10 Non-routine Problems To Sharpen Your Mathematical Thinking

Transcription

1 Have you thought about it differently? 0 Non-routine Problems To Sharpen Your Mathematical Thinking Wee Wen Shih PGDE (Credit), MSc (Research) Page

2 Introduction This electronic book presents the second set of 0 non-routine problems across different mathematical themes to broaden the student s horizons on problem solving. Again, enjoy the process of learning how to crack these problems via unepected means! Wee Wen Shih, Singapore mathematics educator weews@yahoo.com, wenshih.wordpress.com September 07 Page

3 Note to the solver: Problems are printed on pages 3 through 8. Hints and Numerical Answers are provided separately, upon payment of purchase. Problem : The floor function of, denoted by, is the greatest integer less than or equal to. For eample,.7 = and 3.9 = 4. What is the eact value of 07 r=.5r 0.5? Problem : a Guess a function f ( ) that satisfies the functional equation f ( a) f ( b) = f, b and b are any positive real numbers. Give the domain of your choice of function. where a Problem 3: The solution set of the inequality of a and b. sin > cos is ( a b),. Find the eact values Problem 4: To prove a mathematical statement by contradiction, we assume that what we want to prove is not true, and then show that the consequence(s) of this is/are not possible. 5 Take, for eample, the statement: for all real > 0. 5 Assume that there is a positive real value of for which 9 + < 30. Multiplying both sides by, < 0 ( 3 5) < 0. This is impossible since ( 3 5) 0 for all real > 0. Hence must hold for all real > 0. Now prove the following statement by contradiction: + for all non-zero real. Page 3

4 Problem 5: In the first electronic book, we came across the hyperbolic functions e e sinh =. We now eplore the calculus of these functions. e + e cosh = and (i) By considering the derivatives with respect to of these functions, show that d ( sinh ) = cosh d, ( cosh ) = sinh, and d ( tanh ) =. d d d cosh (ii) Verify that ln y = is a solution of the differential equation cosh dy y + = sech. d coth e + e (iii) Let y =. By making the subject, show that cosh = ln +. Apply the same idea to find a logarithmic epression for the inverse hyperbolic sine function. (iv) Show that ( cosh ) d = and d sinh =, d a + a d where a is a constant. (v) Use integration by parts to prove that sinh. a + a d = + a + a + c Problem 6: Use coordinate geometry to prove that the three heights of a triangle meet at a common point as shown in the figure. Page 4

5 Problem 7: The inequality + t over the interval 0 < t, show that is valid for 0 < t. By integrating both sides of this inequality for 0 <. tan Problem 8: Linear momentum is a vector quantity that is a product of mass and velocity of an object travelling in a straight line. It is conserved in collisions between two or more objects. Consider two objects before they collide. The object with mass m has a momentum of mv while the object with mass m has a momentum of mv because it is travelling in the opposite direction. Taken together, the total mv + mv = mv. momentum of the system is ( ) After hitting each other, the objects travel in opposite directions as shown. (i) Calculate the total momentum of the system after collision and verify that the law of conservation of linear momentum holds. (ii) What would happen if the objects merge and move together? Show your working. Now apply the law of conservation of linear momentum for this situation: Three objects, P, Q and R, lie in a straight line. P, with mass m, is moving towards Q at velocity u. Q has mass m and is moving with velocity u towards R which is stationary. P 4 collides with Q, and P and Q then collide with R. If all three objects move together eventually with velocity, 5 u find the mass of R in terms of m and m. Page 5

6 Problem 9: A mathematical statement is a sentence that is either true or false. Often we encounter a mathematical statement, called a conditional statement, that is epressed in the form if A, then B, where A represents the assumption we make and B represents the conclusion. 5 In Problem 4, the statement for all real > 0 can be rephrased equivalently 5 as follows: if is real and positive, then State the assumption and conclusion for this statement. (i) Rewrite + for all non-zero real as a conditional statement. (ii) Rewrite Tourists at the Eiffel tower are in France as a conditional statement. An inverse statement is one in which we negate the assumption and conclusion of the original conditional statement. This statement has the form if not A, then not B. (iii) Epress the conditional statements (i) and (ii) as inverse statements. A converse statement is one in which we interchange the assumption and conclusion of the original conditional statement. (iv) Rewrite the conditional statements (i) and (ii) as converse statements. A contrapositive statement is formed by negating both assumption and conclusion of the original conditional statement and interchanging these negations. (v) Reformulate the conditional statements (i) and (ii) as contrapositive statements. To prove a mathematical statement if A, then B by contrapositive, we assume not B and then proceed to prove not A. (vi) Revisit Problem 4 and prove both statements by contrapositive. Page 6

7 Problem 0: Linear programming is a problem solving approach that is typically used in economics and business contets to look at the optimum allocation of limited resources among competing activities within a set of constraints imposed, such as financial, technological, marketing, organisational, or many other factors. Let us study an eample of a situation in which linear programming applies: A company makes two products (X and Y) with two machines (A and B). unit of X that is made requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. unit of Y that is made requires 4 minutes processing time on machine A and 33 minutes processing time on machine B. At the start of the current week there are 30 units of X and 90 units of Y in stock. The available processing times on machine A and on machine B are predicted to be 40 hours and 35 hours respectively. The demands for X and for Y in the current week are predicted to be 75 units and 95 units respectively. Let the number of units of X made in the current week be and the number of units of Y made in the current week be y. (i) Show how the above information can be modelled by the following inequalities (we call them constraints): 5 + y 00, 0 + y 700, 45, y 5. To solve any linear programming problem involving two variables by the graphical method, we need to sketch the feasible region corresponding to the system of constraints and then find the vertices of the region. (ii) Visit and use the linear programming grapher to sketch the feasible region to identify the three vertices. Refer to the following screenshots: Page 7

8 How may the vertices be obtained without the linear programming grapher? The company s manager wishes to maimise the combined sum of the units of X and the units of Y in stock at the end of the week. This sum is given by the epression + y 50, which we call the objective function. (iii) Eplain how this objective function is derived. (iv) With the vertices found in (ii), calculate the possible values of + y 50 and decide which pair of, y values would theoretically maimise the manager s objective function? (v) Comment, in this contet, on the practicality of the theoretical solution obtained. (vi) Now use the linear programming grapher to solve this problem: Maimise the objective function 3 + y subject to the constraints + y 30, + y 40, y 5, 4, y. (a) What is the solution if the objective is to be minimised instead? (b) What is the range of values satisfied by the objective function + 3 y? Page 8