Practice Questions for Midterm PDF

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COMP 1113 Applied Mathematics for CST

Practice Questions for Midterm 1. Convert the following values from their current base to the indicated base. a. 1458 to Binary and to e. 132334 to base 10 Hexadecimal f. AF116 to octal b. 5647 to base 10 g. 0.11 kibibits to decimal. c. 133.410 to binary d. 104.55 to binary 2. Do the indicated arithmetic operation in the base of the numbers. a. AFF16 c. 110111012 + DED16 -010011002 b. 13445 - 4425

d. ED216 +F2816

3. Use determinants (Cramers’ Method) to find the solution to the following system of equations. 0.344a  0.21b  0.448

1.34 a  2.34 b  1.07 4. Perform the following matrix algebra operation given the matrices or explain why it can not be done: 1 0 3   2 1 1  1 5 0 A   2 5 3 , B   , C        2 0 3   2 0 1  0 2 2  a) BA+3C b) BC 5. A simple to implement (and unfortunately simple to break) encryption method involves taking information (in some numerical form such as binary) rearranged in the form of a m  n matrix, I, and multiplying it by a standard encoding matrix, E, to recode it into a new matrix of numbers. The encoded matrix to be transmitted is then given by T  EI . To decode the information you can multiply the transmitted matrix by the matrix inverse of the encoding matrix, E 1 and you should get back the original information, I  E 1T . a. In the following problem the binary string (representing the secret information) 1011001011 is made into an information matrix I  1 1 0 1 1  . Notice that this was 0 1 0 0 1  just the result of taking each group of 2 bits and putting them into a column. Use an encoding matrix

 1 4 E   2 7 

to find the matrix to be transmitted, T. 5 4 1   , find the original information matrix  5 7  2

b. If you receive a transmitted matrix T  

that had been transmitted. I  E  1T (No it is not the one from part a). In finding the decoding matrix you must clear show and label all steps in finding E  1 .

COMP 1113 Applied Mathematics for CST

6. Solve the following system of linear equations using the method of inverting the coefficient matrix i.e X=A-1B. You must show all your work. Make sure you label each row operation being done in the matrix inversion.

x  3 y  7z  0 2 x  y  z  10 2 x  y  5z  16  cos( ) sin( )  7. In 2D there is a single rotation matrix given by R ( )    . In addition we   sin( ) cos( )  can define a matrix for deforming the shape in a fashion called a shear transformation. In 1 0  this case we will consider a shear transformation given by S   . 2 1  0 0 3 0  We are given a shape defined by point matrix P    3 2    0 4 (which is in the form [x y]) and shown in the figure a) Find the resulting figure after a rotation of +45° and then a shear, S, in that order. Find the new points matrix and plot the result on the given graph. b) Given that the inverse of a matrix can be thought of as undoing any graphic transformation, find the one matrix that will undo the two transformations in part a) (R(45°) and S ) and perform the calculation to confirm that you do get back to the original points P.

COMP 1113 Applied Mathematics for CST

Answers 1) a) 001100101 6516 b) 29110 c) 10000101.0110 d) 1101000.100011 e) 495 f) 53618 g) 112.64 2) a) 18EC16 b) 4025 c) 10010001 d) 1DFA16 3) ( a, b)  (1.1718,  0.2138)  7 18 5 4) a)   b) can’t as dimensions are wrong   8  6 3

 1 5 0 1 5  b) E  1  1 7 4 I  1 0 1 5) a) T    15 2 1  1 1 0   2 5 0 2 5   6

6) A1  1 12  30

 0

10  8 15  . X   5    5 5   1  8 9

 0  9 2  2 7) Post rotation and shear P    11 2  2   2 2

0   3 2 2   5 2 2   2 2 

 2  2 Reversal:   2   2

2  2  3 2  2 

...