THE UNITED REPUBLIC OF TANZANIA

PRESIDENT’S OFFICE,
REGIONAL ADMINISTRATION AND LOCAL GOVERNMENT
NJOMBE REGION

FORM SIX PRE – NATIONAL EXAMINATION

CODE: 142/2 ADVANCED MATHEMATICS 2
(For Both School and Private Candidates)

TIME: 03: 00 HOURS MONDAY, 18TH March 2024 A.M

INSTRUCTIONS
 This PAPER consists of section A and B with eight (8) questions.
 Answer all questions in section A and two (2) questions from section B
 All work done and answer must be shown clearly
 Mathematical tables and non-programmable calculators may be used.
 Cellular phones and other unauthorized materials are not allowed in the examination room
 Write your examination number on every page of your answer booklet(s)

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 SECTION A (60 MARKS)

1. (a) A certain eruptible pandemic disease occur 1% of the population. A sample screening procedure is
available and in 8 out of 10 cases where patient has a disease, if 0.05 chance that the test will give a
positive result.
i. Find the probability that a randomly selected individual does not have disease but gives a positive
result in the screening test
ii. Find the probability that a randomly selected individual gives a positive result on screening test
iii. If danken has taken the disease test and her result is positive, what is the probability that he has
disease.
𝟏 𝟕 𝟐
(b) Given that 𝑷(𝑨′) = 𝟐 , 𝑷(𝑩′)= 𝟏𝟎, and 𝑷(𝑨𝑼𝑩)′ = 𝟓. Find the value of;

(i) 𝑷(𝑨 ∩ 𝑩′ ) (ii) 𝑷(𝑨 ∩ 𝑩)

(c) The random variable 𝑥 has probability distribution as follows.

𝒙 2 3 5 7 11

𝑷(𝒙) 0.17 0.33 0.25 𝒙 𝒚

𝟏𝟒
Find the numerical of value of 𝒙 and 𝒚, if 𝑬(𝒙) = 𝟑

2. (a) (i) Fact 1: Islands are surrounded by water

Fact 2: Maui is an Island
Fact 3: Maui was formed by a volcano.
If the first three statements are facts, which of the following statements must also be a fact?
I. Maui is surrounded by water
II. All Islands are formed by volcanoes
III. All volcanoes are on Islands.

(ii) Simplify 𝒓˅[(~𝒑 → 𝒒)˄ 𝒑] using laws of propositions, hence draw an electrical network for the
resulting simplified statement.
b) Given the statement 𝑝 → (𝒒 → ~𝒑).Construct the truth table for the contrapositive
c) Use the laws of algebra of proposition to show that the statement [(𝑝 →∼ 𝑞) ∧ (𝑟 → 𝑞)] → (𝑝 →∼ 𝑟) is a
tautology.
d) Test whether or not the following argument is valid
“If I am illiterate, then I can’t read and write. I cannot read but I can write. Thus I am not illiterate”

𝟐
3.a) If Ṵ = 4ḭ+𝑴𝒋 + 𝟑𝒌 𝑎𝑛𝑑 ṿ = −𝟒𝒊 + 𝟑𝒋 and the projection of Ṵ onto ṿ is 𝟑. Find the value of 𝒎

b) If Q is an angle between two-unit vectors 𝑎 and 𝑏 show that

𝟏 𝑸
[𝒂 + 𝒃] = 𝐜𝐨𝐬 ( 𝟐 )
𝟐

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 c) At a time 𝒕 second, the position vector 𝒓 of a particle 𝒑 is with respect to the origin (o) is 𝒓(𝒕) =
(𝟑 𝒄𝒐𝒔𝟐𝒕)
i. Show that the velocity of a particle is always at the right angle to op
ii. Find the speed of the particle at 𝒕 = 𝟎
iii. Find the acceretion of the particle 𝑷 at 𝒕 = 𝟎
𝟏+𝒛𝟐
4.(a) Find the complex number 𝒛 = 𝒙 + 𝒊𝒚, satisfy the equation =𝒊
𝟏−𝒛𝟐
(b) Prove that 16 cos 5𝜽 = 𝐜𝐨𝐬 𝟓𝜽 + 𝐜𝐨𝐬 𝟓𝜽 + 𝟓 𝐜𝐨𝐬 𝟑𝜽 + 𝟏𝟎 𝐜𝐨𝐬 𝜽
(C)
The point P represents a complex number z on an argand diagram such that |𝑧 − 6𝑖| =
2|𝑧 − 3|, show that as z varies, the locus of p is a circle, state the radius and coordinates of the
centre of the circle.

SECTION B: (40 MARKS)

Answer only two (2) questions.
5. a) Show that
1+cos 𝑥+cos2 𝑥
(i) = cot 𝑥
𝑠𝑖𝑛𝑥+𝑠𝑖𝑛2𝑥

(ii) Cos 2(∝ +𝛽) + 𝑐𝑜𝑠 2 (∝ −𝛽) = 𝑐𝑜𝑠2 ∝ 𝑐𝑜𝑠2𝛽 + 1.

b) Find the general solution in degree for the equation
9𝑠𝑖𝑛2 𝑥 + 10𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 − 2𝑐𝑜𝑠 2 x = 1
1
c) Find x if tan−1(2𝑥 + 1) − tan−1(2𝑥 − 1) = tan−1 (8)

d) Express cos 2𝜃 − 𝑠𝑖𝑛2𝜃 in form 𝑅 cos(2𝜃+∝), Hence or otherwise find the general solution
in radians for the equation sin 2𝜃 = cos 2𝜃 − 1.
e) Find the expression involving 𝜽 that the following approximate to very small value of 𝜽
1+sin 𝜃
5+3 tan 𝜃−4 cos 𝜃

6. a) (i) If one of the roots of the equation 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 is twice the other root, show that
𝟐𝒃𝟐 = 𝟗𝒂𝒄

(ii) Given that (2𝑥 + 1) is a factor of 2𝑥 3 + 𝑎𝑥 2 + 16𝑥 + 6, find the value of a and the real

quadratic factor of 2𝑥 3 + 9𝑥 2 + 16𝑥 + 6

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 b) (i) Given that a is a constant, x is a variable and the fourth term in the expansion of
1 𝑛 5
(𝑎𝑥 + 𝑥) 𝑖𝑠 2. Find the value of a and n.

(ii) Find x if log 𝑥 8 − log 𝑥 216 = 1

c) Solve the system of equation by determinant method
𝟐 𝟑 𝟏𝟎
+ + =𝟒
𝒙 𝒚 𝒛
𝟒 𝟔 𝟓
− + =𝟏
𝒙 𝒚 𝒛
𝟔 𝟗 𝟐𝟎
+ − =𝟐
𝒙 𝒚 𝒛
𝟏𝟔𝒙
d) Express into partial fraction
𝒙𝟒 −𝟏𝟔

𝒅𝒚 𝒚𝟐
7. a) (i) Solve the differential Equation 𝒙 − =𝒚
𝒅𝒙 𝒙
(ii) Find the general solution of the differential equation 𝒚" − 𝟐𝒚′ − 𝟑 = 𝟎

b) From the differential equation given that 𝒚 = 𝑨 𝐜𝐨𝐬 𝟐𝒙 + 𝑩 𝐬𝐢𝐧 𝟐𝒙

c) A copper ball is heated to a temperature of 100oC, then at time t = 0 it is placed in water which
is maintained at a temperature of 30oC. At the end of 3 minutes the temperature of the ball is
reduced to 70oC. Find the time at which the temperature of the ball is reduced to 31oC.
8. a) Show that the equation of the tangent to the parabola 𝑦 2 = 4𝑎𝑥 at the
point (𝒑, 𝒒) is 𝒒𝒚 = 𝟐𝒂 (𝒙 + 𝒑)
b) A man running a race course, notes that the sum of the distances from the two flag posts from
him is always 10 metres and the distance between the flag posts is 8. Find the equation of the path
traced by a man.
c) The equation of hyperbola is given by
𝒙𝟐 𝒚𝟐
𝟐
− =𝟏
𝒂 𝒃𝟐
(i) Find the length of its latus rectum
(ii) The line y = Mx + c is a tangent to the curve, show that 𝐶 = ±√𝑀2 𝑎2 − 𝑏 2
𝟒
d) (i) Prove that the Cartesian of 𝒓 = represents the translated parabola
𝟏+𝐜𝐨𝐬 𝜽
(ii) Draw the graph of 𝒓 = 𝟐 + 𝟒 𝐜𝐨𝐬 𝜽

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