SEMESTER 1 EXAMINATION 2014/2015 ADEDEX424. Access to Science - Mathematics 1. Dr. Anthony Cronin Dr. Anthony Brown. Time Allowed: 3 hours

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1 University College Dublin An Coláiste Ollscoile, Baile Átha Cliath SEMESTER 1 EXAMINATION 2014/2015 ADEDEX424 Access to Science - Mathematics 1 Dr. Anthony Cronin Dr. Anthony Brown Time Allowed: 3 hours Instructions for Candidates Candidates should attempt all questions. Note that not all questions are allocated the same number of marks. Notes for Invigilators Non programmable calculators are permitted. The formula sheet provided is permitted. c UCD 2014/15 1 of 8

2 1. (i) Without using a calculator, calculate the following. Note that you should show enough of your working to demonstrate that you have not simply entered the expression into a calculator. (a) (b) 8 ( 5 2 ) 7 (c) (d) 2 2 ( 3 (e) 4 (f) 3 8 ) 2 (g) ( ) (h) 8 (i) 5+8 ( 7) 4 (j) (3 2) (k) log 4 16 (l) log (m) log 4 2 [13] (ii) Simplify the following expressions by expressing them as a power of x, y and/or z, as appropriate. (a) x 6 x 3 (b) x 2 x 3 (c) x 2 x 1 3 (d) x 2 x 4 (e) x 1 5 x 3 4 (f) (x 3 ) 2 (g) ( x 4 y ) 4 (h) ( x 1 y 1 2z 2 3 ) 1 4 [8] (iii) (a) Approximate to one decimal place. (b) Approximate to four significant figures. (c) Approximate to one significant figure. (d) Express in scientific notation. (e) Express in scientific notation to two significant figures. [5] c UCD 2014/15 2 of 8

3 (iv) Simplify (x 4 2x 3 +2x 2) (x 5 2x 4 +3x 3 x 2 4). (v) Multiply out (x 3 2x)(2x 2 +3). (vi) Perform long division on , giving the quotient and remainder. [3] (vii) Perform long division on x2 x+2, giving the quotient and remainder. x+1 [4] 2 (viii) Evaluate i 4. i= 2 ( ) 9 (ix) Calculate without using a calculator. 3 Note that you should show enough of your working to demonstrate that you have not simply entered the expression into a calculator. (x) Expand (x 3y) 3 using The Binomial Theorem. [4] 2. (i) Find the equation of the line through the points ( 2,5) and (2,3). [3] (ii) Find the equation of the line through the point (2, 3) parallel to the line y = 3x+2. (iii) Solve the simultaneous equations 2x 3y = 7 3x 2y = 3 (iv) Find the length of the line segment between ( 1,2) and (2, 1). [1] (v) Find the midpoint of the line segment joining (2, 3) and (4,0). [1] [3] 3. (i) Write the expression 2x 2 x 10 in completed square form. [3] (ii) Solve the equation 2x 2 x 10 = 0 by using the completed square form you found in Part (i). [3] (iii) Confirm your answer to Part (ii) by solving the equation 2x 2 x 10 = 0 using the quadratic formula. (iv) Sketch the graph of the function y = 2x 2 x 10, showing the y-intercept, the x-intercept(s) (if applicable) and the turning point. [4] (v) If possible, factorize the expression 2x 2 x 10. c UCD 2014/15 3 of 8

4 4. (i) For each of the following: Say whether or not it is a function and if not say why not. If it is a function state the domain and the codomain. (a) f: R R x x+2 (b) f: R R + x x 3 (c) f: R R x x 2 +1 [6] (ii) Sketch the graphs of each of the following functions. (a) f: { 3, 2,0,2,3} {1,2,3} (b) f: {x R: 3 x 2} {x R: 10 x 10} x 2x 3 [4] c UCD 2014/15 4 of 8

5 (iii) Figure 1 contains the graphs of four of the following functions: (a) y = 7 ( x ) x 5 (b) y = 7 ( 3 (c) y = 5 (d) y = log 5 (x) ) x (e) y = log 1/4 (x) (f) y = 4 x Match the functions to the graphs. [4] Figure 1: The functions for Question 4 (iii). c UCD 2014/15 5 of 8

6 (iv) For each of the following functions, say whether they are injective, surjective or bijective. If a function is not injective or surjective then say why not. (a) (b) (c) f: {A,B,C,D} {1,2,3,4,5} A 1 B 3 C 2 D 4 f: {A,B,C,D} {1,2,3,4} A 4 B 3 C 1 D 2 f: R + R x x 3 (v) For any bijective functions you found in Part (iv), find the inverse function. 5. (i) Convert 345 to radians, leaving your answer as a multiple of π. [1] (ii) Convert 5π radians to degrees. [1] 12 ( ) 5π (iii) Using the geometric method, find sin without using a calculator. [3] 4 (iv) Using whichever trigonometric formulae you like, but without using a calculator, calculate the following. Note that you should show enough of your working to demonstrate that you have not simply entered the expression into a calculator. ( ) 2π (a) sin 3 ( ) 5π (b) cos 3 ( (c) tan π ) 12 [5] [6] c UCD 2014/15 6 of 8

7 (v) Find the length of the side a in the triangle in Figure 2. [3] Figure 2: The triangle for Question 5 (v). 6. (i) Find the derivative of f(x) = x 2 +1 using first principles. [3] (ii) Find the derivatives of the following functions. (a) f(x) = 7 (b) f(x) = e cos(2) (c) f(x) = x 4 (d) f(x) = ln(3x) (where x > 0) (e) f(x) = sin( 3x) (f) f(x) = 2 4x 2 +2x 4 5 (g) f(x) = 3cos(2x) 4sin( x) (h) f(x) = 3+4e 5x +4ln( 2x) (where x < 0) [11] 7. Find the following integrals. (i) 6dx. [1] (ii) (iii) (iv) (v) (vi) 2 1 π x 7 dx. cos(4x) dx. sin( 3x) dx. 2 3x 2 +2x 3 4 dx. 5 3e 5x dx. [1] [3] [3] c UCD 2014/15 7 of 8

8 8. (i) For the list of numbers 3,2, 8, 4,8,3,2,5, 7, 5, find the (a) Mean (b) Median (c) Mode(s) (d) Variance (e) Standard deviation (f) Interquartile range [8] (ii) Find the line of best fit using the least squares method with the points ( 4, 1),( 2,0),(0,0),(1,1),(2,2),(5,3) and (6,4). Plot the line of best fit and the points on a graph. [12] o0o c UCD 2014/15 8 of 8