x, 0 ≤ x ≤ π.
(2)
(b)
Hence sketch on a separate diagram the graph of g(x) = csc 2x, 0 ≤ x ≤ π, clearly stating
the coordinates of any local maximum or minimum points and the equations of any
asymptotes.
(5)
(c)
Show that tan x + cot x ≡ 2 csc 2x.
(3)
(d)
Hence or otherwise, find the coordinates of the local maximum and local minimum points
π
on the graph of y = tan 2x + cot 2x, 0 ≤ x ≤ .
2
(5)
(e)
Find the solution of the equation csc 2x = 1.5 tan x – 0.5, 0 ≤ x ≤
π
.
2
(6)
(Total 21 marks)
6.
The angle θ satisfies the equation 2 tan2 θ – 5 sec θ – 10 = 0, where θ is in the second quadrant.
Find the value of sec θ.
(Total 6 marks)
7.
Consider the functions given below.
f(x) = 2x + 3
1
g(x) = , x ≠ 0
x
(a)
(i)
Find (g ○ f)(x) and write down the domain of the function.
(ii)
Find (f ○ g)(x) and write down the domain of the function.
(2)
(b)
Find the coordinates of the point where the graph of y = f(x) and the graph of
y = (g–1 ○ f ○ g)(x) intersect.
(4)
(Total 6 marks)
8.
The diagram shows a tangent, (TP), to the circle with centre O and radius r. The size of
PÔA is θ radians.
IB Questionbank Mathematics Higher Level 3rd edition
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2014 Summer Practice Problems – Juniors of 2014-2015 School year (HL 1)
(a)
Find the area of triangle AOP in terms of r and θ.
(1)
(b)
Find the area of triangle POT in terms of r and θ.
(2)
(c)
Using your results from part (a) and part (b), show that sin θ < θ < tan θ.
(2)
(Total 5 marks)
9.
The graph of y =
(a)
ax
is drawn below.
b  cx
Find the value of a, the value of b and the value of c.
(4)
(b)
Using the values of a, b and c found in part (a), sketch the graph of y =
b  cx
ax
on the axes below, showing clearly all intercepts and asymptotes.
IB Questionbank Mathematics Higher Level 3rd edition
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2014 Summer Practice Problems – Juniors of 2014-2015 School year (HL 1)
(4)
(Total 8 marks)
10.
(a)
Express the quadratic 3x2 – 6x + 5 in the form a(x + b)2 + c, where a, b, c 
.
(3)
(b)
Describe a sequence of transformations that transforms the graph of y = x2 to the graph of
y = 3x2 – 6x + 5.
(3)
(Total 6 marks)
11.
A geometric sequence u1, u2, u3, ... has u1 = 27 and a sum to infinity of
(a)
81
.
2
Find the common ratio of the geometric sequence.
(2)
An arithmetic sequence v1, v2, v3, ... is such that v2 = u2 and v4 = u4.
N
(b)
Find the greatest value of N such that
v
n
 0.
n 1
(5)
(Total 7 marks)
12.
(a)
Consider the set of numbers a, 2a, 3a, ..., na where a and n are positive integers.
a(n  1)
.
2
(i)
Show that the expression for the mean of this set is
(ii)
Let a = 4. Find the minimum value of n for which the sum of these numbers
exceeds its mean by more than 100.
(6)
(b)
Consider now the set of numbers x1, ... , xm, y1, ... , y1, ... , yn where xi = 0 for i = 1, ... , m
and yi = 1 for i = 1, ... , n.
n
(i)
Show that the mean M of this set is given by
and the standard deviation
mn
mn
S by
.
mn
IB Questionbank Mathematics Higher Level 3rd edition
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2014 Summer Practice Problems – Juniors of 2014-2015 School year (HL 1)
(ii)
Given that M = S, find the value of the median.
(11)
(Total 17 marks)
13.
Two players, A and B, alternately throw a fair six–sided dice, with A starting, until one of them
obtains a six. Find the probability that A obtains the first six.
(Total 7 marks)
14.
An arithmetic sequence has first term a and common difference d, d ≠ 0.
The 3rd, 4th and 7th terms of the arithmetic sequence are the first three terms of a geometric
sequence.
3
(a) Show that a =  d .
2
(3)
(b)
Show that the 4th term of the geometric sequence is the 16th term of the arithmetic
sequence.
(5)
(Total 8 marks)
15.
(a)
Show that p = 2 is a solution to the equation p3 + p2 – 5p – 2 = 0.
(2)
(b)
Find the values of a and b such that p3 + p2 – 5p – 2 = (p – 2)(p2 + ap + b).
(4)
(c)
Hence find the other two roots to the equation p3 + p2 – 5p – 2 = 0.
(3)
(d)
An arithmetic sequence has p as its common difference. Also, a geometric sequence has p
as its common ratio. Both sequences have 1 as their first term.
(i)
Write down, in terms of p, the first four terms of each sequence.
(ii)
If the sum of the third and fourth terms of the arithmetic sequence is equal to the
sum of the third and fourth terms of the geometric sequence, find the three possible
values of p.
(iii)
For which value of p found in (d)(ii) does the sum to infinity of the terms of the
geometric sequence exist?
(iv)
For the same value p, find the sum of the first 20 terms of the arithmetic sequence,
writing your answer in the form a + b c , where a, b, c  .
(13)
(Total 22 marks)
16.
Consider the function f, where f(x) = arcsin (ln x).
(a)
Find the domain of f.
(3)
(b)
Find f–1(x).
(3)
(Total 6 marks)
17.
The functions f and g are defined by f : x  ex, g : x  x + 2.
IB Questionbank Mathematics Higher Level 3rd edition
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2014 Summer Practice Problems – Juniors of 2014-2015 School year (HL 1)
Calculate
(a) f–1(3) × g–1(3);
(b)
(3)
–1
(f ○ g) (3).
(3)
(Total 6 marks)
18.
Solve the equation 22x+2 – 10 × 2x + 4 = 0, x 
.
(Total 6 marks)
YES CALCULATOR (19-27)
19.
The cubic curve y = 8x3 + bx2 + cx + d has two distinct points P and Q, where the gradient is
zero.
(a) Show that b2 > 24c.
(4)
(b)
1

 3

Given that the coordinates of P and Q are  ,  12  and   , 20  , respectively, find the
2

 2

values of b, c and d.
(4)
(Total 8 marks)
20.
Let f(x) =
(a)
4  x2
4 x
.
State the largest possible domain for f.
(2)
(b)
Solve the inequality f(x) ≥ 1.
(4)
(Total 6 marks)
21.
A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum.
(a)
Write down an expression for the value of the investment after n full years.
(1)
(b)
What will be the value of the investment at the end of five years?
(1)
(c)
The value of the investment will exceed $10 000 after n full years.
(i)
Write an inequality to represent this information.
(ii)
Calculate the minimum value of n.
(4)
(Total 6 marks)
22.
Solve the following system of equations.
logx+1 y = 2
1
logy+1 x =
4
(Total 6 marks)
23.
Determine the first three terms in the expansion of (1− 2x)5 (1+ x)7 in ascending powers of x.
(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition
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2014 Summer Practice Problems – Juniors of 2014-2015 School year (HL 1)
24.
(a)
Simplify the difference of binomial coefficients
 n   2n 
     , where n ≥ 3.
 3  2 
(4)
(b)
Hence, solve the inequality
 n   2n 
     > 32n, where n ≥ 3.
 3  2 
(2)
(Total 6 marks)
25.
(a)
Write down the quadratic expression 2x2 + x – 3 as the product of two linear factors.
(1)
(b)
Hence, or otherwise, find the coefficient of x in the expansion of (2x2 + x – 3)8.
(4)
(Total 5 marks)
26.
The weight loss, in kilograms, of people using the slimming regime SLIM3M for a period of
three months is modelled by a random variable X. Experimental data showed that 67 % of the
individuals using SLIM3M lost up to five kilograms and 12.4 % lost at least seven kilograms.
Assuming that X follows a normal distribution, find the expected weight loss of a person who
follows the SLIM3M regime for three months.
(Total 5 marks)
27.
In a triangle ABC, Â = 35°, BC = 4 cm and AC = 6.5 cm. Find the possible values of B̂ and
the corresponding values of AB.
(Total 7 marks)
IB Questionbank Mathematics Higher Level 3rd edition
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