PRESIDENT’S OFFICE

REGIONAL ADMINISTRATION AND LOCAL GOVERNMENTS

SAME AND MWANGA SECONDARY SCHOOLS EXAMINATION
SYNDICATE (SAMWASSES)

FORM SIX PRE-MOCK EXAMINATIONS-2020

ADVANCED MATHEMATICS 2

CODE: 142/2

TIME: 3 HOURS 22nd September 2020

INSTRUCTIONS:
1. This paper consists of sections A and B with a total of eight (8) questions.
2. Answer all questions in section A and any two questions from section B
3. All work done in answering each question must be shown clearly.
4. Mathematical table and non-Programmable Calculator may be used.
5. Cellular phones and any other unauthorized materials are not allowed in an
examination room
6. Write your examination number on every page of your answer sheet

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 SECTION A
Answer all questions in this section.

1. a) Mwanyika’s family consists of mother, father and their ten children. The
family is invited
to send a group of 4 representatives to a wedding. Find the number of ways in
which the group can be formed, if it must contain;
i. Both parents
ii. One parent only
iii. None of the parents
b) The events that A and B are independent are such that, P ( A )=0.2, P ( B )=0.4
and P ( A ∪ B )=0.35, Find
i. P ( A / B )
ii. P ( B / A ' )
c) i) For the discrete random variable x, Prove that var ( ax+ b )=a2 varx
ii) The discrete random variable has the probability distribution table shown
below
x 1 2 3 4
P( X=x) 3 1 1 1
8 8 4 4

Find the δ ( 4 x +3 )

d) Suppose the rate of change of mass of an ice is defined by the continuous

{
2 3
random variable x is given by the function f ( x )= 12 x −12 x for 0 ≤∧x ≤ 1
0 Ot h erwise

i. verify that the function is P.d.f

1
( 3
ii. Evaluate P 4 ≤ x ≤ 4 )
iii. Calculate the mean.

2. a) Write the contra positive and construct the truth table of the converse of the
conditional
( p → q) → q

b) Let P represent the clouds and q represents it rains. Write a word sentence
that can be represented by;
i. ( p ⋀ q )→ ( p ⋁ q )
ii. ( p ⋁ q)→ ( ( p ⋀ q ))
c) Write the statement M in most simplified form from A, B, and C

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 A T T T T F F F F
B T T F F T T F F
C T F T F T F T F
M F F T T F F T T

d) Draw a simple electrical network corresponding to the compound statement

P ⋀ [ Q ⋀ R ⋁ Q] ⋁ ( P ⋀ R)

3. a) i) If |a|=13 ,|b|=5 and a . b=60 Find the value of |a× b|

a∙b
⏟
a
ii) Prove that the component of vector a in direction is given by b
b⏟
||
b) If a=i−2 j+3 k and b=2 i+3 j−5 k then finda × b the arc cosine between the
two vectors, hence verify that a × 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 ¿ find its area

d) i) Forces of magnitudes 10,12 and 14 Newton act direction of 2 i+ 2 j+ k ,

6 i−5 j+ 4 k and 9 i+4 j−6 k respectively. What is the work done by their
resultant if the particles they move undergo a distance of 16 i+3 j

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

r =t 3 i+ ( t 2 −2t ) j Find the velocity and acceleration of this particle at 2.

4. a) i) If x is real, show that ( 2+i ) p (1+3 i ) x + ( 2+ i ) p(1−3 i ) x is real

ii) Prove that for the complex numbers Z1 ∧Z 2 , Arg ( Z 1 . Z 2 ) =Arg ( Z 1 ) + Arq ( Z 2 )

b) i) Use Demoivre’s theorem to prove that cos 4 θ=8 c os4 θ−8 c os2 θ+1
ii) Use the result b(i) above to evaluate ∫ ( 8 c os 4 θ−8 c os2 θ+5 ) dθ

c) i) Solve the following system of equations where Z and W are complex

numbers
{iZ−W =2 i
iZ +iW =i

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 ii) One of the roots of the equation Z 4−6 Z 3+ 23 Z 2−34 Z +26=0 is i+1; find the
other roots

SECTION B
Attempt any two (2) questions in this section

3 1 1
5. a) i) Prove that 2 cos 3 θ=x + 3 given that 2 cos θ=¿ x + x ¿
x

ii) If A, B and C are angles of triangle, prove that

A B C
sin A+sin B+sin C=4 cos cos cos
2 2 2

α β
b) If tan 2 and tan 2 are roots of the equation 8 x 2−26 x +15=0. Find the value of
cos ( α + B )

c) Prove that
i. tan
−1
( 34 )+ tan ( 13 )=tan ( 139 )=53.3 °
−1 −1

2
1−x
cos ( 2 tan tan ( x ) ) =
−1
ii. 2
1+ x

d) i) Find the general solution of the equation ( 2 tan x−1 )2=3 ( sec 2 x −2 )
A+ B A−B
ii) Show that sin A+sin B=2sin 2 cos 2 ( ) ( )
2
6. a) Express n ( n+1 ) ( n+2 ) in partial fraction and use it to deduce the sum of
n

∑ k ( k + 12)( k + 2 )
i=1

and find its sum when n → ∞

( )
0.5
1+ x
b) Expand 1−2 x as series of ascending powers of x up to and including the
term containing x 2 then state the range of values of x for the expression is
valid.

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 c) Solve the following simultaneous equations by Cramer’s Rule.
2 x+3 y −z=−7
−3 x+ y +2 z=1
3 x−4 y−4 z=−1

7. a) i) Form the differential equation of the equation y= A e2 x + B e−3 x

ii) Sow that y=2−cos x is a particular integral of the differential equation

2
d y
2
+4 y=8−3 cos x
dx

b) Solve the following differential equations

dy 1
i. ( 1+ x ) −2 y=
dx 1+ x
ii. xy
} + {y} ^ {'} - x = ¿

c) In a cultures the bacteria count 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 y 2 +4 y−8 x−4=0 represent a parabola. Find its
focus,
Vertex, equation for its directice of the ellipse and symmetrical axis

ii) Find the eccentricity, foci and equation of directise of the ellipse.
2 2
x y
+ =1
25 9

2
b) i) Change the equation ( x 2 + y 2 ) =x 2− y 2 into polar form.

ii) Express r =4 sin θ cos θ in Cartesian form.

c) i) Show that the line 3 x−4 y=5 is tangent to the hyperbola x 2+ 4 y 2=5 and find
the points of contact.

ii) Write down the equation of asymptotes graphically to hyperbola

4 x −9 y =36 and hence Sketch the asymptotes in the Cartesian plane.
2 2

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