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- Course
- Mathematics II (MATH1034)
- Institution
- University Of The Witwatersrand (wits)
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MATH1034A (D diagonal)– Mathematics I Major Algebra - April 2018(3 pages—page 1) MATH1034A (D diagonal)– Mathematics I Major Algebra - April 2018(3 pages—page 2)Question 1 - 2 marks Question 4 - 4 marks
Choose the correct formula from belowl.’mmmmmmbvvcc for converting degrees to radi- 1.
ans: =
4
arcsin sin −
� � π ��
(a) Multiply by πr
180
(b) Multiply by π
π
(c) Multiply by 180
2.
(d) Multiply by πr2 1
tan arccos =
13
� � ��
(e) None of the above
Question 2 - 2 marks
Which one of the following is FALSE?
(a) sin( π2 − θ) = cos(θ) Question 5 - 3 marks
Find the general solution to cos(3x) = 0
(b) cos(2θ) = 2 sin(θ) cos(θ)
(c) sin(θ) sin(ϕ) = 12 (cos(θ − ϕ) − cos(θ + ϕ))
θ−ϕ
2 ) sin( 2 )
(d) sin(θ) − sin(ϕ) = 2 cos( θ+ϕ
(e) cos(θ + ϕ) = cos(θ) cos(ϕ) − sin(θ) sin(ϕ)
Question 3 - 2 marks
Calculate the value of cos( 11π
12 ) using the Trigonometric Identities.
,MATH1034A (D diagonal)– Mathematics I Major Algebra - April 2018(3 pages—page 3) MATH1034A (E diagonal)– Mathematics I Major Algebra - April 2018(3 pages—page 1)
Question 6 - 7 marks Question 1 - 2 marks
2)2 + y2 = 4 into polar form.
Choose the correct formula from below for converting radians to degrees:
(a) Convert the equation (x −
(5 marks)
(a) Multiply by 12 πr2
180
(b) Multiply by π
π
(c) Multiply by 180
(d) Multiply by πr
(e) None of the above
Question 2 - 2 marks
Which one of the following is CORRECT?
1−cos(2θ)
(a) sin2 (θ) = 2
1+cos(2θ)
(b) sin2 (θ) = 2
1−cos(2θ)
√
(b) Write the expression (c) cos2 (θ) = 2
3 sin(x) + cos(x) in the form R cos(x − θ)
(2 marks)
(d) cos2 (θ) = cos(2θ) − sin2 (θ)
(e) None of the above
Question 3 - 2 marks
Calculate the value of tan( 7π
12 ) using the Trigonometric Identities.
Algebra Total 20 Marks
,MATH1034A (E diagonal)– Mathematics I Major Algebra - April 2018(3 pages—page 2) MATH1034A (E diagonal)– Mathematics I Major Algebra - April 2018(3 pages—page 3)
Question 4 - 4 marks Question 6 - 7 marks
1. (a) Change the polar equation r = 3 sec(θ), sec(θ) �= 0 into a cartesian (rectangular)
11π equation.
arccos cos =
6
� � ��
(2 marks)
2.
1
arctan − √ =
3
� �
3
(b) Solve for x if − cos(x) + sin(x) = 2
�
(5 marks)
Question 5 - 3 marks
Find the general solution to sin(5x) = √1 .
2
Algebra Total 20 Marks
,
MATH1034A– Mathematics I Major - D diagonal Algebra - 28 August 2018(3 pages—page 1) MATH1034A– Mathematics I Major - D diagonal Algebra - 28 August 2018(3 pages—page 2)
Question 1 - 4 marks Question 3 - 10 marks
Let a = (3, 4) and b = 2i − 3j. Let z1 = 2i + 1 and z2 = 2 − i.
(a) Express 3a + 4b as an ordered pair and in terms of the vectors i and j. [2] (a) Find conjugates z¯1 , z¯2 and modulus of z1 , z2 . [2]
(b) Find the magnitude of the vector 3a + 4b and the unit vector in the direction of
3a + 4b. [2]
(b) Find complex number v = z1 · z2 and sketch v on the complex plane. [4]
Question 2 - 6 marks
Consider the parallelogram, P , formed by the vectors u and v.
(a) Give the diagonals of P in terms of u and v. [2]
(b) Show that the diagonals are orthogonal if and only if the lengths of u and v are z1
(c) Find complex number w = z2 and sketch w on the complex plane. [4]
equal. [4]
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