Mathematical Sciences

Transcription

1 Mathematical Sciences Diagnostic Test 2014/15 User: The aim of this test is to assess your current knowlege base in certain areas of mathematics. The material is taken broaly from the Mathematics A -level syllabus an it is expecte that most stuents will perform very well in this test. The ata gathere from this test will help us to juge what material nees aitional revision uring this first year. Refresher materials are available online at an further online resources can be foun via the University of Exeter s Stuent Resources pages on ELE. You may attempt this test as many times as you wish your best score will be recore. It is strongly avise that you keep a note of your solutions as you work through the test. You shoul attempt these questions without using a calculator. Barrie Cooper August 28, 2014 Warning! Warning! If your username oes not appear in the box above then you must reloa this page using the link in ELE. Do not use your browser s back button if you o then you must reloa this page using the link in ELE. Contents A Some avice regaring your answers B Algebra C Calculus D Logic

2 A Some avice regaring your answers This test is computer (not human) marke so you shoul take care to present your answers in the format outline below. You shoul always provie exact answers, not ecimal approximations. So, for example, write 1/3 not You shoul present your answers in simplest possible form, so 1/3 will be marke correct, but 2/6 will not. Similarly, log(9) woul be marke incorrect, because the answer shoul be written as 2*log(3). Some of the answers require are symbolic. Variables in such expressions shoul always be precee by a $ symbol. So the variable x, woul be written as $x. Multiplication is written as *. You nee to make the multiplication explicit in expressions such as 2x, so write 2*$x an not 2$x. Powers such as x 3 can be written as pow($x,3) or as $x*$x*$x. Stanar functions can also be use, so: e 2t shoul be written as exp(-2*$t) sin ( ) ht 2 shoul be written sin($h*$t/2) 3 shoul be written as sqrt(3) ln(x) can be written either as log($x) or as ln($x) The computer will only unerstan what you write, not what you mean, so make every effort to make sure that these coincie! If in oubt, there is no harm in aing extra pairs of brackets to symbolic expressions (but be sure to close any brackets you open, otherwise you really will confuse the computer!). For example, x+1 is not the same as $x+1/$x-1 an shoul be written ($x+1)/($x-1). x 1 Extra spaces in expressions shouln t affect the answer, except in the case of negative numbers: so write -2 rather than - 2. I have trie, wherever possible, to esign the test so that it is marke fairly by the computer, but there will probably be cases where I have misse something my sincere apologies in avance. Likewise, I hope I have caught any errors that might occur, but someone will probably fin a way of writing something that confuses the computer so much that it can t continue! Any errors in this regar shoul be brought to my attention at B.Cooper@exeter.ac.uk. I hope you fin this test useful. Goo luck, Barrie.

3 B Algebra 1. Simplify the following expressions involving fractions: (a) (b) (c) 1 a 1 b 2. Simplify the following expressions involving exponents: (a) t 4 t 7 t 1 t (b) (xy)6 x 3 y 2 y 5 x y (c) e3 ln Simplify the following expressions involving square an n th roots: (a) 9 3 (b) (c) Questions involving polynomial expressions: (a) Expan the expression (1 t) 4 t 4 + t 3 + t 2 + t + (b) Fin the zeros of the polynomial p(x) = 12x 3 17x 2 + 3x + 2 x =, an (c) Simplify the expression 2z2 +7z 15 3z 2 +14z 5 5. Write each of the following expressions as a sum of partial fractions: (a) 1 2x 2 5x 3 x 3 + 2x+1 (b) 3x+7 x 2 1 x 1 + x+1 (c) x 2 +2x 5 + x 3 x 2 5x 3 x 3 x+1 +

4 6. Questions involving trigonometric ientities: (a) Write sin(4t) in terms of sin(t) an cos(t) ( sin(t) cos (t) sin (t) cos(t) ) (b) Evaluate ( cos 2 π 12 sin2 π 12) 2 (c) Simplify sin (n+1)π h sin π h sin nπ h sin 2π h + sin (n 1)π h sin π h 7. Questions involving simultaneous equations: (a) Solve the pair of simultaneous linear equations 3x 5y = 0 x + y = 2 x = an y = (b) Solve the following pair of simultaneous equations for x an y positive: xy = 1 x 2y = 1 x = an y = (c) How many solutions oes the following triple of simultaneous linear equations possess? 7x + 2y 3z = 0 4x + 5y = 0 x + 8y + 3z = 0 None 1 3 Infinitely many

5 C Calculus 8. Evaluate the following expressions: (a) (b) (c) x (x3 ) x t x=1 (x n + 5x) t=π (sin t) 9. Evaluate the following expressions using the chain rule: (a) t t=0 (e 4t ) (b) (3 sin 2t + cos 4t) t t= π 2 (c) t( e 3t 2 2+cos(ht) ) 10. Evaluate the following expressions using the prouct rule: (a) t (t2 sin t) (b) t t= π (cos 3t sin t) (c) x=1 (e 3x5 ln x) x 11. Evaluate the following expressions using the quotient rule: ( (a) z ) z z= 1 z 3 +2 ( (b) sin θ ) θ θ=0 θ 2 +cos θ ( ) (c) tan t t=0 t ln (3t+2) 12. Evaluate the following expressions: (a) 1 0 x3 x (b) 2π sin(3t) + 2 cos t t π (c) 1 1 x2 sin 7x 5x cos 2x x

6 13. Evaluate the following expressions using integration by parts: (a) π 0 x cos(3x) x (b) 0 t 2 e 4t t 1 2 (c) 1 1 x5 sin(2x 3 ) x 14. Evaluate the following expressions using integration by substitution: (a) 0 π/9 sin( 3t + π 3 ) t (b) x (2x 5) 3 (c) π 2 sin x π 2 /4 x x 15. Evaluate the following expressions using partial fractions: (a) 5 4 (b) 4 2 (c) x 2x 2 5x 3 3x+7 x x 2 1 x 2 +2x 5 x x 3 x 2 5x 3

7 D Logic Consier the following statement: If it rains then I take an umbrella. 16. Which of these statements are equivalent to the original statement? I only take an umbrella if it is raining. It is raining an I o not have an umbrella. I take an umbrella whenever it is raining. If it is not raining then I o not take an umbrella. If I o not take an umbrella then it is not raining. 17. Which of these statements is the negation of the original statement? I only take an umbrella if it is raining. It is raining an I o not have an umbrella. I take an umbrella whenever it is raining. If it is not raining then I o not take an umbrella. If I o not take an umbrella then it is not raining. Consier the following statement: Every natural number (1, 2, 3,... ) is the square of a natural number. 18. Which of the following hol? The statement is true. The statement is false. Whether the statement is true or false epens on the number chosen. The statement oes not make logical sense.

8 Consier the following statement: There exists a natural number that is the square of every natural number. 19. Which of the following hol? The statement is true. The statement is false. Whether the statement is true or false epens on the number chosen. The statement oes not make logical sense. Consier the following statement: In every street in Exeter there is a house all of whose winows are broken. 20. Which of the following statements must be true if the statement above is false: There is a street in Exeter with no houses. There is at least one street in Exeter in which every house has all its winows intact. In no street in Exeter is there a house all of whose winows are broken. There is at least one street in Exeter all of whose houses have at least one unbroken winow. Exeter has only winowless houses. Submit Answers!