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Cork Institute of Technology
Bachelor of Engineering in Biomedical Engineering – Stage 1
(NFQ – Level 7)
Autumn 2007
Mathematics & Computing
(Time: 3 Hours)
Instructions Answer FIVE questions. Answer at least ONE question from Section B. Question 8 in Section C is compulsory. Use a separate answer book for Section C. All questions carry equal marks.
Examiners: Mr. D. O'Shea Mr. A. McGibney Mr. M. Murphy Prof. P. McHugh
SECTION A
Q1 (a) Using the laws of indices simplify the following, giving your final answers with positive indices.
(i) 4 x^ +^1 .8^2 x^ +^4 .2^3 −^2 x
(ii)
3 6 (^3 89 8 ) 27
A BC
A B −^ C −
(iii)
3 5 3 1 2 3 2
x x x x
− −
(b) Solve for x :
(i) log 2 ( 2 x + 3 ) − log 2 ( x − 2 )= 4
(ii) log 3 ( x + 5 ) + log 3 ( x + 3 ) = 1
(iii) 32 x^ −^3 = 2 x +^2 (8 marks)
(c) Transpose the formula
2
2
c h a s d h
to make d the subject. (4 marks)
Q2 (a) The table below shows values of I and P that are believed to be related by the law P = RIn , where R and n are constants.
I 1 1.2 1.4 1.6 1.8 2 P 2.4 3.5 4.8 6.3 7.9 9.
(i) Verify the law is as stated. (ii) Determine the approximate values of R and n. (iii) Hence state the law. (iv) Find the value of P when I = 5. (v) Find the value of I when P = 8.4. (12 marks)
(b) The temperature, T, is given by the formula T = T e 0^ kt , where T is the temperature in minutes and T 0 is the temperature at time t = 0. If the initial temperature is 12 ° , find: (i) The value of T 0. (ii) The value of k if the temperature after 3 minutes is 24 °. (iii) The temperature after 9 minutes (iv) How many minutes it will take the temperature to reach 300 ° (8 marks)
Q3 (a) Solve for x, y and z:
2 3 3 2 2 3 5 5
x y z x y z x y z
(9 marks)
(b) Expand ( )
12 1 − 3 x as far as the term in x^4. (7 marks)
(c) Convert: (i) 140 36° ' to radians. (ii) 5.25 radians to degrees and minutes. (4 marks)
SECTION B
Q6 (a) Differentiate each of the following:
(i) (^35)
y x x x x
(ii)
sin 5 2
e^ x y x
= −
(iii) y = ( 3 x^3^ + 2 x − 1 )( ln(2 x^2 − 8 x +5))
(iv) y = 7 x^2 − 2 x − 3 (12 marks)
(b) The distance x metres traveled by a body in t seconds is given by: x = 23 t^3 − 5 t^2 + 8 t − 6. Find: (i) Expressions for velocity and acceleration in terms of t. (ii) The velocity and acceleration after 5 seconds. (iii) The times when the body is at rest. (iv) The distance traveled after 10 seconds. (8 marks)
Q7 (a) Determine each of the following integrals:
(i) (^33)
x x dx x x
∫^ +^ +^ +^ −
(ii) ∫(5 x −12)^7 dx
(iii) ( )
2
0
4sin 2 x 3cos 3 x dx
π
(iv)
0.5 2 25
∫ (4^ x^ +5)(4^ e^ x^ + x ) dx (14 marks)
(b) Find the area enclosed between the curve y = x^2 − 3 x − 10 and the x-axis.
(6 marks)
SECTION C
Use a SEPARATE ANSWER BOOK for Question 8
8 (a) Study the following program:
void main() { int num1 = 1; int result = 0;
while(num1 < 4) { result = result + num1; num1++; } cout << "The Result is: "<< result << endl; }
What will the output of this program be; explain your reason step by step. (5 marks)
8 (b) Write a program that displays the 3 options listed below. Allow the user select an option until Exit is selected. Program must report on an invalid selection.
- Up
- Down
- Exit
If Up option is selected, print “What’s up” on the screen. If Down is selected, print “What’s down” to the screen. (10 marks)
8 (c) Look at the following program:
#include using namespace std;
void main() { int x = 4; int y = 7; int ans = y+x / 2 ; cout << "The Result is : "<<endl ; cout<< ans + 7 <<endl; }
Explain each line of the code individually, and indicate what would be output to the screen for cout statements. (5 marks)
