Algebra II CIPHERING 2014

Transcription

1 PLACE ALL ANSWERS IN THE APPROPRIATE BOX. Only answers within the appropriate box will be considered for scoring. All answers should be exact and in simplest form Include units where appropriate No calculators are allowed DO NOT TURN THE PAGE UNTIL TOLD TO DO SO. YOU MAY NOT RETURN TO A PAGE ONCE TIME IS CALLED. NAME SCHOOL

2 1.Themeanoffivenumbersis76,themedianis 75,themodeis81,andtherangeis11.Compute thevalueofthesecondsmallestnumber. 2.Simplifythefollowingexpressionintoone containingnotrigonometricfunctionsandinsimplest form. sin 15 + cos (15 ) sin 75 + cos (75 ) 3.Giventhesystemofequationsbelow,computethe orderedtriple(x,y,z). x+2y=5 2x+z=8 y+2z=2 4.HelenworksattwicetherateofJim.Sigmund worksattwicetherateofhelen.ifworkingtogether, Helen,JimandSigmundcandoajobinonehour, computethenumberofhoursitwouldtakejimtodo thejobifheworkedalone.

3 5.Computehowmanyanglesx,measuredinradians, satisfysin(x)=0and0 2006? 6. IfIf(log 9)( log 4) log 25 ) =,and is an integer, compute the value of x. 7. Thecostofanitemisreducedby20%.Thisnewcostis thenincreasedby20%tomakethefinalcost$60. Computethenumberofdollarsintheoriginalcostofthe item. 8.Compute 5 1 ) )

4 9.If =2 x,computex. 10.Computethesumofthefirst200positiveeven integers. 11.Threefairdicearethrown.Giventhatthesum oftheirfacesis8,computetheprobabilitythatthe threefacesaredifferentnumbers. 12.If + = 5, compute +.

5 13.Computeallvaluesofxsuchthat 6 = If =x h,computethevalueofh Computex,iflog + 2 = log (2 1). 16.Thepoint(k,0)isequidistantfromtheoriginand thepoint(2,6).computethevalueofk.

6 17.Computethenumberofpossiblewaystoplace6 identicalstonesona6 6chessboardsothatno twolieinthesameroworcolumn. 19.Computeallvaluesofxsuchthat =3. 18.Ifx= 3,computethevalueoftheexpression: (x+1)(x 2 +1)(x 2 +x+1)(xw1)(x 4 x 2 +1)(x 2 Wx+1). 20.Computetheareaofthesmallestcirclepassing throughthepoints(2,7)and(8,w1)?

7 Answers to Algebra 2 Ciphering 2014 Questions 1. Based on the range, median, and mode we know the numbers are 70, N, 75, 81, 81. Now we use the mean to find the second number is Because sin(15) = cos (75) and sin (75) = cos (15).. we can conclude that the answer is Adding the equations, we obtain 3x + 3y + 3z = 15 or x + y + z = 5. We can substitute the original expressions into the latter expression and obtain (5 2y) + y + (1 -.5y) = 5. Solving this equation, we get y = 2/3. Using the first equation we get x = 11/3. Finally, z = 2/3. 4. Let r be Jim s rate, 2r be Helen s rate, and 4r be Sigmund s rate. Working together, their rate is 7r. There for r = 1/7. Working alone, it takes Jim 7 hours. 5. Sin(x) = 0 for all x = " where n is an interger. So " "", meaning ". Because n can also be equal to zero the final answer is 639 values Using the change of base formula, we can rewrite the expression as " " " " " " " = = =. " " " " " " Let x be the original cost in dollars. Then = ",so x = $ Let x =.Then the given expression equals (x 1) (x2 + x + 1) = x3 1 = 5 1 = *+*212**+**212**+*212***=*4* 212**=**214***x*=*14.* 10. The sum is double the sum of the first 200 integers. ("" ") = ", "". 11. The ways that 8 can be the sum: (1, 1, 6) (3 ways), (1, 3, 4) (6 ways), (1, 2, 5) (6 ways) (2, 2, 4) (3 ways), (2, 3, 3) (3 ways). Probability the faces are different = " " = = + + ( + ) ( + = " + ) = 23 = ". So, = " If x = 6, then both sides are = 0, if x 6 then x = = " + = 110. = Let the common value of two logarithms be called M. Then the equations may be rewritten as 4m = x + 2, 2m = 2x -1. Since 4m is the square of 2m, we see that x + 2 = (2x -1)2 = 4x2 4x -1. The solution to this is x = 15. We can begin by rewriting 3 " =xhas. =.**Work*it*out*to* " find*that*h=**". 16. Using the distance formula, we know that k = + ". Solving for k, yields k = 10. ( ) + ( ) " = " +

8 17. 6 = ". In the first row the stone has 6 options( columns) it can be place in; the second row has 5columns it can be placed in; the third row has 4, etc, etc. 18. Answer = 8. Pairing the first term with the sixth, the second with the fifth and the third with the fourth gives (x 3 + 1)(x 6 + 1)(x 3 1). Pairing the first and last term in this new product and making the given substitution for x gives the answer of (3 + 1)(3 1) = First solve it straightforward with out the absolute value bars to find that x =. Now solve two equations -2x ± (x + 2) = 3. This leads to x = -1 and x = -. Only x = -1 checks with the original equation so the final two answers are x = & x = In order to minimize the area (which is " ),we need to minimize the radius. We can accomplish this if the two points on the circle are endpoints of a diameter. Therefore, " = ( ) + ( ) = " + " = ". Area = 25.