THE EXAMINATION PAPER

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 Spe[r]

TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn THE EXAMINATION PAPER EXCELLENT STUDENTS USE SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS IN APRIL, 2010 Chief of Organization : The Minh Tran Mathematics Specialist – VietnamCalculator Company To calculate the following value of expression : + + + + 7 + 8 + 9 + 10 10 To find the following remainder of expression : 30 x + x + 19 x + 75 x − 2010 3x − For the following expression : f ( x) = ( x + x + 2) 20 To write processes to press the keys to calculate the expression f (α ) with α =3 1+ 1− + 2 A student have made the deposit of VND 60,000,000 at the bank with interest rates is 0.65 % /month How much money has the student got in the account deposit after years ? To write the grand formula and to apply to calculate Using the programming on the VietnamCalculator Vn - 500RS & Vn - 570RS scientific calculator to calculate : + + 38 + + + + + + + 10 Good luck TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn 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 parts and calculate each one First, to calculate A = + 8 + 9 + 10 10 , we have the Pressing key process: x ( + x ( + x x ( + 10 10 ) ) ) = The calculator notified the result: 1.353494786 To continue, we have: 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: B := B − 1; A := B ( B + A) Continuous pressing key process: 10 SHIFT x 10 SHIFT STO A , 10 SHIFT STO B TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn ALPHA B ALPHA = ALPHA B − ALPHA : ALPHA A ALPHA = ALPHA B Press key “=” until we have result x ( ALPHA B + ALPHA A ) = B = B −1 Press key “=” one more time to get the TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn Problem 2: * Result: 26.71628647 * Detail of solution and Pressing key process: To find the following remainder of expression: 30 x + x + 19 x + 75 x − 2010 3x − 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: 19 x + x + 25 x + 10 x − 670 30 x + x + 19 x + 75 x5 − 2010 3 = 3x − x− 19 Therefore, we only want to calculate x9 + x + 25 x5 + 10 x − 670 at 3 * Use the function CALC of Vn - 570RS: ÷ × ALPHA X ^ + 19 ÷ × ALPHA X ^ + 25 × ALPHA X ^ + 10 × ALPHA X −670 CALC ÷ = We have 26,71628647 We also can put whole expression into calculator and get a similar one TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn Problem 3: * Result: 359426628, * Detail of solution and Pressing key process: - With Vn – 570RS, we assign α = 1+ 1− to variable A and use + 2 function CALC to calculate f (α ) : SHIFT ( (1+ ) ÷ ) + SHIFT ( (1− ) ) SHIFT STO A ( ALPHA A ^ + ALPHA A + ) ^ 20 = Because we rounded the expression two times, the result may be not correct So we should direct calculate after some changes: Easy to have:  1+ 1−  1− 1+ 1− 1+  = α3 =  +3 + + 33 α = − 3α  2  2 2   3 ⇒ α + 3α − = ⇒ α + 2α + = − α So f (α ) = (α + 2α + 2) 20 = (3 − α )20 = (3 − + − 20 − ) 2 Pressing key process: , (3 − ( SHIFT ( (1+ We have 359 426 628,4 ) ÷ ) + SHIFT ( (1− ) ) )) 20 TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn Problem : * Result: 88.507.100 VND * Detail of solution and Pressing key process: - Make the general function: Let U0 is the value of deposit at first (VND), a is the interest rates every month (%), n is amount of months (months), Un is the value of deposit after n months (VND): The interest rates per month is a% so after n month, the value of deposit a a n increase (100 + a )% or + So, after n months, we have: U n = U (1 + ) 100 100 (VND) - Apply to calculate: With U0 = 60 000 000, a = 0,65; n = 60 (5 years = 60 months), we have: 0, 65 60 60000000.(1 + ) ≈ 88507100 (VND) 100 Pressing key process: 60000000 × ( + 0.65 ÷ 100 ) ^ 60 = We have 88 507 078.17, round this result, we have the value of deposit 88 507 100 VND TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn 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 : + + 38 + + + + + + + 10 With Vn – 570RS: it’s easy to use variables to calculate the expression base on one of two recursive formula: u1 = 110 , un = un −1 + n11− n , ≤ n ≤ 10 or u1 = 110 + 101 , un = un −1 + (n11− n + (11 − n)n ), ≤ n ≤ - Assign to B - Assign to A and add all of results that we have when value of B is increases SHIFT STO A , SHIFT STO B ALPHA B ALPHA = ALPHA B + ALPHA : ALPHA A ALPHA = ALPHA A + ALPHA B ^ ( 11 − ( ALPHA B ) Press key “=” until we see B = B +1 10 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 to B - Assign to A and add all of results that we have when value of B is increases SHIFT STO A , SHIFT STO B ALPHA B ALPHA = ALPHA B + ALPHA : ALPHA A ALPHA = ALPHA A + ALPHA B ^ ( 11 − ( ALPHA B ) + ( 11 − ALPHA B ) ^ ALPHA B Press key “=” until we see B = B +1 Press key “=” one more time to get the result That’s 49 863 With VN – 500RS: we use key “REPLAY” to calculate: Way : We have algothihm: - Assign to A, assign to B - Assign A + to A, assign B + A(11 – A) to B TRẦN MINH THẾ - CHUYÊN VIÊN TOÁN HỌC SINH GIỎI MÁY TÍNH CẦM TAY USING SCIENTIFIC CALCULATORS TO SOLVE MATHEMATICAL PROBLEMS EXPERT - HO CHI MINH CITY Website : http://maths-minhthe.violet.vn - Press ∆ to return A +
A Press SHIFT ∆ to Copy express, we have: A +
A : B + A(11 – A)
B A +1 → A Press “=” until we see and press key “=” one more time to get the 10 result SHIFT STO A , SHIFT STO B ALPHA A + SHIFT STO A , ALPHA B + ALPHA A ^ (11 − ALPHA A ) SHIFT STO A , ∆ SHIFT ∆ = Way : We also have a difference algothihm but use combination key ∆ = : - Assign to A, assign to B - Assign B + A(11 – A) to B - Assign A + to A - Press combination key ∆ = several times until we see on the screen A +1 → A 10 , press ∆ = one more time to get the result THE END