Maximum Values - Mathematics - Old Exam Paper, Exams for Mathematics. Chhattisgarh Swami Vivekanand Technical University

Mathematics

Description: Main points of this past exam are: Maximum Values, Mclaurin Expansion, Maximum and Minimum Values, Area Between Curve, Least Square Line, Standard Deviation, Find Probability, Confidence Limits, Graph Sheet, Scales and Axes
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Cork Institute of Technology
Higher Certificate in Engineering in Civil Engineering – Award
(National Certificate in Engineering in Civil Engineering – Award)
(NFQ – Level 6)
Summer 2005
Mathematics
(Time: 3 Hours)
Instructions
Answer FIVE questions; a minimum of TWO
questions in each of Sections A and B should be
attempted.
All questions carry equal marks.
Examiners: Mr. L. O Hanlon
Mr. P. Anthony
Mr. J. Murphy
Section A
Q1. Differentiate with respect to x:
(a)
y = e-2x + 4x (4 Marks)
(b)
y = eSin x (4 Marks)
(c)
x3y3 – x3 y3 = 0 (4 Marks)
(d)
x = a (Cost + bSint)
y = c(Sint + dCost) a,b,c,d are constants (4 Marks)
(e) y = xSinx (4 Marks)
Q2. (a) Write down the first four terms of the McLaurin expansion of f(x) = ex.
Hence calculate 1.
ecorrect to four places of decimals. (10 Marks)
(b) Find the maximum and minimum values of y = lnx – x2 + 2x. (10 Marks)
Q3. Integrate
(a) dx
x
ex
2
1
5.0
1
(4 Marks)
(b)
+2
16
1
x
dx (4 Marks)
(c)
()
xdxx .1
3
2
3
2
+ (4 Marks)
(d) xdxx ln
2
(e)
()()()
dx
xxx
xx
432
12
2
(4 Marks)
2
Q4. (a) Find the area between the curve y2 = 4x and the line y = x. (5 Marks)
(b) If the velocity v in m s-1 of a body, is given by 1+= tv , find the average value of v
from t = 1 to t = 4 s. (5 Marks)
(c) Find the root mean square value of
y = 1 + 2 Sin x between x = 0 and x = 2π. (5 Marks)
(d) The mass per unit length of a rod at xm from one rod is (1 + .04x) kg m-1. If the rod is 3
m long, find its mass. (5 Marks)
Section B
Q5. (a) Solve the differential equations
(i) 2
1x
dx
dy = (3 Marks)
(ii)
()
1
22 = xy
dx
dy (3 Marks)
(iii) 3
2
2
x
dx
yd = (3 Marks)
(iv) 2
when3r and rSin
d
dr
Π
===+
θθ
θ
0 (3 Marks)
(b) For a body at temperature
θ
above its surroundings the rate of all of temperature is given
by the differential equation
θ
θ
k
dt
d= where k is a constant.
Find the solution of this equation given
θ
= 60 when t = 0. (8 Marks)
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