MATHEMATICS ALGEBRA CE LICENSURE EXAMINATION PROBLEMS ALGEBRA
DIVISION OF POLYNOMIALS POLYNOMIALS .
If 3x3 – 4x2 y + 5xy2 + 6y3 is )i/i )i/i)e )e)) 4 x2 – 2xy +3y 25 the remain)er is2 "+'( M 6& a. $ c. % b. d. # ,
!.
If 4y3 + 8y + 18y 2 – 4 is )i/i)e) 4 2y + 35 the remain)er is2 "+'( M %%& a. c. $% $, b. $$ d. $#
".
.in) the remain)er when 2x3 – 4x2 is )i/i)e) 4 x + 3* "M'' M $& a. c. 7'# $%# b. 3# d. 7(0
#.
When the e8pression x4 + ax3 + 5x2 + bx + 6 is is )i/i)e) 4 x – 25 the remain)er is 16 * When it is )i/i)e) 4 x + 15 the remain)er is 10* What is the /alue of the constant a! "+#, M ,& a. c. 70 ( b. 7' d. -
NUMBERS 1.
Which of the following is not a prime numer! "M#$ M $%& c. $#' (( $#( d. ('
a. b.
SIGNIFICANT FIGURES 2.
Roun) off 108.288886 to to four significant figures* "+#, M %& a. $#-*%--' c. $#-*% b. d. $#-*, $#-*%---
EXPONENTS 3.
.in) the /alue of x in (35 )(96 ) = 32x* "+#% M $& a. c. -*0 '*0 b. ' d. -
BINOMIAL EXPANSION RATIONALIZATION 4.
$.
.in) the 6 th term in the e8pansion of2 "+'3 M '& $3 $ ; , %a a. c. %%$$, 33,,' 7 7 $$ %03a %03a$$
"
Rationali1e the following2 "M## M $3& :a : a ; : a a.
$ ; : $<
c.
a ; a : a;
b.
a < a : a;
d.
$ < : $;
b.
1%.
7777777 $ 7777777
7
%%$$, $%-a$$
&
d.
7
33,,' $%-a$$
9etermine the sum of the numerical coefficients in the e8pansion of (a + b)8* "M#% M %'& a. $%c. %03 b. d. 0$% $#%6
MATHEMATICS ALGEBRA 11.
What is the sum of the numerical coefficients of the e8pansion of (a + b) 6 * "M#, M %& a. 36 c. (' b. d. 30 (6
1.
Compute the /alue of x from the gi/en )eterminant2 "+'3 M %& 6 % 8> $# $6
COMPLEX NUMBERS 12.
13.
The e8pression 3 + 4i is a comple8 numer* Compute its asolute /alue* "M'6 M 0& a. , c. 0 b. d. 6 3
1!.
.in) the /alue of x in the e=uation (x + yi)(1 – 2i) = 7 – 4i * "M#, M 6& a. c. $ 6 b. , d. %
MATRIX 14.
a. b.
? @
? @
$ % # 70
5 elements of matri8
, $ $ 0 c. d.
7%%-
What is the /alue of the )eterminant whose elements are shown elow! $ 0 7% % $ 7, , 7% $ a. c. 766 763 b. 76% d. 76-
1".
.in) the /alue of y in the following e=uations2 "+'0 M -& ,8 ; %4 < w > $$ 8 < 04 ; %w > 7' %8 < 4 ; ,w > 73 a. c. % 7% b. 7$ d. $
1#.
.in) the /alue of w in the following e=uations2 "M'( M $6& ,8 ; %4 < w > $$ 8 < 04 ; %w > 7' %8 < 4 ; ,w > 73 a. c. $ % b. d. , 6
, 3 6 $ .in) the elements of the pro)uct of the two matrices5 matri8 BC * "M'3 M %$& C=
b.
% % # 6
SYSTEM OF E&UATIONS
Elements of matri8 B =
a.
%3 7%3
7$ # , %
? @ ? @ $$ 7%# 70
c.
$0 ' 7%% 6
d.
? @ ? @ $% $# %# 76
$0 $0 7$( 73
7777777 % 7777777
MATHEMATICS ALGEBRA 1$.
Sol/e for the sum of A5 B5 an) C from the following e=uations2 "+#$ M %,& %A ; %B < ,C > %6 A < ,B ; %C > 7$0 ,A < 6B < ,C > 7% a. c. % 70 b. 6 d. $
24.
What is the /alue of x in log x 625 = 4 ! "+'0 M $6& a. c. , 0 b. 6 d. 3
2.
og 6 845 =! "+'( M $'& a. b.
21.
.in) the /alue of A in the e=uation2 "M'3 M $(& B "%8 < %& C 8% < 68 < $# A > 8 < % < % , % 8 < %8 < 0 8 < %8 < 0 8 < %8 < 08 a. c. % 7$% $ b. 7% d. %
a. b.
$ 7,
c. d.
76 %
LOGARIT'MS
2".
The logarithm of the =uotient of two numers is #0.066946789 while the logarithm of their pro)uct is 1.62324929* 9etermine the first numer* "+'- M %& a. c. , 6 b. 3 d. 0
2#.
Sol/e for y if y = l$ a. b.
2$.
If log a 10 = 0.25 5 what is the /alue of log 10 a! "M'6 M '& a. c. % 3 b. d. 0 6
23.
og ! – log " is e=ual to2 "+'6 M (&
e8 e8 ; %
* "+'' M 3&
$ %
c. d.
, 6
c. d.
0*$( %*-(
og 8 975 = x * .in) x* "M## M (& a. b.
3%.
22.
,*(3$ 3*,6-
The logarithms of the =uotient an) the pro)uct of two numers are 0.352182518 an) 1.5563025015 respecti/el4* .in) the first numer* "M'- M ,& a. c. $$ ' b. $# d.
%$Sol/e for B in the gi/en partial fraction2 "+## M %,& C 86 ; $08, ; ,%8% ; $%8 ; $6 > A < B < % 8 < , 8 < $ 8 ; % "8 < $&"8 ; %&"8 < ,&"8 < %& 98 < E < 8% < %
c. d.
2!. PARTIAL FRACTIONS 2%.
0*'$% 6*,6-
,*,$ 6*
%$If log x 12 = 2.262 5 fin) the /alue of log x 3* "M#, M $& a. c. #*(0 #*0 b. d. #*%0 $*#
ARIT'METIC PROGRESSION a. b.
log M+ log "M ; +&
c. d.
log "M +& log "+ ; M&
31.
7777777 , 7777777
How man4 terms of the progression 3% 5% 7& must e in or)er that their sum will e 2%600! "M'6 M %& a. c. 3# 0$ b. 0# d. 6-
MATHEMATICS ALGEBRA 32.
33.
How man4 terms of the progression 3% 5% 7% 9& must e taen in or)er that their sum will e 2%600! "+'6 M $& a. 6' c. 6b. d. 0# 6( What is the sum of the progression 4% 9% 14% 19& up to the 20th term! "M'0 M $& a. c. $#60 $#,# b. $#,0 d. $#6#
34.
Which of the following numers shoul) e change) to mae all the numers form an arithmetic progression when properl4 arrange)! "M'( M '& %( $65 ,,%-5 60%-5 an) %#$6 %( 60 a. $6 c. %,, %# b. d. %$6
3.
The sum of the progression 55 85 115 14 is 1%025* How man4 terms are there! "+'( M %& a. c. %% %, b. %0 d. %6
3!.
3".
3#.
There are se/en arithmetic means etween 3 an) 35* .in) the sum of all the terms* "M'- M %& a. $3' c. $3( b. d. $($ $(, There are nine arithmetic means etween 11 an) 51* The sum of the progression is2 "M'' M %& a. c. %(' ,(3 b. d. ,6$ %06 .in) the %0 th term of the progression 115 145 17 * "+'' M $& a. c. -, 'b. d. (6 36
3$.
'36%000 is to e )i/i)e) among Arturo5 Bernar)o5 an) Calo4 such
that their shares5 in the same or)er5 form an arithmetic progression* Bernar)oDs share is three times that of ArturoDs* How much is Bernar)oDs share! "+#% M 6& a. 35### c. $-5### b. d. $%5### $#5###
GEOMETRIC PROGRESSION 4%.
In a enefit show5 a numer of wealth4 men agree) that the first one to arri/e woul) pa4 10 centa/os to enter an) each later arri/al woul) pa4 twice as much as the prece)ing man* The total amount collecte) from all of them was '104%857.50* How man4 wealth4 men pai)! "+'6 M %& a. c. %# %0 b. $0 d. ,#
41.
The numers 285 x + 2 5 112& form a geometric progression* What is the 10th term! "M'0 M %& a. $,5,$% c. $35,,3 b. d. $65,,3 $05,,3
42.
.in) the sum of the series 35 125 485 up to the 9th term* "+'' M %,& a. c. %%,53$6 %,653$% b. ,635%$% d. %3%5$6,
43.
There are four geometric means etween 3 an) 729* .in) the fourth term* "M## M 3& a. c. -$ %6, b. d. %( '
44.
The geometric mean of two numers is 8 an) their arithmetic mean is 17 * What is the first numer! "+## M %%& a. c. 60 ,% b. d. ,3 6-
7777777 6 7777777
MATHEMATICS ALGEBRA 4.
A an) B are positi/e numers* A5 #45 B form a geometric progression* 145 A5 B form an arithmetic progression* .in) the /alue of A* "M#, M ,& a. b.
3 '
c. d.
RELATIONS %.
(
is 12* ol)er than while is 11* ol)er than , * B4 how much5 in percent5 is ol)er than , ! "+#% M ,& a. b.
%0*6, %$*(-
c. d.
%6*,% %,*33
INFINITE GEOMETRIC PROGRESSION PROPORTION 4!.
4".
9etermine the sum of the infinite geometric series of 1 – 1 5 + 1 25 + &! "M'- M $$& 0 6 a. c. ( 3 6 0 b. 0 d. 3 .in) the sum of the infinite geometric series of 16 + 4 + 1 + 1 4 + &* "+'- M ,#& -( 30 a. 6 c. , 36 -3 b. d. , 6
1.
CLOC( PROBLEMS 2.
In how man4 minutes after 2-00 oDcloc will the han)s of the cloc e8ten) in opposite )irections for the first time! "M'0 M ,& a. c. 6%*3,3 6#*3,3 b. 6$*3,3 d. 6,*3,3
3.
How man4 minutes after 7-00 oDcloc will the han)s of a cloc )irectl4 opposite each other for the first time! "+'0 M 0& a. 0*$(, c. 0*606 b. d. 0*%(, 0*$36
4.
In how man4 minutes after 3-00 / will the minute han) of the cloc coinci)e with the hour han)! "M'( M %0& a. c. $(*%(, $3*,36 b. d. $-*$-% $0*600
'ARMONIC PROGRESSION 4#.
What is the /alue of x in the harmonic progression 15 1 55 1 95 1 x5! "+#% M %& a. $# c. $$ b. d. $% $,
A line in a map was )rawn at a scale of 1-25000* An error of 0.02 in the )rawing is e=ui/alent to how man4 meters in actual! "+## M %#& a. c. 0m #*#0 m b. #*0 m d. 0# m
SERIES 4$.
The sum of the first $ terms of a series is 3$+2 – 6 * .in) the fifth term of the series* "M#$ M %%& a. c. $6,6 $60b. d. $66% $6%3
7777777 0 7777777
MATHEMATICS ALGEBRA )OR( PROBLEMS .
A Fo coul) e )one 4 ele/en worers in 15 )a4s* .i/e worers starte) the Fo* The4 were reinforce) with four more worers at the eginning of the 6 th )a4* .in) the total numer of )a4s it too them to finish the Fo* "+'- M %,& a. %,*%% c. %%*,3 b. d. %#*03 %$*6%
!.
Twent4 men can finish a Fo in 30 )a4s* Twent47fi/e men starte) the Fo* If ten men =uitte) the Fo after 18 )a4s5 fin) the total numer of )a4s to finish the Fo* "+'' M %0& a. %( c. %3 b. d. %%'
".
Twent47eight persons can )o a Fo in 60 )a4s* The4 all start complete* .i/e persons =uitte) the Fo at the eginning of the 16 th )a4* The4 were reinforce) with 10 persons at the eginning of the 45th )a4* How man4 )a4s was the Fo )ela4e)! "+## M $& a. c. 0*(- )a4s $*'( )a4s b. $*$6 )a4s d. %*60 )a4s
!%.
At 2-00 /5 an airplane taes off at 340 /h on an aircraft carrier* The aircraft carrier mo/es )ue south at 25 /h in the same )irection as the plane* At 4-05 /5 the communication etween the plane an) the aircraft carrier was lost* 9etermine the communication range in miles etween the plane an) the carrier* "M'' M ,& a. 303 miles c. 00( miles b. d. (-0 miles 6$% miles
!1.
A man wals from his house to the office* If he lea/e at 8-00 oDcloc an) wal at the rate of 2 /h5 he will arri/e 3 minutes earlier5 ut if he lea/e at 8-30 an) wal at 3 /h5 he will arri/e 6 minutes late* What time shoul) he arri/e in the office! "M#$ M $#& a. c. '2#3 oDcloc -206 oDcloc b. '2,% oDcloc d. -26, oDcloc
!2.
A man wals from his house to the office* If he lea/e at 8-00 oDcloc an) wal at the rate of 2 /h5 he will arri/e 3 minutes earlier5 ut if he lea/e at 8-30 an) wal at 3 /h5 he will arri/e 6 minutes late* .in) the )istance from his house to his office* "+#$ M %%& a. c. %*$ m %*6 m b. $*- m d. $*6 m
MOTION PROBLEMS #.
$.
An airplane5 fl4ing with the win)5 too 2 hours to tra/el 1000 an) 2.5 hours in fl4ing ac* What was the win) /elocit4 in /h! "+'6 M ,& a. c. ,# 0# b. d. (# 3# A oat taes 2 3 as much time to tra/el )ownstream from C to 5 as to return* If the rate of the ri/erDs current is 8 /h5 what is the spee) of the oat in still water! "M'- M %#& a. c. 6# ph 6$ ph b. d. ,- ph ,' ph
PERMUTATION !3.
If $ ' 5 = 6 $ ' 35 what is the /alue of $! "M#% M ,& a. c. ( $ b. d. 3
COMBINATION !4.
7777777 3 7777777
How man4 cominations consisting of 4 )igits can e ma)e using the )igits from 0 to 9! "+#$ M $%& a. c. %$# $%# b. ,3# d. 6%#
MATHEMATICS ALGEBRA PROBABILITY !.
How man4 four7)igit numers can e formulate) 4 the use of )igits 15 25 35 45 55 an) 7 if one )igit is use) onl4 once* "+'0 M (& a. %3# c. 6-# b. d. ,-# ,3#
!!.
A ag contains 3 white an) 5 lac alls* If two alls are )rawn in succession without replacement5 what is the proailit4 that oth alls are lac! "M'3 M 6& 0 0 a. %c. ,% 0 0 b. d. $3 $6
!".
How man4 4#)igit numers can e forme)5 without repeating an4 )igit5 from the following )igits 15 25 35 45 an) 6 * "+'3 M ,#& a. c. $0# $%# b. $6# d. $,#
!#.
A face of a coin is either hea) or tail* If three coins are tosse)5 what is the proailit4 of getting three tails! "+'- M $-& $ $ a. c. 6 $ $ b. d. % 3
7777777 ( 7777777
