Preview Mathematics Analysis and Approaches for the IB Diploma Higher Level Worked Solutions by Ibrahim Wazir, Tim Garry (2019) Preview Mathematics Analysis and Approaches for the IB Diploma Higher Level Worked Solutions by Ibrahim Wazir, Tim Garry (2019) Preview Mathematics Analysis and Approaches for the IB Diploma Higher Level Worked Solutions by Ibrahim Wazir, Tim Garry (2019) Preview Mathematics Analysis and Approaches for the IB Diploma Higher Level Worked Solutions by Ibrahim Wazir, Tim Garry (2019) Preview Mathematics Analysis and Approaches for the IB Diploma Higher Level Worked Solutions by Ibrahim Wazir, Tim Garry (2019)
WORKED SOLUTIONS Exercise 1.1 In each of the following, make the indicated letter the subject of the left side of the equality: (a) (b) (c) (d) (e) if r is a length, then f g h k fk gh k gh f (f) (g) (h) Factor out k at the denominator of the fraction: In parts (a) to (d), find m, the slope of the line, using , then find the y-intercept c, by choosing one of the given points and substituting its coordinates and the value of the slope into the slope-intercept form of the equation of a line, (a) (b) EITHER: OR: The given points have the same y-coordinate, so the line is horizontal, the equation of the line is © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (c) (d) By inspection, the given points have the same x-coordinate, so the line is vertical, and its equation is (e) Parallel lines have the same slope, consequently the slope of the required line is (f) The slope of a line perpendicular to another line with slope is In each of these, substitute the coordinates of the given points in the relevant formulae: (a) (b) (c) (d) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free Substitute the coordinates of the given points in the distance formula and set it equal to 5: (a) (b) Use the distance formula to show that properties regarding the length of sides for the indicated shapes hold true (a) Let , and The lengths of the sides are: be the vertices of triangle If the theorem , then is a right-angled triangle is a right-angled triangle (b) Let , and be the vertices of triangle is isosceles, because two of its sides have the same length (c) Let If , , and and , then be the vertices of quadrilateral is a parallelogram, is a parallelogram © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (a) Subtract the two equations to eliminate x: Substitute into either of the original equations to find x: The solution is (b) Multiply the second equation by and add it to the first equation to eliminate y: Substitute into either of the original equations to find x: The solution is (c) Multiply the first equation by and the second equation by 3, then subtract the two equations, to eliminate y: Substitute into either of the original equations to find x: The solution is (d) Multiply the second equation by and add it to the first equation to eliminate x: This is not true, so the system of equations has no solution © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (e) Multiply the first equation by and the second equation by 7, then add the two equations, to eliminate y: Substitute into either of the original equations to find y: The solution is (a) Make y the subject in the first equation: Substitute the expression of y in the second equation and solve for x: Substitute into either of the original equations to find y: The solution is (b) Make y the subject in the second equation: Substitute the expression of y in the first equation and solve for x: Substitute into the expression of y: The solution is (c) Divide the first equation by 2: Make x the subject in this equation: Substitute the expression of x in the second equation and solve for y: true for all This means that there are an infinite number of solutions, due to the fact that the two equations are multiples of each other (the lines representing the two equations are coincident) It follows that the solution of this system is the set of all points on the line with equation (or , or ) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (d) Make y the subject in the second equation: Substitute the expression of y in the first equation and solve for x: Substitute into the expression of y: The solution is (e) Make y the subject in the second equation: Substitute the expression of y in the first equation and solve for x: Substitute into the expression of y: The solution is (f) Multiply the second equation by 10: Make y the subject in this equation: Substitute the expression of y in the first equation and solve for x: Substitute into the expression of y: The solution is © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (a) The solution is (b) The solution is (c) The solution is (a) The solution is © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (b) The solution is (c) infinite solutions (d) The solution is © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free 10 (a) The solution is (b) The system of equations has no solution (c) The solution is (d) Infinite solutions © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (a) (b) kg percentage remaining % (c) (d) Half-life occurs at half initial amount kg From observation, approximately 20 days © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free Option A: increase of Option B: increase of Option C: increase of Option A provides the best increase on the initial investment (a) Given that (b) is the growth rate per minute % 10 (a) Gradient (b) Gradient (c) Gradient © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (d) Gradient 11 (a) After 10 years: (b) years (c) years (d) The answers are the same, as the time to double depends only on the interest rate and the amount of compounding periods, and these are constant between (b) and (c) Exercise 4.4 (a) (b) (c) (d) (e) (f) (g) (h) (i) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (a) (b) (c) (d) (e) (f) (g) (h) (i) (a) (b) (c) (d) (e) (f) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (q) (r) (s) (t) (a) (b) (c) (d) (e) For the function (a) , domain: is a translation of by two units to the right OR domain: (b) which is true for all real numbers not equal to zero domain: (c) is a translation of by two units downward (which has no effect on the domain) domain: (d) (e) Due to square root: Due to logarithm: © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (f) Due to square root: Due to logarithm: (a) Due to denominator: Due to logarithm: Function can never take the value of zero due to the numerator range: (b) Function can never take a negative value due to absolute value range: (c) Due to denominator: Due to logarithm: Function can never take the value of zero as a zero numerator is outside the domain range: All functions in this question take the form (a) From the graph: (b) From the graph: (c) From the graph: © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (d) From the graph: (a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f) 10 (a) (b) (c) (d) (e) (f) 11 (a) (b) (c) (d) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free 12 (a) (b) 13 14 15 For decibels 16 (a) (b) In the form © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free Exercise 4.5 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (l) (a) If Resubstituting gives Either (impossible) OR (b) If Resubstituting Either gives OR (c) If Resubstituting Either gives OR © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (d) If Resubstituting Either gives OR (a) (b) years It will take years to double the investment years Doubling every hour implies a population after hours hours for the population to exceed million bacteria (a) years or years years or years (b) (c) years or years years Thus, in years © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (a) (b) years (a) 37 dogs (b) years years 10 (a) (b) litres minutes and minutes seconds (c) minutes minutes 11 (a) (b) kilograms years 12 (a) (b) (c) (d) ; due to nature of the function © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free (e) (f) (g) (h) due to nature of the function (i) (j) due to nature of the function (k) By observation , OR © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free 13 (a) OR (b) (c) OR -value (d) (to s.f.) © Pearson Education Ltd 2019 Copying permitted for purchasing institution only This material is not copyright free ... applied by analysing the given equation, then transform the important points of the original graph: the two ends, and , the maximum point, , and the minimum point, , then plot and join these images... either of the original equations to find x: The solution is (c) Multiply the first equation by and the second equation by 3, then subtract the two equations, to eliminate y: Substitute into either... 11 The last equation is: In order for the system to have no solutions, the coefficient of z must be 0, and the right side of the equation should not be equal to When , the right-hand side of the