ZAGAZIG UNIVERSITY FACULTY OF COMPUTERS & INFORMATICS MATHEMATICS Practical EXAM: Oct. 2013 Exam Duration: 50 minutes
Q1: a. Letf(x) = x2+1 , g(x) = đ?‘Ľ 2 − 1 Find: i.f-1(x)and fOg(x) ii. is fOg(x) even function ?Explain your reasoning. iii. find xand y-intercepts,if any for f(x). b.True or False, and Justify Circle T or F for each of the following statements to indicate whether the statement is true or false, respectively. If the statement is correct, briefly state why. If the statement is wrong, explain why. 1. is the following matrix is in row-echelon
ďƒŠ1 –8 4 –8ďƒš form ďƒŞďƒŞ0 1 0 –3ďƒşďƒş . ďƒŞďƒŤ0 0 1 1 ďƒşďƒť
2. If A and B such that AB = -BA, then A and B are said to be anti-commute. 3. Every homogeneous linear system is consistent 4. Let A, Bbe n Ă— n matrices. If the product AB is nonsingular, then A and B are both always also nonsingular. 5. If A is n Ă— n upper triangular matrix with nonzero diagonal entries. Then A-1must also be an upper triangular matrix. 6. the graph of the function f(x)=5x2cosx is symmetric eith respect to the y-axis. Q2: a. Use the following matrices to perform the following operations: A ď€ F , AT, BA , A-1, (FT)T, tr(A) ,F-1F,(A-1)-1and (AT)-1if possible. ďƒŠ1 1 3 ďƒš A  ďƒŞďƒŞ 4 2 2 ďƒşďƒş ďƒŞďƒŤ 4 0 4 ďƒşďƒť
ďƒŠ 2 2 4ďƒš F  ďƒŞďƒŞ –1 1 0 ďƒşďƒş ďƒŞďƒŤ 2 –3 –4 ďƒşďƒť
ďƒŠ –9 16 ďƒš B= ďƒŞďƒŞ –2 6 ďƒşďƒş ďƒŞďƒŤ 2 –1ďƒşďƒť
i.Is A is symmetric matrix? Justify your answer ii.Is F is skew-symmetric? Justify your answer. iii.Is A invertible matrix? Justify your answer. iv.Is Aorthogonal matrix? Justify your answer. ďƒŹ
x  2y
B .Solve the following system of linear equations ďƒŻďƒď€2 x  4 y  8 z ďƒŻ ďƒŽ
whether the system is consistent.
1
3x ď€ 3z
 3  10 .  –6
and determine