PRESIDENT’S OFFICE
REGIONAL ADMINISTRATION AND LOCAL GOVERNMENT
MACHAME GIRLS’ SECONDARY SCHOOL
FORM SIX EXAMINATION
ADVANCED MATHEMATICS 1
Time: 3:00 Hours
INSTRUCTIONS
i. This paper contains 10 (ten)compulsory questions
ii. Answer ALL questions
iii. Show working clearly for each questions attempted except question no 1.
iv. Scientific calculator may be used.
1. Use non programmable calculator to compute the value of the following mathematical
expressions correct to four decimal places
0 . 6984 cos 0 . 6745 tan 0 . 5678 cosh 0 . 1867
4 1 1
e
(i) (ii)
0 . 4187 ln 8 tan 1 . 69921 cos 1 0 . 5
3 3
0 .5
4 0
0 . 06 e
3 2
ln 32 e ln 5 tan 48 . 7 1
(iii) (iv) (v) cosh 1
ln 4 tanh
1
0 1 6 !7 ! 0 . 0326 5
tan tan 55 6
4
2. Use non programmable calculator, to find the values of the following correct to four decimal
places
log 5
3
tan
3
48
0
7
ln 9 . 4926 10
8
85 . 34
0 1 11
(a) (b) (c) 24 6 31 rad (give the
sin
1
log e 5
sec he
sin 46
0
3 cos
9
answer in radians)
(d)Cholera, an intestinal disease is caused by a cholera bacterium that multiples exponentially by
cell division as modeled by N N 0 e 1 . 386 t where N is the number of bacteria present after t hours
and N0 is number of bacteria present at t=0.If we start with 1 bacteria, use calculator to find
(i)How many bacteria will be present in 12 hours
(ii)time required to produce 1,020 bacteria
3.A cooperative society has two storage depots. At depot Q there are 300tons of beans and
at depot R there are 200tons of beans. The beans has to be sent to three stores A,B and C.
The demand at store A is 200tons,at store B is 150tons and at store C is 150tons.How much
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beans should be sent from each depot to each of the stores in order to minimize the total
transport cost, if the transport cost per ton from a depot to a store is as given in the table
below
From/To A B C
Q 80/= 100/= 40/=
R 50/= 150/= 70/=
4.(a) Find the maximum and minimum values of z = 3x + 4y subject to the following constraints
x + 2y ≤ 14, 3x − y ≥ 0, x − y ≤ 2
(b)A manufacture firm manufactures chairs and tables.Each requires the use of three machines
A,B and C.Producing one chair requires 2 hours on the use of machine A,1 hour on machine B and
1 hour on machine C.Each table requires 1 hour on machine A and B and 3 hours on machine
C.The profit realized by selling one chair is 30,000/= Tshs and for a table is 60,000/=Tshs.The
total time available per week on machine A is 70 hours ,on machine B is 40 hours while on
machine C is 90 hours.Find the number of chairs and table that should be made per week so as to
maximize the profit.What is the maximum profit?
5.(a) The mean and standard deviation of 20 observations are found to be 10 and 2 respectively.
On rechecking it was found that an observation 8 was incorrect. Calculate the correct mean and
standard deviation if wrong item is omitted.
(b) The sum of 20 numbers is 320 and the sum of squares of these numbers is 5840.
i. Calculate the mean and standard deviation of 20 numbers
ii. If one number is added to the 20 numbers so that the mean is unchanged, find this
number and show whether the standard deviation will change or not
(c)Table below gives age distribution of individuals starting new companies
Age 20-29 30-39 40-49 50-59 60-69
Frequency 11 25 x 7 3
If the mean age is 38.83, find the standard deviation by coding method with assumed mean of 34.5
and correct all the answers to 2 significant figures
6.(a)Consider the distribution table below
Class Interval 0-10 10-20 20-30 30-40 40-50
Frequency 2 a 10 3 2
If the second Decile of the distribution is 13.75, find the value of a
130
(b)The arithmetic mean and variance of 3,a,12,b,21,4 is 10 and respectively.Find the value of
3
a and b.
(c) The masses in grams of 10 pain killing tables were 5.99, 5.95, 5.96, 6.02, 5.97, 5.99, 6.01,
5.97, 6.03, and 6.04. Find the median mass, the range and IQR.
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7.(a) If A and B are two non- empty sets, shade the regions presented by the following sets
i. (A n B' ) U ( B n A') ii. A' U ( AUB)'
(b) (i) Show that A
(ii) Let A a , b , B 2 , 3 and C 3 , 4 .Find A B C and AxB AxC
(c)Simplify as much as possible the following set expressions using the algebra of set
(i) A 1 A C (ii) A B A C A 1
(d) .A survey of 100 students, the number of students studying various language were found to be
English only 18,English but not Hindi 23,English and Spaniola 8,English 26,Spaniola 48,Spaniola
and Hindi8,no language 24.Display the information on venn diagram and then find how many
students were studying (a) Hindi (b) Hindi and English
8. (a) Out of 17 girls guides and 15 boy scouts, 22 play handball, 16 play basketball, 12 of the boy
scouts play handball, 11 of boy scouts play basketball, 10 of the boy scouts play both and 3 of
the girls play neither of the two
(i)How many girls play both, handball and basketball
(ii)How many in the groups play (a) Handball only (b) Basketball only
(iii)How many in the group play handball and basketball
(b)Simplify using the laws of algebra of sets
(i) A A B C C B (ii) A B A B
1 1
(c) Suppose that 100 out of 120 mathematics students take at least one of the subject
Physics,Geography and Chemistry.65 study Physics,45 study Geography and 42 study
Chemistry.20 study both Physics and Geography,15 study both Geography and Chemistry,25 study
both Physics and Chemistry.Find the number of students who study
(i) Exactly one subject (ii)Geography but not Chemistry
9.(a) Given the function f defined as ʄ 𝑥 = 5x + 4 and g(x) = 6x – u
i. Find the value of u for which fog (x) = gof (x)
ii. Find the expression for fofof(x)
(b)Find the domain, range and inverse of given function f ( x ) 3 4 x 2
x 3
(c)Show that fog x x if
1 1 1
g of f (x) and g (x) 2 x 3
x 2
2x
g : x ax bx x 4
2
10.(a)A function is defined by and f : x for . Given that g(3)=9
x 4
and gof(2)=14,find the value of a and b
(b) Find g(x) if (fog)(x) = x2-2x+1 and f(x) = x+3
(c)Identify the following functions if they are odd,even or neither (i) f(x)=2x+x2
(ii) f(x)=3-2x2+x4 (iii )f(x)=tanx
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