These solutions have been extracted from the 2016s2 maple lab practice test. These notes are in no way affiliated with UNSW's School of Mathematics nor UNSW's MathSoc. I highly recommend you try doing the practice questions yourself and use these notes if your really stuck. Maple 18 was used to generate the solutions.
with(LinearAlgebra): # Include the LinearAlgebra package, if not already loaded.
A := <<6534, 37886, 17462, -3738, 4430> | <32490, -26590, -49930, 21570, 51050> | <21912, -6552, -10284, -84984, -72360> | <-14745, -25505, -21035, 6315, 14575> | <26985, 26065, -20645, 16605, 24025>>;
Eigenvalues(A);
with(LinearAlgebra): # Include the LinearAlgebra package, if not already loaded.
A := <<2490, 186, -538, 2294, 958> | <7200, 576, -2424, 7764, 1668> | <1680, 240, -328, 1640, 640> | <540, -36, -492, 876, -228> | <-7680, -192, 3008, -8308, -1436>>;
Rank(A);
sum((5*x -3*k)^k, k=110..1900):
product(x^5 -k +5*x,k=150..1900):
with(LinearAlgebra): # Include the LinearAlgebra package, if not already loaded.
u:=<11,49,-9>; v:=<40,-8,-14>;
CrossProduct(u,v);
p := -3*x^12-6*x^11+409*x^10-1882*x^9-1546*x^8+12188*x^7-36316*x^6+117442*x^5-149243*x^4+273136*x^3-234223*x^2+144906*x-110270;
q := x^13-x^12-72*x^11+110*x^10+1309*x^9-1941*x^8-1338*x^7-5616*x^6-45444*x^5+21928*x^4-117880*x^3+84000*x^2-88000*x+70000;
convert(p/q, parfrac, x);
9 -3 x + 3 -x + 3 9 8
- ----- + ------------ + --------- + ----- - --------
x + 5 2 3 x + 7 3
x + 2 x - 2 / 2 \ (x - 5)
\x + 2/
f := (x,y)-> 8*x^8*y^3*sin(3*x-5*y);
D[1$8, 2$8](f)(2,2) # where x is 1$ and y is 2$
ODE := y(x)*diff(y(x),x$2) + diff(y(x),x)^2 = 0;
dsolve({ODE, y(0)=4, D(y)(0)=7}, y(x));
(1/2)
y(x) = 2 (4 + 14 x)
ODE := y(x)*diff(y(x),x$2) - 1/2*diff(y(x),x)^2 = 0;
dsolve({ODE, y(0)=1, D(y)(0)=6}, y(x));
2
y(x) = 9 x + 6 x + 1
ODE := y(x)*diff(y(x), x$2) -2/3*diff(y(x),x)^2 = 0;
dsolve({ODE, y(0)=2, D(y)(0)=7}, y(x));
343 3 49 2
y(x) = --- x + -- x + 7 x + 2
108 6
ODE := y(x)*diff(y(x),x$2) + 4*diff(y(x),x)^2 = 0;
dsolve({ODE, y(0)=4, D(y)(0)=3}, y(x));
(1/5)
y(x) = 2 (120 x + 32)
restart; # clear memory as we need to reuse the A variable OR use A := 'A'
with(geom3d): # Load the geom3d package, if not already loaded.
point(A, [5,-4,-4]); point(B, [-2,2,2]); point(C, [-5,3,-1]);
line(L1, [A, [2,4,2]]);
plane(P, [B, [-4,-4,2]]);
intersection(E, L1, P);
sphere(S, [A,B,C,E]);
center(F, S);
line(L2, [C,F]);
evalf[10](FindAngle(L1,P));
-0.7483271725
coordinates(F);
[-1947 -239 -2577]
[-----, ----, -----]
[ 910 70 910 ]
distance(A, L2);
1 (1/2) (1/2)
-------- 1724588486 43625067
43625067
restart; # clear memory as we need to reuse the A variable OR use A := 'A'
with(geom3d): # Load the geom3d package, if not already loaded.
point(A, [-3,0,1]); point(B, [3,6,6]); point(C, [19,19,17]);
sphere(S1, [A,12]);
sphere(S2, [B,C])
intersection(T, S1, S2);
center(E, T);
line(L1, [B,E]);
line(L2, [A, [4,1,1]]);
coordinates(E);
[ 761 235 1172]
[-----, ----, -----]
[ 185 37 185 ]
evalf[10](FindAngle(L1,L2));
0.07183271725
distance(L1, L2);
0
restart; # clear memory
a := proc(n) # shift + enter
local a,i; # shift + enter
a[1]:=2; # shift + enter
a[2]:=0; # shift + enter
a[3]:=-1; # shift + enter
for i from 3 to n-1 do # shift + enter
a[i+1]:=a[i]-4*a[i-1]+a[i-2] # shift + enter
end do; # shift + enter
return a[n] # shift + enter
end proc; # shift + enter
a(90);
-6124977648188652773170728
>>>
restart; # clear memory
Digits := 30; # shift + enter
f := proc(m) # shift + enter
local a,i; # shift + enter
a[0]:=0; # shift + enter
for i from 1 to m do # shift + enter
a[i] := evalf(sin((1+a[i-1]/4)^2)) # shift + enter
end do; # shift + enter
if abs(a[m]-a[m-1]) < 10^(-18) then # shift + enter
a[m] # shift + enter
else # shift + enter
-1 # shift + enter
end if # shift + enter
end proc; # shift + enter
f(5);
-1
f(13);
0.999965406070608925874302546481