Tuesday, 25 September 2018

9:07 PM

Machine generated alternative text:
A function f is defined by 
y) = 8 .T6 y4 sin (7 x 
Enter a Maple expression for the function f using the arrow operator "->" in the box below. 
(Do not enter "f at the beginning or ";" at the end of your answer.) 
—5y)

 

Machine generated alternative text:
f: (sy) *8x6y4 sin(7x 5y)

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Given that 
Find the partial derivative 
and enter your answer, in Maple syntax, in the box below 
5) 
sin 2 y +6m 
33

 

Machine generated alternative text:
f: = (x , y) •sin ; 
f: 615 ) 
-10800 cos(2y6 — 1440 sin(2y6

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Find the partial fraction decomposition of 
where 
p (x) = 8 x8 +46 x7 — 211 x6 — 1015m + 4784 A — 11534 x3 + 8707 x 
2 
and 
x +6m —37 —77 x + 633 x 
5 
— 1777 x4 + 2085 x3 — 771 
and enter it in the box below. To prevent typing errors, you can copy and paste 
p 01 1 869*X-91 56; 
q *xA2-1 323; 
into your Maple worksheet and copy and paste the Maple output into the answer box. 
- 1869 - 9156 
- 1386 + 1323

Machine generated alternative text:
x-9156; 
x+1323 ; 
convert (p/ q, parfrac , x) ; 
4617 —21116 101515 478414 — 1153413 
633 e 
17771 2085 e 
2 
870712 1869 x 
77112 13861 
9156 
1323 
12—31 3

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Use Maple to find the solution of the initial value problem 
2 
d2y 1 dy 
y 
dx2 
0 with initial conditions y (0) 
1 and y' (0) — 
7 
Using Maple syntax, type in your answer in the box below, or copy (Ctrl-C) from your Maple worksheet and paste (Ctrl-V) in the answer box the solution Do NOT enter the y(x)= part of the 
Maple output.

 

Machine generated alternative text:
dsolve ({y (x) *diff (y (x) ,x$2) —1/2* (diff (y (x) , x) ) y (O) —I, D (y) (O) =7}) ; 
49 
y(x) = x-2 71 1

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Suppose that a function f has derivatives of all orders ata The the series 
00 
is called the Taylor series for f about a, where f (n) is the n th order derivative of f 
Suppose that the Taylor series for e2x sin (4 x) about 0 is 
k! 
ao al.T4-a,2x + 
Enter the exact values of and a8 in the boxes below 
32/15 
_kav + .

 

Machine generated alternative text:
taylor (exp (2*x) *sin (4 *x) , 
4x 812 — 1614 — x: 
15 
+ 176 x-6 
45 
2224 
315 
32 
15

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Suppose that 
111 — 
420 
256 
-540 
565 
-989 
-437 
150 
-442 
900 
266 
-480 
-442 
113 
-39 
309 
-475 
-681 
-721 
-481 
815 
-155 
and 
-79 
-280 
331 
376 
114 
-699 
262 
-44 
-241 
177 
496 
-396 
242 
To avoid typing errors; you can copy and past the following sequences to your Maple worksheet to form entries of the vectors 
420, 256, -54€, 565, -989, -437, 150, 
90€, 266, -48e, 
-721, -481, 
-699, 262, 
815, 
-44, 
-442, -39, 3€9, -475, 
-155, -79, -28€, 331, 
-241, 177, 496, -396, 
-442 
-681 
376 
242 
(a) Find the dot product of and 112_ Enter your answer in the box below. 
111 • 112 
(b) Suppose that A is the matrix whose columns are these four vectors in order, 
Number 
Let v 
39 
be the vector 
-72 
-64 
Hence Av 
A = (ul 1 114) 
is a linear combination of the form 
Enter the values of Al, 
(c) Suppose that Av = (bl b2 
A 4 in the boxes below 
39 
b8)T 
-72 
Enter the value of bl in the box below 
bl = 128808

 

Machine generated alternative text:
with (Linearmgebra) : 
ul : *420 , 
u2 : *900 , 
256, 
-540, 
266, 
-480, 
-481, 
815, 
262, 
-44, 
565 , 
-989, 
-442, 
-39, 
-155, 
-79, 
-241, 
177, 
-437, 
150, 
309, 
-475, 
-280, 
331 , 
496, 
-396, 
-442>: 
-681>: 
376>: 
242>: 
Transpose (8) [I] ; 
128808 
26446 
- 70804 
5391 
-238 
3526 
-18063 
-66025 
128808

 

 

 

 

 

 

 

 

 

 

 

 

Let A be an m x n matrix. The kernel of A is the setof vectors 
ker(A) = {x 
which is a vector space. 
The dimension of ker(A) is called the nullity of A, denoted by nullity(A) 
(a) Find the nullity of the matrix 
-76 
88 
-96 
176 
and enter in the box below 
-210 
-40 
-58 
-46 
196 
120 
-90 
14 
-180 
44 
-186 
-8 
-618 
-24 
-230 
178 
nullity(A) — 
To avoid typing errors, you may copy and paste the following sequence of entries 
-76, 88, -96, 176, -210, -40, -58, -46, 196, 120; -90; 14, -180, 44 -186, -8, -618, -24, -230, 178

 

(b) For the matrix A in (a); select all the statements below which are true. 
0 
is in ker(Ab 
-86 
-42 
is in ker(A)_ 
-5 
-67 
You may copy and paste the entries 
-86, -42, -5, -67 
-2 
is in ker(A) 
is in ker(Ab 
is in ker (Ab 
-124 
is in ker(A). 
-53 
21 
You may copy and paste the entries 
-124 2, -53, 21 
D ker(A) is subspace of R4 
4.1 ker(A) is subspace of R5

 

with (LinearAIgebra) : 
A: , 88, 
-96, 
s ion (A) 
Nu IISpace (A) ; 
LinearSoIve (A , 
120 , 
-90 , 
-180, 
44, 
-186, 
2 
-2 
-618 , 
-24, 
-230, 
—Rank (A) ;

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Let A be an m x n matrix. The image of A is the setof vectors 
im(A) 
which is a vector space. 
The dimension of im(A) is called the rank of A, denoted by rank(A) 
(a) Find the rank of the matrix 
and enter in the box below 
Ax for some x e IR n} 
170 
-158 
-94 
-150 
133 
-275 
-171 
-225 
-172 
-54 
174 
32 
3 
220 
-142 
-298 
-182 
821 
-59 
-867 
-353 
rank(A) 
Note: The Maple command for find the rank of A is Rank(A), upper case R. 
To avoid typing errors, you may copy and paste the following sequence of entries 
170; -158; -94 -150; 133, -275, -171, -225, -172, -54, 174, 32, 220; -142; -298; -182, 821, -59, -867, -353

 

Machine generated alternative text:
(b) For the matrix A in (a); select all the statements below which are true. 
D Im(A) is subspace ofR5 
24 
-98 
is in im(A)_ 
-62 
-75 
You may copy and paste the entries 
24, -98, -62, -75 
0 
is in im(A) 
is in im(Ab 
is in im(A) 
4.1 im(A) is subspace ofR4 
is in im(Ab 
87 
-68 
is in im(A)_ 
82 
-92 
You may copy and paste the entries 
87, -68, 82, -92

 

Machine generated alternative text:
-158, 
-142 , 
-150> 1<133 , 
<220 , 
> Rank (A) ; 
-298 , 
-275, 
-59, 
-171, 
-867 , 
-54, 
174, 
> Linearsolve (A, , -98 , -62 , -75>) : 
> LinearSoIve (A, , 4, —2 , 4, —2>) : 
Error. (in LinearAIgeEra : —Linearsolve) number of rows of left hand side 
Matrix. 4. does not match number of rows of right hand. 5 
> LinearSoIve (A, , O , O , O , : 
Error. (in LinearAIgeEra : —Linearsolve) number of rows of left hand side 
Matrix. 4. does not match number of rows of right hand. 5 
> LinearSoIve (A, , —2 , —4, 4, —2>) : 
Error. (in LinearAIgeEra : —Linearsolve) number of rows of left hand side 
Matrix. 4. does not match number of rows of right hand. 5 
> LinearSoIve (A, , O , O , : 
> Linearsolve (A, , -68 , 82 , -92>) : 
Error. ( in LinearAIgeEra : —Linearsolve) inconsistent system

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Suppose that S 
P2(x) — 
P4(x) — 
= {PI' P2' P.3' P4}, where 
3 
147 + 103 + 109 + 
19 x 
39+ +43m —281 x 
— -94+209x-224x -262x +21 x 
5 
5 
- 232 
5 
+ 269 x 
-15—93X+192x +203X +231 X + 166T , and 
— -4287 + 13475 x - 533 + 1696 + 16633x +32087 x 
5 
To avoid typing errors, you can copy and past the following sequences to your Maple worksheet 
147, 103, 109, 272, -19, 3 
39, 272, 43, 43, 
-281, -232 
-94, 209, -224, -262, 21, 269 
-15, -93, 192, 
-4287, 13475, 
203, 231, 166 
-533, 1696, 16633, 32087 
(a) The number of elements of S is less than 
the dimensions of IP5 
Hence S cannot be 
a spanning set of IP5 
(b) The polynomial p is a linear combination of S written in the form 
, the set of polynomials of degree 5 or less 
apl + ßP2 + + 
Find a possible set of values for a, B, 7, ö_ Enter the values ofa, B, 7, as a sequence in the box below 
[a, ß, 7, ö] [27, 4, 78, 721

 

Machine generated alternative text:
PI : *147 , 
p 4: 15 
103, 
272, 
209, 
-93 , 
109, 272, 
-19, 
43, 43, 
-281, -232>: 
-224, -262, 21, 
192, 203, 231, 
13475, 
-533, 1696, 
LinearSoIve (<pl Ip2 Ip3 Ip4> , p) ; 
convert (8 , list) ; 
269>: 
166>: 
16633 , 
27 
4 
78 
72 
32087>: 
[2724: 78: 72]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Suppose that S — {111 , 112, 113, 114, 115, 116} C R 
111 — 
-41 
-28 
56 
53 
-26 
where 
u2 
-91 
-1 
-66 
-60 
93 
113 
578 
89 
162 
141 
-387 
-9 
50 
114 
-43 
-33 
-11 
-16 
91 
27 
-86 
and 
116 
-5 
182 
-323 
-183 
73 
To avoid typing errors, you can copy and past the following sequences to your Maple worksheet 
-41, 
-91, 
578, 
-9, 
-11, 
-5, 
-28, 
—I, 
89, 
-16, 
182, 
56, 
-66, 
162, 
-47, 
91, 
-323, 
53, 
-60, 
141, 
-43, 
27, 
-26 
93 
-387 
-33 
-86 
-183, 73 
Since the number of vectors in S is greater that the dimension of R the set S 
Find a possible setof values for Al, A 4, not all zero, such that 
must be linearly dependent 
Enter the valuesof Al, .\2, as a sequence in the box below 
[Al' •\2' A4, '\5, [3, 5, 1, -3/2, 1, 1/2] 
Hint: There are infinitely many solutions for Al, . The solution given by Maple will be in terms of parameters. To get one possible setof values, not all zero, choose 
some nice values for the parameters.

 

Machine generated alternative text:
with (Linearmgebra) : 
-28, 56, 
91 
u3 : *578 , 
11 
-66 , 
162 , 
89, 
50, 
53, 
-26>: 
-387>: 
, -33>: 
-16, 91, 
-60 , 
141, 
27, 
-86>: 
182, 
-323, 
-183, 73>: 
_ 41 
-28 
56 
53 
_ 26 
_ 91 
-66 
-60 
93 
578 
89 
162 
141 
-387 
2 
2 
-9 
_ 47 
-43 
-33 
-11 
-16 
91 
27 
-86 
182 
-323 
-183 
73 
LinearSoIve (S

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Use Maple to find the eigenvalues of 
-8040 
2160 
-180 
2520 
720 
-25471 
7971 
27472 
25617 
-9238 
-9410 
9930 
1460 
13110 
-2180 
15807 
-11187 
-16224 
-24489 
4506 
21676 
-13056 
-22702 
-23772 
1138 
Enter the eigenvalues in the box below as a Maple set (Eg, ifthe eigenvalues were 1, 2, 3; 4, you should enter {1 
To avoid typing errors; you may copy and paste the following sequence of entries of A 
-8040; 2160, -180, 2520, 720, -25471; 7971 , 27472; 25617, -9238, -9410; 9930, 1460; 13110, -2180; 15807, -11187, -16224, -24489; 4506, 21676; -13056, -22702, -23772; 1138 
and edit it appropriately to create a matrix. 
{-10980, -9150, -7320, -5490, 10980}

 

Machine generated alternative text:
2160, 
-180, 
1<-25471, 7971, 27472, 
2520, 
25617 , 
720> 
-9238> 
1<-9410 , 
1<15807 , 
1<21676, 
9930, 1460, 13110, 
-2180> 
-11187, 
-13056, 
-16224, 
-22702 , 
-24489, 
-23772, 
4506> 
E igenvalues (A) ; 
convert (8 , set) ; 
-5490 
-10980 
- 7320 
-9150 
10980 
-10980: -9150: -7320: -5490: 10980)

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Given the three points A(-2, 4; 5); B(0i 5, 6), C(7, 7, 7), let: 
• Sl be the sphere with centre A and radius 4, 
• S2 be the sphere which has the line segment BC as a diameter, 
• T be the circle of intersection of Sl and S2, 
• E be the centre of T, 
• Ll be the line through B and E, 
• L 2 be the line through A parallel to 
Using the geom3d package, or othenvise: 
4 
1 
3 
(i) Find the coordinates of E and enter them in the box below You should enclose the coordinates with square brackets, eg [1 2,3], and your answer should be exact ie not a decimal 
approximation. To prevent typing errors you can copy and paste the answer from your Maple worksheet 
[137/146, 370/73, 847/146] 
(ii) Find a decimal approximation to the angle (in radians) between Ll and L 2 
0.8527069987 
Your answer should be correct to 10 significant figures 
Enter your answer in the box below. 
(iii) Find the distance between Ll and L 2. Your answer should be exact, not a decimal approximation. 
can copy and paste the answer from your Maple worksheet 
Enter your answer in the box below using Maple syntax. To prevent typing errors you

 

Machine generated alternative text:
with (geom3d) : 
point (A, [—2 , 
4, 
point (E , 
[0, 
point (C , [7 , 
5, 
7, 
sphere ( SI , [A, 
sphere ( S2 , [E , 
intersection (T , 
5]): 
c]): 
sl, S2) : 
point (E , coordinates ( center (T) ) ) : 
line (Ll, [E, E]) : 
line (L2, [A, [4, 1, 
coordinates (E) ; 
137 
146 
evalf (FindAngIe (LI , L2) ) ; 
370 
73 
847 
146 
0.8527069987 
distance (LI , L2) ; 
104 
659 
4613

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
Given the three points A (—2, —3, 0), B (—1, 0, 4), C (—3, 1, —2), let: 
• Ll be the line through A parallel to 
5 
5 
-2 
-4 
• P be the plane through B with normal 5 
• E be the point of intersection of Ll and P, 
• S be the sphere through A, B; C, E; 
• F be the centre of S, 
• L2 be the line through C and F. 
Using the geom3d package, or othepwise: 
(i) Find a decimal approximation to the angle between Ll and P, correct to 10 significant figures. Enter your answer in the box below. To prevent typing errors you can copy and paste the 
answer from your Maple worksheet 
0.1064637912 
(ii) Find the coordinates of F and enter them in the box below You should enclose the coordinates with square brackets; eg [1 and your answer should be exact, ie not a decimal 
approximation. To prevent typing errors you can copy and paste the answer from your Maple worksheet 
[3699/670, 519/670, -979/670] 
(iii) Find the distance beftveen A and L2_ Your answer should be exact, not a decimal approximation. 
can copy and paste the answer from your Maple worksheet 
Enter your answer in the box below using Maple syntax To prevent typing errors you

 

Machine generated alternative text:
restart ; 
with (geom3d) : 
point (A, [—2, —3, O]): 
point (E 
point (C, I, : 
line (Ll, [A, 
plane (P , [E , 
intersection (E , LI , P) : 
sphere (S, : 
point (F , coordinates (center (S) ) ) : 
line (1.2 , [C,F]) : 
evalf (FindAngIe (LI , P) , 10) 
coordinates (F) ; 
dis tance (A , L2) ; 
0.1064637912 
3699 
670 
519 
670 
638170638 
979 
670 
32745803 
32745803

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
The skeleton "for loop" below is used to evaluate the sum 
18 
for k from 4 to 70 
Select the correct item from each drop down box so that the output from the loop contains exactly 67 lines and displays the sum for each integer value of k from 4 to 70 
for k 
, from 4 
, to 70 
, do 
n=18..23) 
end do; 
The terms of a sequence an are generated by the recurrence relation 
an+l = an — 2 an _1 + an _2 forn = 3, 4, 5, ... 
Using your Maple worksheet, write a for loop to find the value of am given that 
al = 4, = 0 and q — 
Copy (Ctrl-C) the correct value of am from your Maple worksheet and paste (Crtl-V) it in th answer box. 
-290694793 
-1

 

Machine generated alternative text:
for n from 4 to 70 do 
end do : 
a[70] ; 
3] 
-290694793

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text:
A simple iteration procedure with ao 
0 and 
2 
an 
2 
is being used to find an approximate solution to the equation x = sin 
Select the correct expressions from the drop down menus to define a procedure which takes a positive integer m and uses a for loop to calculate am. The procedure 
should return am if lam — am 
Digits 
t proc(m) 
local a, i; 
for i from 
—Il < 10—17 
and 
m 
—1 otherwise. All calculations are done using 30 significant figures. 
do 
end do; 
if abs (a [m] 
a[m] 
-a[m-l)) < Ion 
(—17) then

 

Machine generated alternative text:
end if 
end proc; 
Use this procedure to calculate f (5) and f (13) and enter your answers in the box below as decimal approximations correct to 30 significant figures. To prevent typing 
errors, you can copy and paste your answers from your Maple worksheet. 
The value of f (5) is -1 
The value of f (13) is 0.99996540607060892587.

 

Machine generated alternative text:
Digits : ; 
f : —proc (m) 
local a , i ; 
for i from I to m do 
evalf (sin ( / 4) ) 
end do ; 
if abs (a[m] < IOA (—17) then 
else 
end if 
end proc ; 
f (13) ; 
proc(m) local i; a[O] = O; for i to m do a[i] — 
Digits 3D 
1])A2)) end do, if abs(a[m] — a[m 
0.999965406070608925874302546481 
1]) < 1/100000000000000000 then a[m] else 
I end if end proc

 

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