1ZZ5
1st Math Sample Test #4 Name:___________________________________________ (Last Name) (First Name) Student Number:_____________________ Tutorial Number:____________________ This test consists of 20 multiple choice questions worth 1 mark each (no part marks), and 1 question worth 1 mark (no part marks) on proper computer card filling. All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Calculators are NOT allowed. 1. Let A= > J ?= > @= >, where J , ?, and @ are differentiable, ?" ! # ?= " ! # ?> " ! ' @" ! $ @= " ! & @> " ! % J? # $ " and J@ # $ "!. Find A> " !. (a) %" (b) & (c) "' (d) #) (e) $%
2. Which of the following differential equations is satisfied by the function ? lnB# C# ? (a) ?BC ?BB ! (b) #?BC ?BB ! (c) ?BB ?CC ! (d) ?BB ?CC ! (e) #?BB ?CC ! 3. Find the linearization of 0 B C C ln B at # ". (a) " B ln #C " (b) " B ln #C " (c) " B ln #C " # # # (d) #B " (e) #B ln #C " 4. Find the equation of the tangent plane to the surface D & B# C# at the point # " #. (a) D %B #C ) (b) D %B #C ) (c) D %B #C ) (d) D %B #C ) (e) D %B #C ' 5. If D B# BC $C# and B C changes from $ " to #*' !*&, find the value of the differential .D . ($ "( ($ "( "( (a) "!! (b) "!! (c) "!! (d) "!! (e) "! 6. Find the directional derivative of 0 B C BsinBC at the point # ! in the direction indicated by the angle 1$. (a) $ " (b) #$ " (c) #$ (d) $ (e) "
# # # # #
7. Estimate the volume of the solid that lies below the surface D BC and above the rectangle V B C l ! B ' ! C % Use a Riemann sum with 7 $ 8 #, and take the sample point to be the lower left corner of each square. (a) 32 (b) 24 (c) 16 (d) 48 (e) 96 8. Set up a double integral which represents the volume under the function D #BC and above the triangular region with vertices ! ! " # and ! $. " B$ "$ " B$ (a) ! B #BC .C .B (b) ! #B #BC .C .B (c) ! ! #BC .C .B " B " B$ (d) ! #B #BC .C .B (e) ! #B #BC .C .B 9. Evaluate
BC C/ .C.B # " ! !
(a) " /# $ (b) " /# # (c) /# " (d) /# $ (e) " /# " # % $ 10. Which of the below integrals is equal to
# " ln B
0 B C.C.B (c) !
ln B
11. Express the following complex number in polar form. D $ 3. 1 1 1 1 (a) #cos &' 3 sin &' (b) #cos 1 3 sin 1 (c) #cos %$ 3 sin %$ ' ' 1 1 (d) #cos #$ 3 sin #$ (e) #cos 1 3 sin 1 $ $ 12. Solve the following equation for the complex number D . D" 3 D $ #3 (a) $ &3 (b) ) $3 (c) $ &3 (d) & $3 (e) $ )3 13. Find all complex numbers D such that D$ (a) (c) (e)
" # " # " #
/ C (a) / " 0 B C.B.C "" (d) ! /C 0 B C.B.C
#
ln # # (b) ! /C 0 B C.B.C " ln C (e) ! ! 0 B C.B.C
!
"# 0 B C.B.C
" " 3 # # (b) (d)
" # " #
" 3&1"# 3(1"# / # 3, / " " " " # 3, # # 3 # " 3415 3(15 / # 3, /
" # 3
" # 3, " # 3,
/3""1"# /3"*1"# /3%1& /371&
14. Consider the set of all vectors +" +# with +" !, with the usual scalar multiplication and a new addition: +" +# ," ,# +" ," +# ,# . If this set satisfies Axiom (iv) of a vector space (the existence of a zero vector), what would be the zero vector? (a) " ! (b) ! ! (c) ! " (d) " " (e) ! " 15. Consider the set of all vectors +" +# with +" !, with the usual scalar multiplication and a new addition: +" +# ," ,# +" ," +# ,# . If this set satisfies Axiom (v) of a vector space (the existence of the negative of a vector), what would be the value of +" +# ? " " (a) +# +" (b) +" +# (c) +" +# (d) +" +# (e) +" +# In questions 16-19, determine whether the given set is (a) A subspace (b) Closed under addition, but not closed under scalar multiplication (c) Closed under scalar multiplication, but not closed under addition (d) Not closed under addition, and not closed under scalar multiplication 16. The set of vectors +" +# where +" +# !. 17. The set vectors +" +# where +" +# ". 18. The set of all vectors +" +# +# with +" +# +$ !. 19. The set of polynomials :B -! -" B -# B# where -! -" -# !. 20. Is the function 0 B $B# #B * a linear combination of 1B B# " and 2B B $? (a) yes (b) no (c) $ (d) " (e) # %
21. Correctly fill out the bubbles corresponding to your student number and the version number of your test in the correct places on the computer card. (Use the below computer card for this sample test.)
Math 1ZZ5
2nd Sample Test #4
Name:___________________________________________ (Last Name) (First Name) Student Number:_____________________ Tutorial Number:____________________ This test of consists 20 multiple choice questions worth 1 mark each (no part marks), and 1 question worth 1 mark (no part marks) on proper computer card filling. All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Calculators are NOT allowed. 1. Let 0 B C B# #C# !
#BC
if B C ! ! , if B C ! !
and let 1B C
$BC #B# C#
!
if B C ! ! . if B C ! !
Which of the above functions are continuous at ! !? (a) 0 only (b) 1 only (c) 0 and 1 (d) neither 2. Let D B , B <=/> C </=> . Find `D when < " = " > !. C `< (a) ! (b) # (c) " (d) # (e) " 3. Let AB C D "!BC ln D . Find ABD . # # C# (a) Dln D# "!BC lnD BCD " ln
C ln"! C (c) Dln D# B lnln"! " D
# # # #
(e) C Dln"! "!BC 2 ln D
#
#
lnD
BClnln"! " D
#
C (d) Dln D# BCD " ln
# #
C ln"! (b) Dln D# "!BC
#
#
lnD
BClnln"! " D
#
4. Find the linear approximation of Q B C ln %B# C# at # %, and use it to approximate Q #" $). (a) %ln # #( (b) %ln # #* (c) %ln # $" (d) #ln % #* (e) #ln % #( # # # # #
5. Matilda wants to use a Riemman sum to approximate the integral 0 B C .E H where H B C l ! B % ! C %. To do so, she divided H into four equal squares and chose a sample point from the middle of each square. The sum of the four pieces was '! "# "# #) ""#. This approximation is appropriate if her function 0 B C was (a) %/$B C ' (c) "#BC CB "# (d) lnB# C# * % (e) '" B$ C% $ (b) 6. Evaluate *B# C# .C .B $ ! ! "B
# "
#
*sin1B#sin1C# BC
(a) ln $ (b) ln ' (c) ln % (d) ln * (e) ln # 7. A contour map is given for a function 0 B C.
Based on the contour map, what are the signs of 0B and 0C at the point " $? (a) 0B ! 0C ! (b) 0B ! 0C ! (c) 0B ! 0C ! (d) 0B ! 0C ! (e) 0B ! 0C !
8. Find the equation of the tangent plane to D at the point " # " . $ (a) $B C *D " (d) B C D " 9. If /BCD B C D then
" (a) "/BCD
`D `B
B " BC
(b) B $C D ! (c) B C *D ! (e) B C *D %
is equal to (b) "BC/BCD (e)
CD/BCD "
(c) "BC/BCD
CD/BCD "C
(d) CD/BCD " C
BC/BCD D # "C "/BCD
10. Let 1> 0 >sin #> cos #>. Suppose that 0B ! " # and 0C ! " $. Find the value of 1w !. (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 11. Which of the following complex numbers is equal to (a) " 3 12. Find " # (b) # 3
$ '! # 3 . $ #
(c) " (3
$ #
(d) # &3
% #3 ' (3 ? "3 #3 (e) $ %3
(a) " (b) " (c)
" 3 (d) #
" 3 (e) #
$ #
"3 #
13. Consider the set of all vectors +" +# with the operations +" +# ," ,# +" ," +# ,# and 5+" +# 5+" " 5+# . Recall axioms (vii)-(ix) of a vector space: (vii) 5 x y 5 x 5 y (viii) 5" 5# x 5" x 5# x (ix) 5" 5# x 5" 5# x Which of these axioms are true? (a) all of them (b) (vii) only (c) none of them (d) (vii) and (viii) only (e) (ix) only In questions 14-19, determine whether the given set is (a) A subspace (b) Closed under addition, but not closed under scalar multiplication (c) Closed under scalar multiplication, but not closed under addition (d) Not closed under addition, and not closed under scalar multiplication
# 14. The set of vectors +" +# , where +# +" .
15. The set of all polynomials of the form :B -$ B$ -# B# -" B $. 16. The set of all functions of the form 0 B -" sin B -# cos B. 17. All functions 0 such that + 0 B.B ".
,
18. All functions 0 such that + 0 B.B !.
,
19. All functions 0 such that + 0 B.B !
,
20. Determine whether the following set of vectors is linearly independent or dependent. " # $ " # "$ " # ( in V $ . (a) independent (b) dependent (c) neither (d) both (e) #)
21. Correctly fill out the bubbles corresponding to your student number and the version number of your test in the correct places on the computer card. (Use the below computer card for this sample test.)
Answers for 1st Sample Test #4 1. e 2. c 3. a 4. a 5. a 6. c 7. d 8. e 9. a 10. b 11. a 12. b 13. b 14. a 15. e 16. c 17. d 18. b 19. a 20. a 21.
NOTE: On the sample tests, a version number is not given. On the actual tests, it will say "Version X" at the top, where X is the version number that you will have to fill in on the computer card. The sample answer above assumes that the test says "Version 3" at the top. On the actual test you will have to fill in the bubble corresponding to the version number of YOUR test (which may or may not be Version 3). The sample above also assumes that your student number is 8816132. On the actual test, you will have to fill in the bubbles corresponding to YOUR student number (not 8816132).
Answers for 2nd Sample Test #4 1. b 2. e 3. b 4. b 5. b 6. d 7. a 8. e 9. b 10. a 11. d 12. a 13. c 14. d 15. d 16. a 17. d 18. b 19. d 20. b 21. see the answer to #21 on the first sample test above.