Questionnaire for CSET Mathematics subset 1

Transcription

1 Questionnaire for CSET Mathematics subset 1 Below is a preliminary questionnaire aimed at finding out your current readiness for the CSET Math subset 1 exam. This will serve as a baseline indicator for the workshop. The way we expect you to respond is given below For the questions asked below, answer Y - if you are sure that you know the answer very well (i.e you remember the principles behind the question very well, you know how to arrive at the answer and you can answer it well even now) R - if you think you remember having done the principles behind the problem, sometime in the past, but do not remember them really well now but feel that you might be able to have a go at answering this question well with a little bit of revision. V - If you vaguely recall coming across a similar question somewhere but do not remember anything about how to answer it G - If the question seems like Greek to you and is totally above you A. Number systems 1. Do you know of an example of a natural number?. Do you know of an example of an integer which is not a natural number? 3. Do you know of an example of a rational number which is not an integer? 4. Do you know of an example of a real number which is not a rational number? 5. Do you know of an example of a complex number which is not a real number? 6. Are all natural numbers also integers? 7. Are all integers also rational numbers? 8. Are all rational numbers also real numbers? 9. Are all real numbers also complex numbers?

2 Algebra (SMR Domain 1) B. SMR 1.. Polynomial equations and inequalities 1. Do you know what the Factor theorem is?. Can you show that x + 5 is a factor of x 3 + 7x + 7x! 15 = 0? 3. Do you know what the Rational Root theorem is? 3 4. You are given that the equation 6x! 13x! 13x + 30 =0 has a solution which is a rational number. Do you know how to guess what this solution might be? Do you know how to find all the other solutions? 5. Do you know what the Conjugate roots theorem is for polynomials with real coefficients? 6. Do you know the Quadratic formula to solve the equation ax + bx + c = 0? 7. Do you know how to, without actually solving the equation, determine the nature of the roots of a quadratic equation? 8. Can you for example show that the equation x! x + = 0 has two complex roots? 9. Do you know what the Binomial Theorem is? 5 ( x + y) using the Binomial 10. Do you know how to expand Theorem? 11. Without expanding, do you know how to find the 5 th term of 7 ( x + y)? 1. Do you know how to solve a linear inequality like x! 5 > 1? 13. If you were to graph the above solutions in the XY plane, what type of area would they represent? 14. Can you graphically represent the area in the XY plane that is the solution to the following system of inequalities? 3y! x "1 y + 3x >18

3 C. Functions (SMR 1.3) 1. Do you know what a relation is?. Do you know what a function is? 3. If f ( x) = x and g ( x) = x + 3, can you find f o g(x)? 4. Do you know what a one-to-one function is? 5. Can you give an example of a one-to-one function? 6. Can you give an example of an onto function? 7. Do you know what properties a function must satisfy in order to have an inverse function?!1 8. If f ( x) = x +, can you find the inverse f? 9. If you are given the graph of a function f which has an inverse, do!1 you know how to draw the graph of f? 10. Do you know the shapes of the graphs of the following functions? f ( x) = 3x f ( x) = x 3 f ( x) = x 11. Do you know what a rational function is? 1. Do you what the vertical asymptotes and horizontal asymptotes of a rational function are and can you find them in a given function? 13. Do you know what the graph of the function x a (exponential function with base a) looks like for different values of a? 14. Can you graph the function y = log x? 15. Do you know the properties of the logarithm function that will help you to do a calculation such as the following Solve log 4 ( x + 3) + log 4 ( x! 3) = a

4 D. Linear Algebra (SMR 1.4) 1. Consider the following matrices! A = 1 $! # & B = 7 5 $ # & C = 3 9 " 3 4% " 3 % F = 1 5 7! 1 $! $ # & # & G = # 3 4 & "'1 0 5% # & " 7 '1% ( ) D = # 10!' 1 5 $! $ # & & E = # & " 1% # & " 1 '1% Out of the above matrices, can you determine which pairs can be added and which pairs can be multiplied? Can you carry out these additions and multiplications?. Can you find the determinant of the matrix E (above)? 3. Once you calculate the determinant of E, do you know any result which will help you to easily calculate the determinant of the matrix 3E? 4. Look at the matrix H obtained by interchanging the first two rows of E. " % $ ' H= $! 1 5 ' $ ' # 1!1& Can you say at once what the determinant of H is? Consider the following system of linear equations x + y + z = 6 x + y + 3z = 14 x + 4y + 9z = Do you know how to represent this system as a matrix equation? 6. Once expressed as a matrix equation, do you know how to solve this using Cramer s rule?

5 7. There is another way (other than Cramer s rule) of solving a system of equations such as above where you use the concept of row operations. Do you know this method? Consider the following three dimensional vectors u = i + j v = i + 3 j! 4k 8. Can you perform the operation 7 u! 8v? 9. Can you perform the dot product u v between the two vectors given above? 10. Do you know the value of the dot product of two vectors which are at 0 90 to each other? 11. Can you perform the cross product u! v between the two vectors given above? 1. Geometrically, can you position u! v with respect to u and v? E. Algebraic Structures (SMR 1.1) 1. Have you heard of the term Group in a Mathematical sense?. Have you heard the term Ring in a Mathematical sense? 3. Have you heard the term Field in a Mathematical sense? 4. If you are given a set and two operations, do you know the conditions that they have to satisfy in order to form a field? 5. Do you know an example of a ring which is not a field? 6. Have you heard of the term ordered field?

6 Number theory (SMR Domain 3) F. Natural numbers (SMR 3.1) 1. Can you write down the first 10 prime numbers?. Do you know that any natural number is a product of prime numbers? 3. Can you write down the prime factorization of 180? 4. Do you know how many natural numbers divide 5 10? 5. Do you know what the division algorithm is? 6. Do you know what is meant by the greatest common divisor (g.c.d) of two numbers and the least common multiple (l.c.m) of two numbers? 7. Do you know what the Euclidean Algorithm is? 8. Do you know how to use the Euclidean Algorithm to calculate the g.c.d of two numbers such as 5767 and 4453? 9. Once you have the g.c.d., do you know how to find the l.c.m of the above two numbers? 10. Do you know what the Principle of Mathematical Induction is? 11. Do you know how to use the principle of Mathematical Induction to prove statements like those shown below? n( n + 1) a) n = for every natural number n n b) n < for all natural numbers greater than 4.