University of Dublin

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1 University of Dublin TRINITY COLLEGE Faculty of Enginering & Systems Sciences School of Engineering Junior Freshman Engineering Trinity Term 2015 Computer Engineering I (1E3) Date Location Time Dr L. Hederman Attempt THREE questions. 1 of 8

2 1. (a) Write a program to compute the minimum and average of an input sequence of numbers which is terminated by the sentinel NB The is not part of the sequence and should not be counted. (6 marks) (b) Write a program to compute the perimeter of a polygon. The program should first ask the user how many vertices the polygon has, then read in the (x, y) coordinates of those vertices, and compute and print out the total perimeter length. Recall that the distance between (x 1, y 1 ) and (x 2, y 2 ) is (x 2 x 1 ) 2 + (y 2 y 1 ) 2 (15 marks) (c) Write a program to list all the perfect numbers between 1 and 500. A positive whole number is perfect if the sum of its factors (apart from itself) equals the number. For example, the number 6 is perfect because its factors are 1, 2, and 3, and 1+2+3=6. Recall that if (n%i==0), i is a factor of n. (12 marks) (Total 33 marks) 2 of 8

3 2 (a) (i) (ii) (iii) Write a function called gcd to return the greatest common divisor of two integers. Use the following algorithm: Start at the smaller of the two integers, work back towards 1, until you reach an integer that divides into both of the parameters. Return this integer as the gcd. (7 marks) Using the gcd function of part (i), write a function to simplify fractions. The function should take two integer parameters, representing the numerator and denominator, respectively, and should update those parameters to represent the simplified fraction. For example if n is 49 and d is 84, then simplify (n, d) should result in n having the value 7 and d the value 12, since 49/84 simplified is 7/12. (6 marks) Using the function you wrote for part (ii), write a short main program that allows a user to enter fractions as two integers, and prints the fraction in simplified form. For example: (b) (i) (ii) Enter numerator and denominator: In simplified form that is 7/12 (5 marks) Declare a function isprime which takes a positive integer and returns true if the integer is prime and returns false otherwise. (There is no need to define this function.) (3 marks) Declare and define a function printnstars which takes a positive integer n and prints n asterisks followed by a new line. (5 marks) 3 of 8

4 (iii) Write a main program to draw a horizontal bar plot of the prime numbers between 1 and 50, preceded by a scaled axis. The sample extract below shows only the primes 1, 2, 3, 5 and 47, 49. You must use the isprime and printnstars functions of parts (i) and (ii) * ** *** *****... *********************************************** ************************************************* (7 marks) (Total 33 marks) 4 of 8

5 2. (a) Write a C++ function bool close_enough (double x, double y, double precision); which returns true if x and y are within precision of each other. The value for precision will typically be a very small number such as (b) Write a C++ function (5 marks) double newtonsqrt (double n, double init, double precision); which returns an approximation to the square root of n, obtained using Newton s method (see below), starting with an initial approximation of init. Use the function close_enough from part (a) to obtain an approximation that is within precision of the previous approximation in the Newton s method process. (NB There are no marks for terminating when approx squared is close to n.) Use other functions as appropriate. Newton s method for square roots is based on the fact that given a number n and an approximation x for its square root, a closer approximation to the square root can be obtained using the formula new approximation = (n / x) + x 2 For example if n is 2 and the initial approximation is 1, the new approximation is ((2/1)+1)/2 or 1.5. Since 1.5 is realtively far from the previous approximation 2, we use the formula to obtain a better approximation, ((2/1.5)+1.5)/2 or , and so on. (15 marks) Continued on next page of 8

6 (c) Newton s method for general functions uses the fact that, if x 0 is an approximation to the root of a function f, then x 1 = x 0 f(x 0 ) (where Df(x) is the derivative of f at x) Df(x 0 ) is a better approximation. Write a C++ function newtonf which returns an approximation to the root of the function f(x) = x 3 + 5x - 3. Use a function for f, a function for Df which is (3x 2 + 5), and reuse functions from parts (a) and (b) as appropriate. (13 marks) (Total 33 marks) 6 of 8

7 4. This question asks you to complete the program on the next page to count occurrences of index terms in a piece of text. The program already reads in the text (i.e. series of words), and you don t need to provide the corresponding function. The program also provides the 10 index terms of interest as an array called terms. (a) What is the implication of the keyword const in the declaration of the functions count and maxpos? (3 marks) (b) Provide a definition of the function count, which counts the occurrences of a given word in an array of words (7 marks) (c) Provide a definition for the function maxpos which returns the position of the highest integer in an array of integers (assume there are no ties). (8 marks) (d) Using the functions defined in parts (b) and (c), and using another array to store the counts for each term, complete the program to count the occurences in the text array of each term in the terms array, where the count is greater than 2, print the term and its count, and assign to topterm the most common term in the text, so that it is printed out by the last line of the program. Note that numwords stores the number of words of text to be processed; it is assigned a value by the call to readtosentinel. (15 marks) (Total 33 marks) 7 of 8

8 #include <iostream> #include <iomanip> #include <string> using namespace std; int count (string s, const string a[], int size); int maxpos (const int nums[], int size); void readtosentinel (string sentinel, string s[], int& size, int capacity); //no need to provide this function const int MAXTEXTSIZE = 1000; const int NUMTERMS = 10; int main () { string text[maxtextsize]; //holds text string terms[numterms] = {"array", "compute", "declare", "define", "for", function, "if", "loop", "variable", "while"}; int numwords; readtosentinel ("XXX", text, numwords, MAXTEXTSIZE); // Your answer to part (d) goes here cout << "The most common term in that text is \"" << topterm << "\"\n" << endl; } return 0; //Your answers to parts (b) and (c) go here University of Dublin of 8