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Mechanical Engineering Exam: Mathematics for Higher Certificate (Autumn 2007), Exams of Mathematics

West Bengal State UniversityMathematics

The questions and answers for a mathematics exam for students enrolled in the higher certificate engineering in mechanical engineering program at cork institute of technology. The exam covers topics such as calculus, integration, and statistics. Students are required to answer five questions within a 3-hour time frame.

Typology: Exams

2012/2013

Uploaded on 03/28/2013

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1
Cork Institute of Technology
Higher Certificate Engineering in Mechanical Engineering – Award
(NFQ – Level 6)
Autumn 2007
Mathematics
(Time: 3 Hours)
Answer FIVE questions Examiners: Ms. J. English
Dr. D. Cremin
Mr. J. Connelly
Dr. P. Delassus
Q1. (a) A curve is described by the parametric equations 9)sin(7 +
=
tx ,2)cos(4
−
=ty .
Find the equation of the tangent to the curve at 1.9t
=
radians.
[7 marks]
(b) If 06)cos(45 2=++
−yyxx , find dy
dx at the point (3,2). [6 marks]
(c) Show that the equation 2
() 5fx x
=
−+ has a root between x=2 and x=3. Use the
Newton-Raphson method with three iterations to find the root correct to two
decimal places.
[7 marks]
Q2. (a) Given m= 3x2y + 4x -2y-2 +6 find
,
mm
x
y
∂∂
∂∂
and
2
2
m
x
∂
∂
[6 marks]
(b) You are given that
3
kr
ym
= where k is a constant and r and m are variables.
Use a calculus method to find the approximate percentage error in y due to errors
of +2.5% in r and –3.1% in m. [8 marks]
contd…/
pf3
pf4
pf5

Partial preview of the text

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Cork Institute of Technology

Higher Certificate Engineering in Mechanical Engineering – Award

(NFQ – Level 6)

Autumn 2007

Mathematics

(Time: 3 Hours)

Answer FIVE questions Examiners: Ms. J. English Dr. D. Cremin Mr. J. Connelly Dr. P. Delassus

Q1. (a) A curve is described by the parametric equations x = 7 sin( t )+ 9 , y = 4 cos( t )− 2.

Find the equation of the tangent to the curve at t = 1.9radians. [7 marks]

(b) If 5 x −^2 + 4 x cos( y )+ 6 y = 0 , find dydx at the point (3,2). [6 marks]

(c) Show that the equation f ( ) x = − x^2 + 5 has a root between x=2 and x=3. Use the Newton-Raphson method with three iterations to find the root correct to two decimal places. [7 marks]

Q2. (a) Given m= 3x 2 y + 4x -2y -2^ +6 find ∂∂ m (^) x ,∂∂ my and 2 2

m x

[6 marks]

(b) You are given that kr^3 y = (^) m where k is a constant and r and m are variables.

Use a calculus method to find the approximate percentage error in y due to errors of +2.5% in r and –3.1% in m. [8 marks] contd…/

(c) Locate the turning points on the curve y = 3 2 t +^ t 2 − 2 t + 4 and establish whether they are maximum or minimum points. [6 marks]

Q3. Determine each of the following integrals:

(i)

(^14) 0

∫^3 xe^ xdx

(ii)

6 4 2

x (^) dx x

(iii)

3 1 3

t t

e (^) dt

∫ + e

(iv) r^13 dr r

∫   [20 marks]

Q4. (a) Find the position of the centroid of the figure bounded by the curve y = 4x -x^2 , the x-axis, the y-axis and the ordinate at x = 0 and x= 4. b ab

a

X

xydx

ydx

=

b

ba a

Y

y dx

ydx

=

[8 marks]

(b) Calculate the area enclosed between the curves y = x 2 + 4 and y + x =10. [6 marks] (c) Find the root mean square of the function y = 7 x − x^2 over the interval 0 ≤ x ≤ 7 [6 marks]

Q7. (a) Warranty records show that the probability that a new car needs a warranty repair in the first 50 days is 0.05. If a sample of 8 cars is selected, what is the probability that in the first 50 days (I) 3 need a warranty repair? (ii) at least two needs a warranty repair? (iii) four or less need a warranty repair? [7 marks]

(b) A chemical manufacturer produces aspirin tablets having a mean mass of 5.8g and a standard deviation of 0.3g. What is the probability that a tablet chosen at random will have a mass (i) less than 6.1 g? (ii) greater than 5.7g? (iii) between 5.7g and 6.4g? [7 marks]

(c) Cork Airport has recently analysed its passenger listings and found that, on average, out of every thousand, three are known to be Italian citizens. Calculate the probability that if a random sample of 3,500 tourists were interviewed, that: (i) five (ii) more than two would be Italian citizens.

[6 marks]

Probability Distributions

Binomial Distribution:

P r ( ) = n^ C p qr r^ n^ − r

Poisson Distribution:

( ). !

e m^ mr P r (^) r

−

Normal Distribution:

Standard units, Z = x σ^ − X