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PRESIDENT’S OFFICE
REGIONAL ADMINISTRATION AND LOCAL GOVERNMENT
BRIGHT DIAGNOSTIC JOINT EXAMINATION SYNDICATE
FORM SIX PRE NATIONAL JOINT EXAMINATION
ADVANCED MATHEMATICS 02
CODE: 142/2
TIME: 3 HOURS WEDNESDAY 20 FEBRUARY, 2019 PM
INSTRUCTIONS
I. THIS PAPER CONSISTS OF SECTIONS A AND B WITH A TOTAL OF EIGHT (8)
QUESTIONS
II. ANSWER ALL QUESTIONS IN SECTION A AND ANY TWO (2) QUESTIONS
FROM SECTION B
III. ALL WORKDONE IN ANSWERING EACH QUESTION MUST BE SHOWN

CLEARLY
IV. NON PROGRAMMABLE CALCULATORS MAY BE USED
V. CELLULAR PHONES ARE NOT ALLOWED IN THE EXAMINATION ROOM
VI. WRITE YOUR EXAMINATION NUMBER ON EVERY PAGE OF YOUR ANSWER

BOOKLET(S)
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SECTION A(60marks)
Answer All questions in this Section
(1)(a)(i) If is real, show that is real.

(ii) Prove that for the complex numbers
(b) (i) Use de Moivre’s theorem to prove that

(ii) Use the result b (i) above to evaluate
(c) (i) Solve the following system of equations where z and w are complex numbers
(ii) One of the roots of the equation is , find the

other roots.
(2) (a) (i) Differentiate between Tautology and Contradiction

(ii) Show that the following statement is valid:
If I study hard, then I will not fail Mathematics.
If I do not fail to manage my time, then I will study.
But I failed Mathematics
Therefore I failed to manage my time.
(b)(i) Write down the statement corresponding to the following network
(ii) Given the statement P Q

 Make the truth table for the converse
 Make the truth table for the contrapositive
(c) A sentence M is formed from the statements A, B and C as shown in the table below.
Write M in its most simplified form and use it to draw an electric circuit which will allow
current to flow.
A B C M
T T T F
T T F F
T F T T
T F F T
F T T F
F T F F
F F T T
F F F T
(3) (a) (i) If = 13, = 5 and Find the value of
(ii) Prove that the component of a vector in the direction is given by
(b) If and + 3j 5k, then find a x b and the arc cosine between the two
vectors, hence verify that a x b are perpendicular to each other
(c) (i) Derive the formula for the area of the triangle by using vector product technique
(ii) The triangle ABC has the vertices A(-2, 3, 1), B (2, 4, -1) and C(1,-1, -1).
Find its area
(d)(i) Forces of magnitudes 10, 12 and 14 Newtons act direction of .

and respectively. What is the work done by their resultant if the particles
they move undergo a distance of
(ii) The position vector of a particle moving in the plane is given by .

Find the velocity and acceleration of this particle at .
(4) (a) Express in partial fraction and use the result to deduce the sum of
and find its sum when
(b) Expand as a series of ascending powers of up to and including the term

containing then state the range of values of for which the expansion is valid
(c) Solve the following simultaneously equation by cramers rule

2x + 3y -Z= -7
-3x + y + 2z= 1
3x – 4y – 4z = -1
SECTION B(40 MARKS)

Answer only TWO Questions from this Section
(5) (a) Prove that (i) tan-1( ) + tan-1( ) = tan-1( ) = 55.3o

(ii)
(b) Express + 12 in the form giving the values of r and .
Hence show that + 12 +7 20 and find the minimum value of
+ 12 + 7.
(c) Factorize completely the trigonometric expression

– – +
(d) Find the general solution of the equation

- = 1 by using half angle formulae
(6) (a) The events that A and B are independent are such that
P(A) = 0.2, P(B) = 0.4 and P(A ) = 0.35. Find (i) (ii)
(b) A bag contains 10 red, 3 green balls, while another bag contains 3 red and 5 green
balls. Two balls are drawn from the first bag and put into the second and then a ball
is drawn from the later. Find the probability that it is red
c) A committee of 4 must be chosen from 3 women and 4 men. Calculate :

(i) The number of ways that the committee can be chosen
(ii) The number of ways that 2 men and women can be chosen
(d) Suppose the rate of change of mass of an ice defined by the continuous random
variable is given by the function
(i) Verify that the function is a p.d.f

(ii) Evaluate P ( )
(iii) Calculate the mean
(7) (a) (i) Form the differential equation of the equation +B

(ii) Show that is a particular integral of the differential equation
4y = 8 – 3Cos x
(b) Solve the following D.E(s)
i.
ii.
(c) In a cultures the bacteria counts in 100,000. The number is increasing by 10%
in 2 hours. In how many hours will the count reach 200,000, if the rate of growth
of bacteria is proportional to the number present?
(8) (a) (i) Show that the equation + 4y – 8x – 4 = 0 represent a parabola. Find its focus,
vertex, equation for its directrix and symmetrical axis.
(ii) Find the eccentricity, foci and equation of directrix of the ellipse
+ =1
(b) (i) Change the equation into polar form

(ii) Express r= 4sin cos in Cartesian form
(c) (i) Show that the line is a tangent to the hyperbola + =5

and Find the point of contact.
(ii) Write down the equation for the asymptotes graphically to hyperbola
and hence sketch the asymptotes in the Cartesian plane
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Visit: www.africona.net/Teacher pizo for Free Past Papers and Notes.

III. ALL WORKDONE IN ANSWERING EACH QUESTION MUST BE SHOWN

IV. NON PROGRAMMABLE CALCULATORS MAY BE USED

V. CELLULAR PHONES ARE NOT ALLOWED IN THE EXAMINATION ROOM

VI. WRITE YOUR EXAMINATION NUMBER ON EVERY PAGE OF YOUR ANSWER

(1)(a)(i) If is real, show that is real.

(b) (i) Use de Moivre’s theorem to prove that

(ii) One of the roots of the equation is , find the

(2) (a) (i) Differentiate between Tautology and Contradiction

(b)(i) Write down the statement corresponding to the following network

(ii) Given the statement P Q

(3) (a) (i) If = 13, = 5 and Find the value of

(ii) Prove that the component of a vector in the direction is given by

(d)(i) Forces of magnitudes 10, 12 and 14 Newtons act direction of .

(ii) The position vector of a particle moving in the plane is given by .

and find its sum when

(b) Expand as a series of ascending powers of up to and including the term

(c) Solve the following simultaneously equation by cramers rule

SECTION B(40 MARKS)

(5) (a) Prove that (i) tan-1( ) + tan-1( ) = tan-1( ) = 55.3o

(c) Factorize completely the trigonometric expression

(d) Find the general solution of the equation

c) A committee of 4 must be chosen from 3 women and 4 men. Calculate :

(i) Verify that the function is a p.d.f

(7) (a) (i) Form the differential equation of the equation +B

(b) (i) Change the equation into polar form

(c) (i) Show that the line is a tangent to the hyperbola + =5

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