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H*************\\F/$!3#>?8CD$zzi!#D-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%%V
IEWG6$%(DEFAULTGFden" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "2. Con
sider the polynomial p," }}{PARA 0 "" 0 "" {TEXT -1 79 "      p = 6*x^
2*a^2 - 11*x^2*a + 3*x^2 - 4*x*a + 2*x*a + 6*x + 2*a^2 - 5*a + 3," }}
{PARA 0 "" 0 "" {TEXT -1 48 "where a is a parameter.  Give the Maple c
ommands" }}{PARA 0 "" 0 "" {TEXT -1 50 "  (a) to compute the roots of \+
the polynomial in x;" }}{PARA 0 "" 0 "" {TEXT -1 141 "  (b) to find th
e special value(s) of the parameter for which there are fewer than two
 solutions, also find the corresponding solution for x;" }}{PARA 0 "" 
0 "" {TEXT -1 117 "  (c) to compute the value(s) of the parameter a fo
r which there are infinitely many solutions; also give this value." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "p := 6*
x^2*a^2 - 11*x^2*a + 3*x^2 - 4*x*a^2 + 2*x*a + 6*x + 2*a^2 - 5*a + 3;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,4*&)%\"xG\"\"#\"\"\")%\"aGF
)F*\"\"'*(\"#6F*F'F*F,F*!\"\"*&\"\"$F*F'F*F**(\"\"%F*F(F*F+F*F0*(F)F*F
(F*F,F*F**&F-F*F(F*F**&F)F*F+F*F**&\"\"&F*F,F*F0F2F*" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 13 "Answer to (a)" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sol := solve(p,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$solG6$,$*&,&%\"aG\"\"$\"\"\"!\"\"F,,(\"\"#F+*&F
.F+F)F+F+*&F.F+-%%sqrtG6#,&F)\"\"'*&F.F+)F)F.F+F,F+F+F+#F+F.,$*&F(F,,(
F.F+*&F.F+F)F+F+*&F.F+F1F+F,F+F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
13 "Answer to (b)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "special := solve(denom(sol[1]),a);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%(specialG#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 66 "So we have one special value for a, we substitute and sol
ve for x:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "q1 := subs(a=special,p); solve(q1,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#q1G,&%\"xG#\"#c\"\"*#\"#9F)\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6##!\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "H
ere we have only one solution when a = 1/3 and this solution is x = -1
/4." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Answer \+
to (c)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "sol := solve(p,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG6$#\"
\"$\"\"#*&,(\"\"\"F+*&F(F+%\"xGF+F+*$)F-F(F+F+F+,(F.F'*&F(F+F-F+!\"\"F
+F+F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "q := subs(a=sol[1]
,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 51 "So there are infinitely many solutions for a = \+
3/2." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Altern
atively:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "collect(p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,(%\"aG!#6\"
\"$\"\"\"*&\"\"'F))F&\"\"#F)F)F))%\"xGF-F)F)*&,(F+F)*&F-F)F&F)F)*&\"\"
%F)F,F)!\"\"F)F/F)F)F(F)*&F-F)F,F)F)*&\"\"&F)F&F)F5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "solve(coeff(p,x,2),a);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$#\"\"$\"\"##\"\"\"F$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 242 "Computing the roots of the coefficient with x^2 gives th
e two special values for the parameter.  After substitution and solvin
g for x we see that one special values leads to a linear equation whil
e the other will lead to the trivial equation." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "22 0 1" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }