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Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering – Stage 1
(Bachelor of Engineering in Structural Engineering – Stage 1)
Summer 2005
Mathematics
(Time: 3 Hours)
Answer FIVE questions Examiners: Mr. T. Corcoran All questions carry equal marks. Prof. P.O’Donoghue Ms. F. Wood
1.(a) Find dx
dy for each of the following functions: (i)
− 2
1 1
tan x
x , (ii) e x^ +^ y = 3 x^2 y.
Simplify your answer fully in part (i). (8 marks)
(b) A rectangular hyperbola is defined by the parametric equations x = 4cosh t , y = 4sinh t.
Find dx
dy for the function and hence find the equation of the tangent to the curve at t = 0.2.
Given that cosh 2 t − sinh^2 t = 1 , express the function in terms of x and y and state the values of x for which the function is defined. (5 marks)
(c) A function is defined as
f ( x ) =
x if x
x if x
2 cos( )
2 sin( )
(i) Show that the function is continuous over the interval [0, 2 π ].
(ii) Deduce whether or not the function is differentiable over the interval.
Evaluate the integral ∫
(^32)
0
π f x dx. (7 marks)
2.(a) If f ( x )= sin x .cos x show that f " ( x )= − 2 sin( 2 x ).
Use Maclaurin’s theorem to obtain the first three non-zero terms of a series for the
function. Hence find x
x x x
sin. cos lim → 0
Find also the first three terms of a Taylor series for the function about the point x = 4
π .
Note: 2
cos 4
sin (^) =
π^ π. (10 marks)
(b) Use L’Hôpital’s Rule to find ( 1 sec )
.sin lim (^0) x
x x x → −
. (5 marks)
(c) Show that the function e −^2 x^ = 5 − x^2 has a root in the interval [2,3]. Use three iterations of the Newton-Raphson method to find the root correct to three decimal places. (5 marks)
3(a) Show that Z = ln x^2 + y^2 satisfies the equation (^2)
2 2
2
y
Z
x
Z
If x = r^2 − 4 p and p
r y = , use a tree diagram to find expressions for r
Z
and p
Z
and simplify an expression for r
Z
p
Z
when r = 2 and p = 1.
(8 marks)
(b) The deflection y at the centre of a circular plate suspended at the edge and uniformly
loaded is given by the formula (^3)
4
t
kwd y = where w = load, d = diameter of plate,
t = thickness and k is a constant. Calculate the approximate percentage change in y when w is increased by 1.5%, d is decreased by 0.8% and t is increased by 1.2%. (6 marks)
(c) Use the method of least squares to find the equation of the straight line which best fits the data:
x 0.0 1.1 3.2 3.9 7.1 8.
y 1.1 1.6 1.6 2.8 2.9 3.8 (6 marks)
6.(a) Given A = (^)
and B = (^)
show whether the following relationships are
true or false: (i) ( A -1^ ) T^ = ( A T ) -1^ (ii) ABT^ = BA T. In the event of a relationship being false, can you give the correct version? (4 marks)
(b) Given the matrices A and B where A =
and B =
find A −^1 and hence find the matrix D where AD = B. (7 marks)
(c) Explain the terms linear dependence and linear independence in relation to simultaneous equations. Using Gaussian Elimination reduce the following set of simultaneous equations into an upper triangular array:
x y az b
x y z
x y z
Solve the equations when a = − 2 , b =− 1. What values should a and b have if the system is to have an infinite number of solutions? Find the solutions in this case. What values of a and b would lead to an inconsistent system? (9 marks)
7.(a) Define the scalar product of two vectors a and b. (i) For what values of m are the vectors a = m i –2 j + k and b =2 m i + m j – 4 k perpendicular? (ii) Find the angles which the line joining the points (3,2, – 4) and (1, –1,2) makes with the co-ordinate axes. (6 marks)
(b) Let A (1,2,3), B (2, –1,5) and C (4,1,3) be three sides of a parallelogram ABCD in which the point C lies diagonally opposite point A. Find (i) the coordinates of D (ii) the area of ABCD. If e = 3 i – 4 j +2 k , determine whether or not e lies in the plane of ABCD and if not, find a unit vector lying in the plane. (7 marks)
(c) Given A = i + 4 j – 4 k , B = 2 i – 3 j + 2 k and C = 3 i + j – 2 k show that (i) the vectors form a triangle (ii) ( A – B) x ( A + B ) = 2( A x B ) (iii) C x ( A x B ) = ( C.B ) A – ( C.A ) B (7 marks)
