1 Using elimination method to solve the set of simultaneous equations:

X - 2y + ƻ = 5
2X - Y +ƻ= 15
3X + 2Y - 2ƻ= 6

(10 Marks)㎡

Two resistors of resistance R1 and R2 have a effective resistance of 9ῼ when

connected in series and 2ῼ when connected in parallel

Show that R12 - 9 R1 = 18 = 0;

Hence determine the values of r1 and r2.

(10 Marks)

2 a) Determine the middle term of the binomial expansion of [2x + y]6,

Hence evaluate the value of the term when x = 3 and y =1/2

(3 Marks)

B) Using substitution , solve the simultaneous equations:

4x + 6y + 7ƻ = 7
2x + 3y + 8ƻ = -10
x + 9y = 22

(10 Marks))

3 a) Given the matrices

4 3 3 4 1 1
M =2 1 5 and N=1 3 1
1 2 1 2 1 3
Show that MTNT = (NM)T
(8 Marks)

C) Three voltages v1,v2 and v3 in a d.c circuit satisfy the simultaneous equations:

ZV1 - ZV2 - 4V3 = 10

4V1 + 4V2 - 2V= = -14
-v1 + 5v2 + 3v3 = -24
Use inverse matrix method to solve the equations
(12 Marks)

4 a) Given the vectors B =2i + 3j + k and I = 4i + j - 2K , I =4i + j + 2k and I =4i + j +

2k,determine;
Angle between the vectors;
Unit vector perpendicular to both vectors.
(10 Marks)

C ) Given the vector field function V = x2yi + xyi -x3ƻk and the scalar potential
function V = 4x2ƻ + 3yƻ2 , determine at the point ( 1,2,1 )

▼.E

▼XE

▼V

2 B) Expand up to the term in x2

Hence by setting x = 1/5 , determine the value of √2/3, correct to two decimal places

(7 Marks)
1 a)

i. Use the binomial theorem to expand

√ 1+5 x as far as the term in x3
√1 −5 x
1
ii. By setting x = /20 determine the approximate value of correct to 5
decimal places
(10 Marks)

b) Use elimination method to solve the equations

3x + 5y + 2z = 38
6x - 2y + 5z = 34
4x + 3y + z = 26

(10 Marks)

2 a) Given the scalar field ø(x,y,z) = x4 + y4 + ƻ4. Determine the directional

derivative of ø at the point A( 1, -2,1) in the direction of AB, where B is ( 2,6,-1).
(8 Marks)

b) Determine the angle between the vectors (6

Marks)
2 c) Solve the following quadratic equation using completing square method
9y2 - 10y + 1 = 0
(6 Marks)
B) Use the inverse matrix method to solve the equations;
x+y+z =4
2x + 3y + 4z = 33
3x - 2 - 2z = 2

(13 Marks)
3a) Given matrices

4 3 5 1 2 3
A= 2 6 − 1 and B = 6 3 4 Evaluate
8 9 −6 4 2 5
5A - 3B
BA
(7 Marks)

b)Use the inverse matrix method to solve the equations;

x + y + z= 4
2x - 3y + 4z = 33
3x - 2y -2z = 2
(13 Marks)
4a)Three forces F1,F2 and F3 in newtons, necessary for the equilibrium of a certain
mechanical system satisfy the simultaneous equations;

F1 - 2F2 + F3 = 1
F1 + 3F2 - 2F2 = 2
F1 + F2 + F3 = 7

Use substitution method to solve the equations (10 Marks)

b) Expand 1/( 4- x) 2 in ascending powers of x as far as the term in x3, using the
binomial theorem
What is is the limits of x for which the expansion in (i) above is valid
(10 Marks)
5 a) Given the vectors A = 10i + j = 2k,
B = 2i + 2j = 4k and C = 5i + 2j = 3k, determine;
AxB
A. (B x C)
Determine the unit vector perpendicular to both P = 2i + j + 2k and Z = 3i +2 j = 6k.
 (5 Marks)

c) An electric field E = xzi + y2zj + xyzk exists in a region of space

Determine ;
∇ .E
∇XE
at the point (1, -1,2 ) (8 Marks)
Cvl/600/s21
1 a)
i. Differentiate f(x) = 4x2 - 2x + 5 from the first principle
ii. Hence determine the gradient of the curve at x = -3
(8 Marks)
b) Given that z = In(x2 + y2 ), show that
2 2
d z d z
2 + 2 =0 (12 Marks)
dx dy

2 a)Given that z = In ( xy ) (x2 + y3)

2 2
d z d z
Determine
dx
2 + dy
2 in terms of z (10 Marks)

b)The radius of a cylinder increases at the rate of 0.2cm per second, which the height
decreases at the rate of 0.5cm per second. Find the rate of change at which the volume
is changing at an instant when r is 8cm and h is 12cm.
(10 Marks)

3 a) Evaluate the integration using partial fractions;

2
3x
❑

4a)Locate stationary points of the function z = x3 - 6xy + y3 and determine their

nature
(10 Marks)
b)Using integration to determine the area of the region bounded by the curve y = 2x
- x3 and a straight line y = x (10 Marks)

5a) Given the complex numbers z1 = 2 - 5j , z2 = 5 + 7j and z3 = 6 + 2j , determine

z 1+ z 2
the value of z1 = z3 + z 1z 2 in polar form (8 Marks)
 b)Use Demoivres theorem to express cos4θ in terms of cosines of multiples of cosθ
(6 Marks)
c) Use Demoivres theorem to find an expression for cos3θ and sin3θ in terms of
multiple of cosθ (6 Marks)
3
x
d)Determine the equations of the tangent and normal to the curve y = the point (-
5
−1
1, )
5
EEE/TEL/EIC/ART M23 1
SECTION A - PART 1
SECTION A
1) In this section , you are required to provide short oral response

2) What are the three methods of solving quadratic equations (3 Marks)

3) What is the value of any number to base Zero ( 1 Mark)
4) State the three formula for solving simultaneous equations with three unknown(1
Mark)
5) Name 3 special angles as referred to in trigonometry ( 3 Marks)

WRITTEN ASSESSMENT (10 MARKS)

Indicate as to whether the following statements are TRUE or FALSE (4 Marks)

X2 x 3 = x6
6
x
3 ≡ x2
x
x ≠ x ≡ 23√ x
3 1/ 3

(xn)m≡ xnm

2 a) You may use the binomial theorem to obtain an expansion of √ 16+ 4 x

(1Mark)
b)There are five base trigonometric identities ( 1 Mark)

a) Cos( A +B) = cos AcosB - sinAsinB (1 Mark)

b) Sinά + sinβ = 2sin1/2(ά + β) cos 1/2 ( ά -β) ( 1 Mark)

c) 9/sinA + b/ cosB = C/sinC = 2R (2 Marks)

SECTION B

1 a Use indices to simplify

log64 - log1024 + log 4096

log7 512 +log32768 + log7 262144 (5 Marks)

b)Solve the equations;

(x2 + 5x)log39 + ( X + 2)log216 = -12
(7 Marks)

2)Solve the equation using the formula method

1 4
+ =2 (7 Marks)
x −3 x −1

b)Three forces F1,F2 AN F3 in newtons, necessary for the equilibrium of a certain mechanical

system satisfy the simultaneous equations;

Use elimination method to solve the equations

F1 - 2F2 + F3 = 1
F1 + 3F2 - 2F3 = 2
F1 + F2 + F3 = 7
Solve the following by factorization (7 Marks)

8, and find its value

3 Determine the middle term in binomial expansion of (3x + y)

when x = 1 and y = 1 . (6 Marks)

2 3
4 a) Expand 1/(4 - x)2 in the ascending power for x as far as the term in x3, using

binomial theorem and state the vales of x for which the expansion is valid .

( 7 Marks)

5 a) Simplify the expressions;

log 125− log 25 − log 5 (7 Marks)

log 625+1/2 log 25

6 Solve the equations

2 + 3 =2 (6 Marks)
x+2 x+3

7 .Use the binomial expansion to obtain the first three terms of

√ 1+ x (5 Marks)
√1 − x
8 Solve the equation

Log x + 4 log 4 - 5 = 0 ( 6 Mrks)

4 x

9 Solve the equations

( 5 log10x) + (2 -log10x) = 30 ( 8 Marks)

59 5

10 The sum of three numbers in arithmetic progression is 27 and the sum of their squares

is 293 .Determine the :

Numbers
I.
II. Sum of the first 10 term of the series (10 Marks)
11 In a geometric progression , the sum of the second and third terms is 9, and the

seventh term is eight times the fourth .Determine the

First term
i.
ii. Common ratio ( 6 Marks)
C2 EET/TEL/EIC/ART M23 - 1

SECTION A

In this section ,you are required c to provide short oral response

What an arithmetic operations s in mathematics (Give an example) ( 2 Marks)

1) Define indices ( 2Marks)

2) State the three main formula for solving instantaneous equation with three
unknown ? ( 3 Marks)

3) State any the formula of solving quadratic equations ( 3 Marks)

Section A part II

Write short response questions ( 10 Marks0

Indicate whether the following statement is TRUE or FALSE. ( 3 Marks)

3x 2≡ 6
x x x

m/ n ≡ m-n
x x x

If n = y then log y =n
x x

2 You can use binomial expansion to find the expansion of

√ 1+ X (1 Mark)
√1 − X
1
3 Logab = a (1
logb
Mark)

b
logn
4 Logab = a (1 Mark)
logn

5 The equation below can be reduced into a quadratic equation

32x - 3x+1 = 0 ( 1 Mark)

6. The equation below can be solved using binomial theorem :

8x2 + 8x + 144 = 0 (1 Mark)

n −1 n−2 2
n= n+ na + (n −)a . b is a quadratic formula
7 (a + n) a
1 2
(1 Mark)

5 Simplify the expressions:

1
log 1024 − log256+3 log16
2
(7
1
log64 − log 16+ log 4
2
Marks)

6 Solve the equation 13.2 ( 122x + 4) = 16 ( 6 Marks)

7 Simplify
3 4 2 2
3 x 5 +3 x 5 ( 4Marks)
3 3
3 +5
8 Solve the equation

-
7 +6=0 ( 6 Marks)
2 x +1 2x
2

9 Determine the middle term in the binomial expansion of

3 x+ 4 y , and find the value and find
8

its value when x = 1 and y = 1 ( 6 Marks)

2 3
10 Solve the equation

1 , + 2 correct to two decimal places ( 6 Marks)

x −2 x −1
SEC B -50 MARKS

1. Solve the equation

9 -3
X 2 X +1 + 8 = -10 (6 Marks)

2 The application of Kirchott’s law to a d.c circuit yielded the simultaneous equations;

I - 2I - 2I = -1
1 2 3

2I + 3I + I = -1
1 2 3

3I - I - 3I = -4
1 2 3

Where I ,I and I are currents in amperes .Use substitution method to solve the equations (8
1 2 3

Marks)

3) Solve the equations

log 4 + 4 logx - 5 = 0 ( 6 marks)

x 4

Three current I , I and I in amperes flowing in a d,c network satisfy the equations;
4) 1 2 3

3I - I + 3 = 7
1 2

I + 2I + 3I = 20
1 2 3

2 + 4I + I = 20
1 2 3

Use elimination method to solve the equations; ( 7 Marks)

Prove the trigonometric identities

1−sinθ ( 8 Marks)
1+sinθ
= secθ−tanθ 2
PRD/AUT/EET/TELS20/S21 Sup 2

1 a) Reduce the equation below and find the value of x

9X+1 - 3X = 3(X+2) - 1
9X - 3(2X +1 ) + 8 = -10 ( 10 Marks)

b)Three currents I1,I2 and I3 in amperes flowing in a n electric circuit satisfy the
simultaneous equations;

2I1 + 3I2 - 4I3 = -4

3I1 + 4I2 - I3 = 8
I1 - 5I2 + I3 = -6

Use the method of elimination to determine the values of the currents ( 10 Marks)

2 a) Solve the equations below using quadratic formula method ;

1 4
+ =2 ( 12
X −3 X −1
Marks)

b)Three forces F1, F2 and F3 in Newtons ,acting at a point in a system satisfy the
simultaneous equations

2F1 - F2 + 3F3 =4
-F1 + 2F2 + F3 = 5
F1 + F2 + 4F3 = 1
( 8 Marks)

Use substitution method to solve the equations

3 a) Given the matrices

 2 −1 3 1 −1 1
A=1 3 4 and B = −2 2 1
1 −2 −2 0 1 3

(10 MARKS)
Determine (AB)-1

B)Three magnetic fields B , B AND B in a Tesla satisfy the linear equation ;

1 2 3

8B + 5B + 3B = 11
1 2 3
2B + 7B + 6B = 30
1 2 3
4B + 2B - 2B = -6
1 2 3
Use crames rule to solve the equations (10 Marks)

4 a) Find the middle term in the binomial expansion of ( 5x -2y)10 and determine its
1 1
value when x= and y= (5 Marks)
2 5

b) Use the binomial theorem to expand

√ 1+ 5 x As far as the term in x3

√1 −5 X
By setting x=1 , determine the approximate value of correct to 5 decimal places
/20 √ 15
( 10 Marks)
Solve the equation

4 - 5 =1
(5 Marks)
x+3 6 x+2 2

C) A Scalar potential Q, and a vector F are given by Q= 2x3y2 + 3xyk .Determine at

point (1,2,1):
∇θ;
∇ . F;
∇ XF ;
(10 Marks)
Given the matrices ;

A=
5A -3B
BA
(10 Marks)