Higher Engineering Mathematics, Sixth Edition

Revision Test 5

This Revision Test covers the material contained in Chapters 14 to 17.The marks for each question are shown in
brackets at the end of each question.

1. Solve the following equations in the range 0◦
   to 360◦.
   (a) sin−^1 (− 0. 4161 )=x
   (b) cot−^1 ( 2. 4198 )=θ (8)


2. Sketch the following curves labelling relevant
   points:
   (a) y=4cos(θ+ 45 ◦)
   (b) y=5sin( 2 t− 60 ◦) (8)


3. The current in an alternating current circuit at
   any timetseconds is given by:

i=120sin( 100 πt+ 0. 274 )amperes.

Determine

(a) the amplitude, periodic time, frequency and

phase angle (with reference to 120sin100πt)

(b) the value of current whent= 0

(c) the value of current whent=6ms

(d) the time when the current first reaches 80A

Sketch one cycle of the oscillation. (19)

4. A complex voltage waveform v is comprised
   of a 141.4Vrms fundamental voltage at a fre-
   quency of 100Hz, a 35% third harmonic com-
   ponent leading the fundamental voltage at zero
   time byπ/3radians, and a 20% fifth harmonic
   component lagging the fundamental at zero time
   byπ/4radians.
   (a) Write down an expression to represent
   voltagev.

(b) Draw the complex voltage waveform using

harmonic synthesis over one cycle of the

fundamental waveform using scales of 12cm

for the time for one cycle horizontally and

1cm=20V vertically. (15)

5. Prove the following identities:

(a)

√[

1 −cos^2 θ

cos^2 θ

]

=tanθ

(b) cos

(

3 π

2

+φ

)

=sinφ

(c)

sin^2 x

1 +cos2x

=^12 tan^2 x (9)

6. Solve the following trigonometric equations in the
   range 0◦≤x≤ 360 ◦:
   (a) 4cosx+ 1 = 0

(b) 3.25cosecx= 5. 25

(c) 5sin^2 x+3sinx= 4

(d) 2sec^2 θ+5tanθ= 3 (18)

7. Solve the equation 5sin(θ−π/ 6 )=8cosθ for
   values 0≤θ≤ 2 π.(8)


8. Express 5.3cost− 7 .2sintin the form
   Rsin(t+α). Hence solve the equation
   5 .3cost− 7 .2sint= 4 .5 in the range
   0 ≤t≤ 2 π. (12)


9. Determine

∫

2cos3tsintdt.(3)