THE SOLUTION OF THE EXAMINATION PAPER EXCELLENT STUDENTS USE SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS IN APRIL, 2010 Chairman of Organization : The Minh Tran Mathematics Specialist – VietnamCalculator Company Problem 1: *Result: 1.654364493 *Detail of solution and Pressing key process: - Way 1: As we know, if we put all of this expression into calculator, it’ll notify us a problem: “Stack ERROR”. So we should divide it into 2 parts and calculate each one. First, to calculate 7 8 9 10 7 8 9 10A = + + + , we have the Pressing key process: 7 ( 7 8 ( 8 9 ( 9 10 10 ) ) ) x x x x + + + = The calculator notified the result: 1.353494786. To continue, we have: 3 ( 3 4 ( 4 5 ( 5 6 ( 6 ) ) ) ) x x x x Ans+ + + + = The calculator notified the result: 1.654364493. That is the value of given - expression. - Way 2 : Base on the structure of this expression, we can create a “Continuous pressing key process”: - Assign 10 10 to variable A. - Assign 10 to variable B. - Do the process: : 1; : ( ) B B B A B A= − = + Continuous pressing key process: 10 10 , 10 x SHIFT SHIFT STO A SHIFT STO B 1 : ( ) x ALPHA B ALPHA ALPHA B ALPHA ALPHA A ALPHA ALPHA B ALPHA B ALPHA A = − = + = Press key “=” until we have 1 3 B B= − . Press key “=” one more time to get the result. Problem 2: * Result: 26.71628647 * Detail of solution and Pressing key process: To find the following remainder of expression: 53 20107519430 579 − −+++ x xxxx Base on one Theorem about Algebra, the remainder of division polynomial P(x) and binomial x – a exactly is P(a), we can solve this problem easier than calculate it with the normal way. Other hand, we have: 9 7 5 9 7 5 4 19 25 10 670 30 4 19 75 2010 3 3 5 3 5 3 x x x x x x x x x x + + + − + + + − = − − . Therefore, we only want calculate 9 7 5 4 19 25 10 670 3 3 x x x x+ + + − at 5 3 . * Use the function CALC of Vn - 570RS: 4 3 ^ 9 19 3 ^ 7 25 ^ 5 10 670 5 3 ALPHA X ALPHA X ALPHA X ALPHA X CALC ÷ × + ÷ × + × + × − ÷ = We have 26,71628647. We also can put whole expression into calculator and get a similar one. Problem 3: * Result: 359426628, 4 * Detail of solution and Pressing key process: - With Vn – 570RS, we assign 33 2 51 2 51 − + + = α to variable A and use function CALC to calculate )( α f : 3 3 ( ( 1 5 ) 2 ) ( ( 1 5 ) ) ( ^ 3 2 2 ) ^ 20 SHIFT SHIFT SHIFT STO A ALPHA A ALPHA A + ÷ + − + + = Because we rounded the expression two times, the result may be not correct. So we should direct calculate after some changes: Easy to have: 3 3 3 3 3 3 3 1 5 1 5 1 5 1 5 1 5 1 5 3 . . 1 3 2 2 2 2 2 2 3 1 0 2 2 3 α α α α α α α α + − − + − + ÷ = + = + + = − ÷ ⇒ + − = ⇒ + + = − So 3 20 20 20 3 3 1 5 1 5 ( ) ( 2 2) (3 ) (3 ) 2 2 f α α α α + − = + + = − = − − . Pressing key process: , 20 3 3 (3 ( ( ( 1 5 ) 2 ) ( ( 1 5 ) ) ))SHIFT SHIFT− + ÷ + − . We have 359 426 628,4. Problem 4 : * Result: 88.507.100 VND * Detail of solution and Pressing key process: - Make the general function : Let U 0 is the value of deposit at first (VND), a is the interest rates every month (%), n is amount of months (months), U n is the value of deposit after n months (VND): The interest rates per month is a% so after n month, the value of deposit increase (100 )%a+ or 1 100 a + . So, after n months, we have: 0 .(1 ) 100 n n a U U= + (VND). - Apply to calculate : With U 0 = 60 000 000, a = 0,65; n = 60 (5 years = 60 months), we have: 60 0,65 60000000.(1 ) 88507100 100 + ≈ (VND). Pressing key process: 60000000 ( 1 0.65 100 ) ^ 60× + ÷ = . We have 88 507 078.17, round this result, we have the value of deposit 88 507 100 VND. Problem 5: * Result: 49863 * Detail of solution and Pressing key process: Using the programming on the VietnamCalculator Vn - 500RS & Vn - 570RS scientific calculator to calculate : 10987654321 23456789 +++++++++ . With Vn – 570RS: it’s easy to use variables to calculate the expression base on one of two recursive formula: 10 11 1 1 1 , , 2 10 n n n u u u n n − − = = + ≤ ≤ or 10 1 11 1 1 1 10 , ( (11 ) ), 2 5 n n n n u u u n n n − − = + = + + − ≤ ≤ . - Assign 0 to B. - Assign 0 to A and add all of results that we have when value of B is increases. 0 , 0 1 : ^ ( 11 ( ) SHIFT STO A SHIFT STO B ALPHA B ALPHA ALPHA B ALPHA ALPHA A ALPHA ALPHA A ALPHA B ALPHA B = + = + − Press key “=” until we see 1 10 B B= + . Press key “=” one more time to get the result. That’s 49 863. We also can cut down that Pressing key process by using the second fomula: - Assign 0 to B. - Assign 0 to A and add all of results that we have when value of B is increases. 0 , 0 1 : ^ ( 11 ( ) ( 11 ) ^ SHIFT STO A SHIFT STO B ALPHA B ALPHA ALPHA B ALPHA ALPHA A ALPHA ALPHA A ALPHA B ALPHA B ALPHA B ALPHA B = + = + − + − Press key “=” until we see 1 5 B B= + . Press key “=” one more time to get the result. That’s 49 863. With VN – 500RS: we use key “REPLAY” to calculate: Way 1 : We have algothihm: - Assign 0 to A, assign 0 to B. - Assign A + 1 to A, assign B + A (11 – A) to B. - Press ∆ to return A + 1 A. - Press SHIFT ∆ to Copy express, we have: A + 1 A : B + A (11 – A) B Press “=” until we see 1 10 A A+ → and press key “=” one more time to get the result. 0 , 0 1 , ^ (11 ) , SHIFT STO A SHIFT STO B ALPHA A SHIFT STO A ALPHA B ALPHA A ALPHA A SHIFT STO A SHIFT + + − ∆ ∆ = Way 2 : We also have a difference algothihm but use combination key ∆ = : - Assign 1 to A, assign 0 to B. - Assign B + A (11 – A) to B. - Assign A + 1 to A. - Press combination key ∆ = several times until we see on the screen 1 10 A A+ → , press ∆ = one more time to get the result. . THE SOLUTION OF THE EXAMINATION PAPER EXCELLENT STUDENTS USE SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS IN APRIL, 2010 Chairman of Organization : The Minh Tran Mathematics. get the result. Problem 2: * Result: 26.71628647 * Detail of solution and Pressing key process: To find the following remainder of expression: 53 20107 519430 579 − −+++ x xxxx Base on one Theorem. + + + = The calculator notified the result: 1.654364493. That is the value of given - expression. - Way 2 : Base on the structure of this expression, we can create a “Continuous pressing key