SEF1124 MATHEMATICS 2
COMPILATION OF MID-SEMESTER EXAM QUESTIONS (CHAP 1 & CHAP 2)
Content
Semester III 2011/2012
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Semester III 2012/2013
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Additional Exercises from Past Exams
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Semester III 2011/2012
QUESTION 1
3
(a) Given that tan 𝜃 = and 𝜃 is an acute angle. Find the exact values of
5
(i)
(ii)
sin θ
sec θ
(b) Without using a calculator, find the exact value of
(i)
sin 383° ∙ csc 23°
(ii)
cos
5𝜋
3
− tan
11𝜋
4
(c) Write sin t in terms of cos t, where 𝜋 < 𝑡 <
1
11𝜋
8
SEF1124 MATHEMATICS 2
QUESTION 2
(a) Given the graph of 𝑦 = 𝐴 sin(𝑘𝑥 − 𝑏) where 𝐴 > 0 and 𝑏 < 0 as shown below.
Determine the values of 𝐴, 𝑘 and 𝑏.
(b) Given the function 𝑓(𝑥) = 2 cos(3𝑥 + 𝜋).
(i)
Determine the period.
(ii)
𝜋
Sketch the graph of 𝑦 = 𝑓(𝑥)for − 6 ≤ 𝑥 ≤
Hence state the range of 𝑓(𝑥).
QUESTION 3
Without using a calculator, find the exact value of:
(a) cos (tan−1 (−
2
1
))
√5
𝜋
(b) sec (cot −1 (3) + 2 )
2
5𝜋
6
. Label the x and y-intercepts.
SEF1124 MATHEMATICS 2
QUESTION 4
1
(a) Given sin α = 2 and α is in the second quadrant. Find
α
(i)
cos 2
(ii)
tan 2𝛼
(b) By writing cos15°in the form cos(𝐴 − 𝐵)and cos75°in the form cos(𝐴 + 𝐵),
determine the angles 𝐴 and 𝐵.
Hence, without using a calculator and by using the sum and difference formula,
show that cos15° − cos 75° = 2 sin 45° sin 30°.
QUESTION 5
(a) Express √3 cos 2𝑥 − sin 2𝑥 in the form 𝑅 cos(2𝑥 + 𝛼) where 𝑅 > 0 𝑎𝑛𝑑 0 ≤ 𝛼 ≤ 2𝜋.
Hence find all the values of 𝑥, with 0 ≤ 𝑥 < 𝜋 such that
√3 cos 2𝑥 − sin 2𝑥 = √3
(b) Prove that
3 1
− (1 + 2 cos 2𝛼) cos(2𝜃 + 2𝛼)
2 2
and 0 ≤ 𝜃 ≤ 2𝜋, solve the equation
sin2 𝜃 + sin2 (𝜃 + 2𝛼) + sin2 (𝜃 + 𝛼) =
Hence, if 𝛼 =
𝜋
4
sin2 𝜃 + sin2 (𝜃 + 2𝛼) + sin2(𝜃 + 𝛼) =
3
7
4
SEF1124 MATHEMATICS 2
Semester III 2012/2013
QUESTION 1
Let 𝑃(𝑎, 𝑏) be the point on the terminal side of 𝜃 on the unit circle as shown above.
(a) Find cos 𝜃.
(b) Show the coordinates of the points for the terminal sides of the following angles
(i)
𝜃+𝜋
𝜋
(ii)
𝜃−2
(iii)
𝜋
Hence find the values of sin(𝜃 + 𝜋) and sin (𝜃 − ).
2
QUESTION 2
1
(a) Given the function 𝑦 = 2 sin(2𝜋𝑥 + 𝜋)
(i)
Find the amplitude, period and phase shift of the function.
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SEF1124 MATHEMATICS 2
(ii)
Hence, sketch the graph over one period starting from the phase shift and
label the x-intercepts.
(b) Without using a calculator, find the exact value of:
2 csc(−570°) + 3 sec 4 495° − tan(−420°)
QUESTION 3
(a) Prove the identity
𝛼
2(1 − cos 𝛼)
sec 2 ( ) =
2
sin2 𝛼
𝑘−tan 𝐵
(b) If sin(𝐴 + 𝐵) = 𝑘 cos(𝐴 − 𝐵)where k is a constant, show that tan 𝐴 = 1−𝑘 tan 𝐵.
QUESTION 4
(a) Without using a calculator, find the exact value of
3𝜋
3𝜋
tan [sin−1 (sin ) + cos−1 (cos (− ))]
5
5
𝐴
(b) Given that sin 2 = −
√5
3
3
𝐴
and tan 2𝐵 = 4 where 2 and 2𝐵 are in the same quadrant.
Find the exact value of:
(i)
sin A
(ii)
𝐵
sin 2 sin
3𝐵
2
QUESTION 5
(a) Solve the equation 2 cos2 𝜃 − sin2 𝜃 = 4 sin 𝜃 − 2 for 0° < 𝜃 < 360°.
(b) Express sin 3𝜃 − √3 cos 𝜃 in the form of
𝑅 sin(3𝜃 − 𝛼) where 𝑅 > 0 and 𝛼 is an acute angle.
Hence solve the equationsin 3𝜃 − √3 cos 𝜃 = 1 for 0 ≤ 𝜃 ≤ 2𝜋.
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SEF1124 MATHEMATICS 2
ADDITIONAL EXERCISES FROM PAST EXAMS
1. If cos 𝜃 = 0.3, find the value of cos 𝜃 + cos(𝜃 + 2𝜋) + cos(𝜃 − 2𝜋).
2. The point (5, −12) is on the terminal side of an angle 𝜃 in standard position. Find
the exact value of sin 𝜃, cos 𝜃 and tan 𝜃.
3. Establish the identity
cos 4𝜃−cos 8𝜃
cos 4𝜃+cos 8𝜃
= tan 2𝜃 tan 6𝜃.
4. Develop a formula for cos 4𝜃 as a fourth degree polynomial in the variable cos θ.
5. Find the exact value of 2 cos −1 𝑥 + 𝜋 = 4 cos −1 𝑥.
6. Establish the identity
cos 𝜃+2 cos 2𝜃+cos 3𝜃
cos 𝜃−2 cos 2𝜃+cos 3𝜃
𝜃
= − cot 2 2
7. Solve the equation in [0°, 180°]
sin 𝑥 + sin 3𝑥 + sin 5𝑥 = 0
8. Prove the identity
cos 𝑥
1 + sin 𝑥
+
= 2 sec 𝑥
1 + sin 𝑥
cos 𝑥
9. Using the sum-to-product formula, solve the equation sin 6𝑥 + sin 2𝑥 = √3 sin 4𝑥 in
the interval [0, 𝜋].
10. Without using a calculator, evaluate cos 200° csc(−70°).
11. Show that 4 sin 4𝑥 cos 2𝑥 cos 𝑥 = sin 7𝑥 + sin 5𝑥 + sin 3𝑥 + sin 𝑥.
12. Solve the equation sin3 𝑥 sec 𝑥 = 2 tan 𝑥 where 𝑥 is in the interval [0, 𝜋].
13. Using the sum-to-product formula, find the general solution for the equation
sin 4𝑥 + sin 2𝑥 = sin 3𝑥
END
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