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JIM 211 ADVANCED CALCULUS

2018/2019

Question 1 ( 2.15 - 2.50 pm )

( a ) Given x² -y²+2u² + 3v² -7 = 0

2x² -3y² -3uv -2 = 0

Find

( i ) ꝺu / ꝺx

( ii) ꝺv/ ꝺy ( 40 marks )

( b ) Find the local extreme values and saddle points
of the function f( x, ),

if there is any,given that

f( x, ) = x² -2xy + ( 1 / 3 )y³-3y
( 30 marks )

( c ) Use the method of Lagrange multipliers to find the extreme values of the

function f( x,y ) = 3x² +2y²-4y + 1

on the region
x ²+ y ²< 16
( 30 marks )

Question 2 ( 2.50 - 3.25 pm )

( a ) Evaluate

( i ) ʃ ₀² ʃ ₀^{[ 4- ( x )² ] 1/2} x( x² + y² )^-1/2dydx

( i i ) ʃ ʃ _D2x dxdy

over the region D enclosed by
the curve y = e^x

the lines x= 0,x = 2 and
y = 0
( 50 marks )

( b ) Evaluate

( i ) ʃ ₁^e ʃ _{π /2}^π ʃ ^1n2y_lny e^x ( 1 / y )sin y dxdydz

( i i ) ʃ ʃ ʃ _V ( x² + y² + y² )^1/2dxdydz

V = { ( x,y,z ) € R³/ 1 < x² + y² + z² < 4 }

( 50 marks )

Queston 3 ( 3.25 - 3.55 pm )

( a ) State the definition for each of the following:

                      ( i ) increasing sequence
                      ( ii ) bounded sequence
                      ( iii ) convergence of a sequence
                      ( iv ) Cauchy sequence
( 40 marks )

( b ) The sequence { a_n } is is defined as
a ₁=1,a _{n + 1}= 3-( 1/a_n) , n> 1
       Show that
                     ( i ) { a_n } is increasing
                     ( ii ) a_n< 3,for all n
                     ( iii ) { a_n } is converging

( c ) Show that the sequence { 1 /n } is Cauchy
( 20 marks )

Question 4   ( 3.55 - 4.30 pm )

( a ) State the Squeezing Theorem for sequences.

       Hence ,use the theorem to evaluate

lim_x->α x sin ( 1/x
( 30 marks )

( b ) Calculate the following limits :

       ( i ) lim_x->1   ( x³ - x² + x -1 ) / ( x + ln x -1)^-1 ^{( ii ) lim_{x->1
[} ln
( x + 1 )^-1 - (
1 / x ) ]

       ( iii) lim_x->1  [
( e^3x- 1 ) / ^{tan x ]
( iv) lim_x->1
[( 3^x -1 ) / ( x )]}}
( 70 marks )

Question 5   ( 4.30 - 5.15 pm )

( a ) Determine whether each of the following series converges or diverges
infinity
( i ) ∑ ( e²ⁿ / nⁿ )
n = 1

( ii ) infinity

∑ ( nⁿ / n ! )

n= 0

_{( iii ) infinity}

∑ ( 1 / ink )
k = 2

_{( iv ) infinity}

∑ [ 3^k / 4^k ( k + 1 ) ]

k = 1
( 60 marks )

( b ) Find the taylor's polynomial of degree 3
about x = 1 and the remainder for
      the function f given by
f ( x ) = xe^x
( 40 marks )

=========END=============

JIM 201 Linear Algebra
2018/2019

1. Consider the matrix

A = 2 3 4
2 1 1
-1 1 2

( a ) Calculate
( i ) determinant A
( ii ) adj ( A )
( iii ) A^-1
^{( iv ) determinant adj ( A )}
^{( v ) A adj( A )}
^{( vi ) Reduced row-echelon form of A}
^{( vii ) Solution of X if 2}
^{AX = -1}
³

^{( 60 MARKS )}
^{( b ) Let}B= [ b_ij ]_3x3be a 3x3 matrix and I is an identity
matrix.

Given that

E₁³( -2 )E₂³( 1 )E₃ (1/2 )E₃² (-1 )E₁²( -1 )E¹₂(-1)B=1

Find B^-1 and B
( 40 marks )

2 ( a ) Given the system of linear equations

x + 2y + 3z = 6
2x -3y +2z = 14
3x + y -z + -2

Solve the system using the method of
( i ) Cramer's Rule
( ii ) Gauss-Jordan elimination
( 60 marks )

( b ) Determine the values of constant k if the system of linear equations

x + 2y -3z = 4
3x -y +5z =2
4x + y + ( k² -14 ) z =k+2

has
( i ) unique solution
( ii ) infinitely many solutions
( iii ) no solutions
( 40 marks )

3 ( a ) State the conditions that a subset W of a vector space is a subspace of V

( 20 marks )
( b ) Show that each of the following vector is a subspace of V

( i ) S = { 3y / y e R } , V = R²
y

^{( ii ) S ={ a -a / a, b, e R} , V =}M_2x2
^{b 0}

^{( iii ) S ={ x / x + y = 0, x,y e}R } , V = R²
^y
^{( 40 marks )}
^{( c ) Let T : V ->W be a linear transformation.}
^State^{the definition for each of the following :-}
^{( i ) Kernel of T}
^{( ii ) Range of T ( 20 marks )}
^{( d ) Given that T :}R² -> R where T(x,y ) = x + y
^{Find the kernel and range of T ( 20 marks )}
^{4 ( a ) Let matric A be diagonalizable.Then there exist two matices P and D}
^{such that}P^-1 AP = D
^{Show that}Aⁿ ⁼PDⁿP^-1
^{by induction}^{( 30 marks )}
^{( b ) Given the matrix}
^{2 1 1}
^{A= 1 2 1}
^{0 0 1}
^Find
^{( i ) the characteristic polynomial}
^{( ii ) eigenvalues}
^{( iii ) eigenvectors}
^{( iv ) the matrix P that diagonalizes A ( 70 marks )}
^{5 ( a ) State the definition for each of the following}
^{( i ) orthogonal set}
^{( ii ) orthonormal set}
^{( iii ) orthogonal basis}
^{( iv ) orthonormal basis ( 40 marks )}
^{( b ) Determine whether each of the following set is orthogonal or orthonormal}
^{( i ) S = { ( 1,0,0),(0,1,0),( 0,0,1) } c}R³
^{^{( ii ) S = { ( 1,1,1),(-2,1,1),( 0,-1,1) } c}R³}
^{^{( c ) Use Gram -Schmidt process to find an orthogonal basis and orthonomal basis from the set}}
^{^{^{S = { ( 1,1,0),(-1,-1,1),( 1,2,4) } c}R³}}
^{^{^{( 40 marks )}}}
^{^{^{=============END=================}}}
^{^{^{JIM 213 Differential Equations I}}}
^{^{^2018/2019}}
^{^{^{1 ( a ) Find the solution for the following differential equation xy^' + 2y =  x³ , y( 1 ) = 2}}}
^{^{^{( 40 marks )}}}
^{^{^{( b ) Show that y = 1 / ( cx²-x^{3 )}}}}
^{^{^{where C is a constant, is the general solution}}}
^{^{^{for the Bernoulli equation}}}
^{^{^{dy/dx + ( 2/x )y}}}= x²y²

2( a ) Verify that the integrating factor of

  x²y^' = 3y + x⁴

is u( x ) = (e³ )^1/x
( 20 marks )

( b ) Show that the following equation

( e^xsinx-2ysinx ) dx = -(2cosx +e^xcosy)dy

is an exact equation.Hence ,find the general solution

( 30 marks )

( c ) Find the explicit solution for the following
initial-value problem

dy /dx - xe^2y  = e^2y sinx, y( 0 ) = 0

( 50 marks )

3 ( a ) The population of a town P( t ) is modelled by
the following differential equation

dP/dt = kP

at time t.The initial population of 5000 increase
by 20% in 5 years .What will be the population
in 12 years ?

( 30 marks )

( b ) A 250-Volt electromotive force is applied to
an RC series circuit where the resistance is
1000 ohms and the capacitance is 5x 10^-6farad.

Find the charge q( t ) on capacitor if the
initial charge is q( 0 ) =0.
Hence,find the current I( t )

Hint : Kirchoff's Second Law, RI + ( 1/C ) q =E( t )

( 30 marks )
( c ) Given LRC series circuit

L d²q / dt² + Rdq/dt + ( 1 / C ) q = E( t )

with L = 1 Henry, R= 20 ohms, C= 0.0005 farad
and E( t ) = 300 volt

( i ) Show that the general solution of the charge
on the capacitor is
q( t ) = e^-10t[A cos( 10t ) + B sin( 10t) ] + 1.5

( ii )Hence,find the particular solution
if q( 0 ) = 1 and I( 0 ) = 0

( iii ) What is the charge on the capacitor when
t ->infinity ?

( 40 Marks )

4( a ) Solve the initial value problem of Cauchy-Euler
differential equation

x²d^{2 y}/dx^{2 - ( 3x )dy/dx}+ 3y = 3x⁴

y ( 1 ) = 6 , y^'( 1 ) = 7

( b ) Use the Laplace transform to solve the following initial-value problem
dy/dt -2y = e^-t, y( 0 ) = 0

( 30 marks )

( c ) Using Laplace transform, prove that the particular

solution of

y^"+ 4y^'+ 4y = e^-2t , y( 0 ) =1 , y^{'( 0 )}= 1

is
y( t ) = ( 1/6 )t³e^{-2t +}e^-2t + 3te^-t

( 40 marks )

5 Given a system of homogenous linear differential equations
dx /dt = x +y +2z

dy/dt = x + 2y +z

dz/dt= 2x + y + z

( a ) Write the system of equations in the form

dX /dt = AX

where x and identify the matric A
A = y
z
( 15 marks )

( b ) Show that the characteristic equation is

  k³-4k²- k + 4 = 0 , k = eigen values

( c ) Find all eigen values and the corresponding
eigenvectors
( 50 marks )

( d ) Hence, write down the general solution of the
given system

( 10 marks )

================END=================

JIM 212 Statistical Method

Answer ALL Four QUESTIONS

Question 1
An insurance company wants to know if the amount of life insurance depends on the incomes of a person.The research deparment at the company collected information on six persons.The table below lists the annual income ( in thousands of RM) and the amount( in thousands of RM ) of life insurance policies for these six persons.

Annual Income 62 78 41 53 85 34
Life Insurance 250 300 100 150 500 75

( a ) Find the regression line y = a + bx with annual
income as an independent variable and amount
of life insurance policy as a dependent variable

( 50 marks )
( b )Give a brief interpretation of the values of a and b
calculated in part ( a )
( 20 marks )

( c ) What is the estimated value of the life insurance
for a person with an annual income of RM55,000 ?

( 10 marks )

( d ) One of the persons in this sample has an annual
income of RM78,000 and RM300,000 of life
insurance.What is the predicted value of the life
insurance for this person ?.Find the error for
this observation. ( 20 marks )

Question 2

( a ) Describe in your own words a test of independence and a test of homogeneity.Give example for each

( 20 marks )

( b ) To make a test of indepence or homogeneity,what should be the minimum expected frequency for each cell ? What are the alternatives if this condition is not satisfied ?

( 20 marks )
( c ) Two drugs were administered to two groups of randomly assigned 60 and 40 patients,respectively to cure the same disease.The following table gives information on the number of patients who were cured and not cured by each of the two drugs

CURED NOT CURED
DRUG I 44 16
DRUG II 18 22

Test at the significance level if the two drugs are similar in curing the patients ( 60 marks )

Question 3

( A ) A university alumni office wants to compare the time taken by graduates from three majors to find their first job after graduation.A random sample of eight business majors,seven computer science majors and six engineering majors who graduated in 2017 were taken

The following table list the time ( in day ) taken to find their full - time job after graduation

Business Comp Science Engineering
36 56 26
62 13 51
35 24 63
80 28 46
48 44 78
27 47 34
76 20
44

At the 5% significance level,can you conclude that the mean time taken to find their first job for all 2017 graduates in these fields are the same ?
( 50 marks )

( b ) The two -way table below gives data for a 2x2 factorial experiments per factor.

Factor B
Low High

Low 29 47
35 42
Factor A
High 12 28
17 22

Construct the ANOVA table for this experiment and do a complete analysis at alpha = 0.05

Question 4
( a ) Two different devices for measuring suphr monoxide in the atmosphere were compared in a test for measuring air pollution.The following data was obtained

Sulphur Monoxide ( parts per million )

Device A Device B
0.96 0.68 0.87 0.57
0.82 0.65 0.74 0.53
0.75 0.84 0.63 0.88
0.61 0.59 0.55 0.51
0.89 0.94 0.76 0.79
0.64 0.91 0.70 0.84
0.61 0.77 0.69 0.63

( 50 marks )

( b ) The following table gives the burning times ( in minutes ) of four fabrics coated with inflammable materials

Fabric Burning Times ( Minutes )
1 18 17 18 17
2 12 11 11 11
3 15 9 13 7
4 14 12 8 13

Are there any differences in the burning times of the four fabrics ? .Use alpha = 0.01

( 50 marks )

=============END===============

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