fx-95MS fx-100MS fx-115MS (fx-912MS) fx-570MS fx-991MS

User's Guide

kfx-912MS owners
Please be sure to read the important notice of this manual.
http://world.casio.com/edu_e/
CASIO ELECTRONICS CO., LTD. Unit 6, 1000 North Circular Road, London NW2 7JD, U.K.
Removing and Replacing the Calculators Cover
Before starting..1 Holding the cover as shown in the illustration, slide the unit out of the cover. After you are finished..2 Holding the cover as shown in the illustration, slide the unit out of the cover. Always slide the keyboard end of the unit into the cover first. Never slide the display end of the unit into the cover.

Safety Precautions

Be sure to read the following safety precautions before using this calculator. Keep this manual handy for later reference.

Caution

This symbol is used to indicate information that can result in personal injury or material damage if ignored.

Batteries

After removing the battery from the calculator, put it in a safe place where it will not get into the hands of small children and accidentally swallowed. Keep batteries out of the reach of small children. If accidentally swallowed, consult with a physician immediately. Never charge batteries, try to take batteries apart, or allow batteries to become shorted. Never expose batteries to direct heat or dispose of them by incineration. Misuse of batteries can cause them to leak and damage nearby items, and can create the risk of fire and personal injury. Always make sure that the batterys positive k and negative l ends are facing correctly when you load it into the calculator. Remove the battery if you do not plan to use the calculator for a long time (fx-95MS/100MS/570MS). Use only the type of battery specified for this calculator in this manual.
Disposing of the Calculator
Never dispose of the calculator by burning it. Doing so can cause certain components to suddenly burst, creating the risk of fire and personal injury. The displays and illustrations (such as key markings) shown in this Users Guide are for illustrative purposes 2
only, and may differ somewhat from the actual items they represent. The contents of this manual are subject to change without notice. In no event shall CASIO Computer Co., Ltd. be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the purchase or use of these materials. Moreover, CASIO Computer Co., Ltd. shall not be liable for any claim of any kind whatsoever against the use of these materials by any other party.

Handling Precautions

Be sure to press the 5 key before using the calculator for the first time. Even if the calculator is operating normally, replace the battery at least once every three years for the fx115MS/fx-570MS/fx-991MS, or at least once every two years for the fx-95MS/fx-100MS. A dead battery can leak, causing damage to and malfunction of the calculator. Never leave a dead battery in the calculator. The battery that comes with this unit discharges slightly during shipment and storage. Because of this, it may require replacement sooner than the normal expected battery life. Low battery power can cause memory contents to become corrupted or lost completely. Always keep written records of all important data. Avoid use and storage in areas subjected to temperature extremes. Very low temperatures can cause slow display response, total failure of the display, and shortening of battery life. Also avoid leaving the calculator in direct sunlight, near a window, near a heater or anywhere else it might be exposed to very high temperatures. Heat can cause discoloration or deformation of the calculators case, and damage to internal circuitry. 3
Avoid use and storage in areas subjected to large amounts of humidity and dust. Take care never to leave the calculator where it might be splashed by water or exposed to large amounts of humidity or dust. Such conditions can damage internal circuitry. Never drop the calculator or otherwise subject it to strong impact. Never twist or bend the calculator. Avoid carrying the calculator in the pocket of your trousers or other tight-fitting clothing where it might be subjected to twisting or bending. Never try to take the calculator apart. Never press the keys of the calculator with a ballpoint pen or other pointed object. Use a soft, dry cloth to clean the exterior of the calculator. If the calculator becomes very dirty, wipe it off with a cloth moistened in a weak solution of water and a mild neutral household detergent. Wring out all excess moisture before wiping the calculator. Never use thinner, benzene or other volatile agents to clean the calculator. Doing so can remove printed markings and can damage the case.

Contents

Removing and Replacing the Calculators Cover.. 1 Safety Precautions.. 2 Handling Precautions.. 3 Two-line Display.. 7 Before getting started.. 7
kModes... 7 kInput Capacity... 8 kMaking Corrections During Input.. 9 kReplay Function... 9 kError Locator... 9 kMulti-statements.. 10 kExponential Display Formats. 10 kDecimal Point and Separator Symbols. 11 k Initializing the Calculator.. 11

Basic Calculations.. 11

k Arithmetic Calculations.. kFraction Operations.. kPercentage Calculations.. kDegrees, Minutes, Seconds Calculations. kFIX, SCI, RND... kAnswer Memory... kConsecutive Calculations. kIndependent Memory.. k Variables.. 17

Memory Calculations.. 16

Scientific Function Calculations.. 18
kTrigonometric/Inverse Trigonometric Functions. 18 kHyperbolic/Inverse Hyperbolic Functions. 19
kCommon and Natural Logarithms/ Antilogarithms.. 19 kSquare Roots, Cube Roots, Roots, Squares, Cubes, Reciprocals, Factorials, Random Numbers, , and Permutation/Combination. 20 kAngle Unit Conversion... 21 kCoordinate Conversion (Pol (x, y), Rec (r, )).. 21 kEngineering Notation Calculations.. 22
Equation Calculations. 22
kQuadratic and Cubic Equations.. 22 kSimultaneous Equations.. 24
Statistical Calculations.. 26
Standard Deviation.. 26 Regression Calculations.. 28
Technical Information.. 32
kWhen you have a problem.. kError Messages... kOrder of Operations.. kStacks... kInput Ranges.. 36
Power Supply (fx-95MS only).. 38 Specifications (fx-95MS only). 39

Two-line Display

The two-line display makes it possible to view both the calculation formula and its result at the same time.
The upper line shows the calculation formula. The lower line shows the result. A separator symbol is displayed every three digits when the integer part of the mantissa has more than three digits.

Before getting started.

k Modes
Before starting a calculation, you must first enter the correct mode as indicated in the table below. The following table shows the modes and required operations for the fx-95MS only. Users of other models should refer to the Users Guide 2 (Additional Functions) for information about modes and their selection.

fx-95MS Modes

To perform this type of calculation: Basic arithmetic calculations Standard deviation Regression calculations Solution of equations 7 Perform this To enter key operation: this mode:

F1 F2 F3 FF1

COMP SD REG EQN
Pressing the F key more than twice displays additional setup screens. Setup screens are described in the sections of this manual where they are actually used to change the calculator setup. In this manual, the name of the mode you need to enter in order to perform the calculations being described is indicated in the main title of each section.

Example:

Equation Calculations
To return the calculation mode and setup to the initial defaults shown below, press A B 2(Mode) =. Calculation Mode: Angle Unit: Exponential Display Format: Fraction Display Format: Decimal Point Character: COMP Deg Norm 1 a b/c Dot
Mode indicators appear in the upper part of the display. Be sure to check the current calculation mode (SD, REG, COMP) and angle unit setting (Deg, Rad, Gra) before beginning a calculation.

k Input Capacity

The memory area used for calculation input can hold 79 steps. One step is taken up each time you press a number key or arithmetic operator key ( +, ,, -, \ ). A A or p key operation does not take up a step, so inputting A D, for example, takes up only one step. You can input up to 79 steps for a single calculation. Whenever you input the 73rd step of any calculation, the cursor changes from _ to k to let you know memory is running low. If you need to input more than 79 steps, you should divide your calculation into two or more parts. Pressing the g key recalls the last result obtained, which you can use in a subsequent calculation. See Answer Memory for more information about using the g key.
k Making Corrections During Input
Use e and r to move the cursor to the location you want. Press [ to delete the number or function at the current cursor position. Press A K to change to an insert cursor t. Inputting something while the insert cursor is on the display inserts the input at the insert cursor position. Pressing A K, or = returns to the normal cursor from the insert cursor.

k Replay Function

Every time you perform a calculation, the replay function stores the calculation formula and its result in replay memory. Pressing the [ key displays the formula and result of the calculation you last performed. Pressing [ again back steps sequentially (new-to-old) through past calculations. Pressing the e or r key while a replay memory calculation is on the display changes to the editing screen. Pressing the e or r key immediately after you finish a calculation displays the editing screen for that calculation. Pressing t does not clear replay memory, so you can recall the last calculation even after you press t. Replay memory capacity is 128 bytes for storage of both expressions and results. Replay memory is cleared by any of the following actions. When you press the 5 key When you initialize modes and settings by pressing A B 2 (or 3) =. When you change from one calculation mode to another When you turn off the calculator.

k Error Locator

Pressing r or e after an error occurs displays the calculation with the cursor positioned at the location where the error occurred. 9

k Multi-statements

A multi-statement is an expression that is made up of two or more smaller expressions, which are joined using a colon ( : ). Example: To add 2 + 3 and then multiply the result by 4 2+3p\g-4=
k Exponential Display Formats
This calculator can display up to 10 digits. Larger values are automatically displayed using exponential notation. In the case of decimal values, you can select between two formats that determine at what point exponential notation is used. To change the exponential display format, press the F key a number of times until you reach the exponential display format setup screen shown below.

F i x Sc i No r m

Press 3. On the format selection screen that appears, press 1 to select Norm 1 or 2 for Norm 2.

u Norm 1

With Norm 1, exponential notation is automatically used for integer values with more than 10 digits and decimal values with more than two decimal places.

u Norm 2

With Norm 2, exponential notation is automatically used for integer values with more than 10 digits and decimal values with more than nine decimal places. All of the examples in this manual show calculation results using the Norm 1 format.
k Decimal Point and Separator Symbols
You can use the display setup (Disp) screen to specify the symbols you want for the decimal point and 3-digit separator. To change the decimal point and separator symbol setting, press the F key a number of times until you reach the setup screen shown below.
Display the selection screen. fx-95MS: 1r Other Models: 1 r r Press the number key ( 1 or 2) that corresponds to the setting you want to use.
1(Dot): Period decimal point, comma separator 2(Comma): Comma decimal point, period separator
k Initializing the Calculator
Perform the following key operation when you want to initialize the calculation mode and setup, and clear replay memory and variables.

A B 3(All) =

Basic Calculations
k Arithmetic Calculations
Use the F key to enter the COMP Mode when you want to perform basic calculations. COMP... F 1 Negative values inside of calculations must be enclosed within parentheses. sin 1.23 S R D 1.23 T 11
It is not necessary to enclose a negative exponent within parentheses. sin 2.S 2.34 e D 5 Example 1: 3(5109) 1.5108 3-5eD9= Example 2: 5(97) 80 5-R9+7T= You can skip all T operations before =.

k Fraction Operations

uFraction Calculations
Values are displayed in decimal format automatically whenever the total number of digits of a fractional value (integer + numerator + denominator + separator marks) exceeds 10. Example 1:

2C3+1C5=

13 15.00

Example 2: 3

3C1C4+ 1C2C3=

2C4= 1 C 2 + 1.6 =

Example 3: Example 4:

1 1.6 2.1 2

Results of calculations that mix fraction and decimal values are always decimal.
uDecimal Fraction Conversion
Use the operation shown below to convert calculation results between decimal values and fraction values. Note that conversion can take as long as two seconds to perform. 12

Example 1: 2.75 2

3 (Decimal Fraction) 4 2.75 = 2.75

Example 2: 11 4

4.4.00
1 0.5 (Fraction Decimal) 2 1C2=

1 2.00 0.2.00

uMixed Fraction Improper Fraction Conversion

Example: 3 1C2C3=

3.3.2 3.00
You can use the display setup (Disp) screen to specify the display format when a fraction calculation result is greater than one. To change the fraction display format, press the q key a number of times until you reach the setup screen shown below.
Display the selection screen. fx-95MS: 1 Other Models: 1 r Press the number key ( 1 or 2) that corresponds to the setting you want to use.
1(a b/c): Mixed fraction 2(d/c): Improper fraction
An error occurs if you try to input a mixed fraction while the d/c display format is selected.
k Percentage Calculations
Example 1 : To calculate 12% of 1500 (180) 1500 - 12 A v Example 2: To calculate what percentage of 880 is 660 (75%) 660 \ 880 A v Example 3 : To add 15% onto 2500 (2875) 2500 - 15 A v + Example 4 : To discount 3500 by 25% (2625) 3500 - 25 A v , Example 5: To discount the sum of 168, 98, and 734 by 20% (800) 168 + 98 + 734 = g A j 1 p 1 - 20 A v , * As shown here, if you want to use the current Answer Memory value in a mark up or discount calculation, you need to assign the Answer Memory value into a variable and then use the variable in the mark up/discount calculation. This is because the calculation performed when v is pressed stores a result to Answer Memory before the , key is pressed. Example 6: If 300 grams are added to a test sample originally weighing 500 grams, what is the percentage increase in weight? (160%) 300 + 500 A v Example 7: What is the percentage change when a value is increased from 40 to 46? How about to 48? (15%, 20%) 46 , 40 A v eeeeee8= 14
k Degrees, Minutes, Seconds Calculations
You can perform sexagesimal calculations using degrees (hours), minutes, and seconds, and convert between sexagesimal and decimal values. Example 1: To convert the decimal value 2.258 to a sexagesimal value and then back to a decimal value 2.258 =

2.258 21528.8 2.258

Example 2: To perform the following calculation: 123456 3.I 34 I 56 I - 3.45 =

432431.2

k FIX, SCI, RND
To change the settings for the number of decimal places, the number of significant digits, or the exponential display format, press the F key a number of times until you reach the setup screen shown below.
Press the number key ( 1, 2, or 3) that corresponds to the setup item you want to change.
1 (Fix): Number of decimal places 2 (Sci): Number of significant digits 3 (Norm): Exponential display format

Example 1: \ 7 - 14 =

(Specifies three decimal places.)

400.00

F.. 1(Fix) 3

400.00000

(Internal calculation continues using 12 digits.)

200 \ 7 =

28.57100 400.00000

- 14 =

The following performs the same calculation using the specified number of decimal places. 200 \ 7 =

(Internal rounding)

28.57100 28.57100 399.99400

AQ - 14 =

Press F.. 3(Norm) 1 to clear the Fix specification. Example 2: 1 3, displaying result with two significant digits (Sci 2)

F.. 2(Sci) \ 3 =

Press F.. 3(Norm) 1 to clear the Sci specification.

Memory Calculations

Use the F key to enter the COMP Mode when you want to perform a calculation using memory. COMP... F 1

k Answer Memory

Whenever you press = after inputting values or an expression, the calculated result automatically updates Answer Memory contents by storing the result. In addition to =, Answer Memory contents are also updated with result whenever you press A v, |, A {, or A j followed by a letter (A through F, or M, X, or Y). 16
You can recall Answer Memory contents by pressing g. Answer Memory can store up to 12 digits for the mantissa and two digits for the exponent. Answer Memory contents are not updated if the operation performed by any of the above key operations results in an error.
k Consecutive Calculations

k Engineering Notation Calculations
Example 1: To convert 56,088 meters to kilometers 56.3 (km) 81.3 (mg) 56088 = J
Example 2: To convert 0.08125 grams to milligrams 0.08125 = J
The EQN Mode lets you solve equations up to three degrees and simultaneous linear equations with up to three unknowns. Use the F key to enter the EQN Mode when you want to solve an equation. EQN. F F 1 (fx-95MS) F F F 1 (Other Models)
k Quadratic and Cubic Equations
Quadratic Equation: ax2 + bx + c = 0 Cubic Equation: ax3 + bx2 + cx + d = 0 Entering the EQN Mode and pressing r displays the initial quadratic/cubic equation screen.

Deg r ee?

Use this screen to specify 2 (quadratic) or 3 (cubic) as the degree of the equation, and input values for each of the coefficients.
Coefficient name Arrow indicates direction you should scroll to view other elements.

Element value

Any time until you input a value for the final coefficient (c for a quadratic equation, d for a cubic equation), you can use the [ and ] keys to move between coefficients on the screen and make changes, if you want. Note that you cannot input complex numbers for coefficients. Calculation starts and one of the solutions appears as soon as you input a value for the final coefficient.
Variable name Arrow indicates direction you should scroll to view other solutions.

Solution

Press the ] key to view other solutions. Use [ and ] to scroll between all of the solutions for the equation. Pressing the t key at this point returns to the coefficient input screen. Certain coefficients can cause calculation to take more time. Example 1: To solve the equation x3 2x2 x + 2 = 0 ( x = 2, 1, 1) (Degree?) 3 (a?) (b?) (c?) (d?) (x1 = 2) (x2 = 1) (x3 = 1) If a result is a complex number, the real part of the first solution appears first. This is indicated by the RI symbol on the display. Press A r to toggle the display between the real part and imaginary part of a solution. 23 1= D2= D1= 2= ] ]

0.75 i

Example 2: To solve the equation 8x2 4x + 5 = 0 ( x = 0.25 0.75 i) (Degree?) (a?) (b?) (c?) (x1 = 0.25 + 0.75i) (x2 = 0.25 0.75i) 2 8= D4= 5= ]

k Simultaneous Equations

Simultaneous Linear Equations with Two Unknowns: a1x + b1y = c1 a2x + b2y = c2 Simultaneous Linear Equations with Three Unknowns: a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3 Entering the EQN Mode displays the initial simultaneous equation screen.

Coefficient name

Unk now ns?
Use this screen to specify 2 or 3 as the number of unknowns, and input values for each of the coefficients.
Arrow indicates direction you should scroll to view other elements.
Any time until you input a value for the final coefficient (c2 for two unknowns, d3 for three unknowns), you can use the [ and ] keys to move between coefficients on the screen and make changes, if you want. Note that you cannot input complex numbers for coefficients. Calculation starts and one of the solutions appears as soon as you input a value for the final coefficient.
Press the ] key to view other solutions. Use [ and ] to scroll between all of the solutions for the equation. Pressing the t key at this point returns to the coefficient input screen. Example: To solve the following simultaneous equations 2x + 3y z = 15 3x 2y + 2z = 4 5x + 3y 4z = 9 ( x = 2, y = 5, z = 4) (Unknowns?) (a1?). (d1?) (a2?). (d2?) (a3?). (d3?) (x = 2) (y = 5) (z = 4) = 3 = D1 = 15 = 3=D2=2=4= 5=3=D4=9=

Statistical Calculations

Standard Deviation

SD REG SD

Use the F key to enter the SD Mode when you want to perform statistical calculations using standard deviation. SD. F 2 (fx-95MS) F F 1 (Other Models) Always start data input with A B 1 (Scl) = to clear statistical memory. Input data using the key sequence shown below. <x-data> S Input data is used to calculate values for n, x, x2, o, n and n-1, which you can recall using the key operations noted nearby.
To recall this type of value: Perform this key operation:

x2 x n o n n-1

AU1 AU2 AU3 AX1 AX2 AX3
Example: To calculate n1, n, o, n, x, and x2 for the following data : 55, 54, 51, 55, 53, 53, 54, 52 In the SD Mode: A B 1 (Scl) = (Stat clear) 55 S
Each time you press S to register your input, the number of data input up to that point is indicated on the display (n value).
54 S 51 S 55 S 53 S S 54 S 52 S
Sample Standard Deviation (n1) = 1.407885953
Population Standard Deviation (n) = 1.316956719 Arithmetic Mean (o) = 53.375 Number of Data (n) = 8 Sum of Values (x) = 427 Sum of Squares of Values (x 2 ) = 22805

AX2= AX1= AU3= AU2= AU1=

Data Input Precautions S S inputs the same data twice.
You can also input multiple entries of the same data using A G. To input the data 110 ten times, for example, press 110 A G 10 S. You can perform the above key operations in any order, and not necessarily that shown above. While inputting data or after inputting data is complete, you can use the [ and ] keys to scroll through data you have input. If you input multiple entries of the same data using A G to specify the data frequency (number of data items) as described above, scrolling through data shows both the data item and a separate screen for the data frequency (Freq). You can then edit the displayed data, if you want. Input the new value and then press the = key to replace the old value with the new one. This also means that if you want to perform some other operation (calculation, recall of statistical calculation results, etc.), you should always press the t key first to exit data display. Pressing the S key instead of = after changing a value on the display registers the value you input as a new data item, and leaves the old value as it is. You can delete a data value displayed using [ and ] by pressing A U. Deleting a data value causes all values following it to be shifted up. Data values you register are normally stored in calculator memory. The message Data Full appears and you will not be able to input any more data if there is no memory left for data storage. If this happens, press the = key to display the screen shown below.

Technical Information

k When you have a problem.
If calculation results are not what you expect or if an error occurs, perform the following steps. 1. Press A B 2(Mode) = to initialize all modes and settings. 2. Check the formula you are working with to confirm it is correct. 3. Enter the correct mode and try performing the calculation again. If the above steps do not correct the problem, press the 5 key. The calculator performs a self-check operation and deletes all data stored in memory if any abnormality is detected. Make sure you always keep written copies of all important data.

k Error Messages

The calculator is locked up while an error message is on 32
the display. Press t to clear the error, or press e or r to display the calculation and correct the problem. See Error Locator for details.

Math ERROR

Causes Calculation result is outside the allowable calculation range. An attempt to perform a function calculation using a value that exceeds the allowable input range. An attempt to perform an illogical operation (division by zero, etc.) Action Check your input values and make sure they are all within the allowable ranges. Pay special attention to values in any memory areas you are using.

Stack ERROR

Cause The capacity of the numeric stack or operator stack is exceeded. Action Simplify the calculation. The numeric stack has 10 levels and the operator stack has 24 levels. Divide your calculation into two or more separate parts.

Syntax ERROR

Cause An attempt to perform an illegal mathematical operation. Action Press e or r to display the calculation with the cursor located at the location of the error and make required corrections.

Arg ERROR

Cause Improper use of an argument 33
Action Press e or r to display the location of the cause of the error and make required corrections.

k Order of Operations

Calculations are performed in the following order of precedence. 1 Coordinate transformation: Pol (x, y), Rec (r, ) Differentials: d/dx* Integrations: dx* Normal distribution: P(*, Q(*, R(* 2 Type A functions: With these functions, the value is entered and then the function key is pressed. x3, x2, x1, x!, Engineering symbols* Normal distribution: t * m, m1, m2, n Angle unit conversions (DRG') Metric conversions** 3 Powers and roots: ^ (xy), x 4 a b/c 5 Abbreviated multiplication format in front of , e (natural logarithm base), memory name, or variable name: 2, 3e, 5A, A, etc. 6 Type B functions: With these functions, the function key is pressed and then the value is entered. 3 , , log, In, ex, 10x, sin, cos, tan, sin1, cos1, tan1, sinh, cosh, tanh, sinh1, cosh1, tanh1, () d*, h*, b*, o*, Neg*, Not*, Det**, Trn**, arg*, Abs*, Conjg* 7 Abbreviated multiplication format in front of Type B functions: 2 3, Alog2, etc. 8 Permutation and combination: nPr, nCr * 9 Dot ()** 0 , ! , @ and* # xnor*, xor*, or* 34

* fx-100MS/fx-115MS/fx-570MS/fx-991MS only. ** fx-570MS/fx-991MS only. Operations of the same precedence are performed from right to left. exIn 120 ex{In( 120)} Other operations are performed from left to right. Operations enclosed in parentheses are performed first. When a calculation contains an argument that is a negative number, the negative number must be enclosed within parentheses. The negative sign () is treated as a Type B function, so particular care is required when the calculation includes a high-priority Type A function, or power or root operations. Example: ( 2)4 = = 16

k Stacks

This calculator uses memory areas, called stacks, to temporarily store values (numeric stack) and commands (command stack) according to their precedence during calculations. The numeric stack has 10 levels and the command stack has 24 levels. A stack error (Stack ERROR) occurs whenever you try to perform a calculation that is so complex that the capacity of a stack is exceeded. Matrix calculations use up to two levels of the matrix stack. Squaring a matrix, cubing a matrix, or inverting a matrix uses one stack level. (fx-570MS, fx-991MS only) Example:

Numeric Stack 5 4

Command Stack
Calculations are performed in sequence according to Order of Operations. Commands and values are deleted from the stack as the calculation is performed.

k Input Ranges

Internal digits: 12 Accuracy*: As a rule, accuracy is 1 at the 10th digit. Functions sinx Input Range DEG 0 x 4.4999999991010 RAD 0 x 785398163.3 GRA 0 x 4.9999999991010 cosx DEG 0 x 4.5000000081010 RAD 0 x 785398164.9 GRA 0 x 5.0000000091010 tanx DEG Same as sinx, except when x= (2n-1)90. RAD Same as sinx, except when x= (2n-1)/2. GRA Same as sinx, except when x= (2n-1)100. sin1x cos1x tan x

0 x x 9.x 230.2585092

sinhx coshx
sinh1x 0 x 4.9999999991099 cosh1x 1 x 4.9999999991099 tanhx

0 x 9.9999999991099

tanh x 0 x 9.999999999101 logx/lnx 0 x 9.9999999991099 10x 9.9999999991099 x 99.99999999 9.9999999991099 x 230.2585092

Functions

Input Range 0 x x x ; x G 0 x 0 x 69 (x is an integer) 0 n 11010, 0 r n (n, r are integers) 1 {n!/(nr)!} n 11010, 0 r n (n, r are integers) 1 [n!/{r!(nr)!}] 110100

x x2 1/x 3 x x!

49 Pol(x, y) x, y 9.99999999910 (x2+y2) 9.Rec(r, ) 0 r 9.99999999910 : Same as sinx a, b, c 110100 0 b, c x110100 Decimal Sexagesimal Conversions 000 x 99999959 x0: 110100ylogx100 ^(xy) x0: yx0: yn, 2n+1 (n is an integer) However: 110100 ylogx100

y0: x G 0

1101001/x logy100 y0: xy0: x2n1, n (n G 0; n is an integer) However: 110100 1/x logy100 Total of integer, numerator, and denominator must be 10 digits or less (including division marks). x 11050 y 11050 n 110100 xn, yn, o, p : n G 0 xn1, yn1, A, B, r : n G 0, 1 37

SD (REG)

* For a single calculation, calculation error is 1 at the 10th digit. (In the case of exponential display, calculation error is 1 at the last significant digit.) Errors are cumulative in the case of consecutive calculations, which can also cause them to become large. (This is also true of internal consecutive calculations that are performed x 3 in the case of ^(x y), y , x!, , nPr, nCr, etc.) In the vicinity of a functions singular point and point of inflection, errors are cumulative and may become large.

Power Supply

(fx-95MS only)
Users of other models should refer to the Users Guide 2 (Additional Functions). This calculator is powered by a single AA-size battery.

Replacing the Battery

Dim figures on the display of the calculator indicate that battery power is low. Continued use of the calculator when the battery is low can result in improper operation. Replace the battery as soon as possible when display figures become dim.
To replace the battery 1 Press A i to turn power off. Screw
2 Remove the six screws that hold the back cover in place and then remove the back cover. 3 Remove the old battery. 4 Load a new battery into the unit with its positive k and negative l ends facing correctly. 5 Replace the back cover and secure it in place with the six screws. 6 Press 5 to turn power on. 38

u Auto Power Off

Calculator power automatically turns off if you do not perform any operation for about six minutes. When this happens, press 5 to turn power back on.

Specifications

Users of other models should refer to the Users Guide 2 (Additional Functions). Power Supply: Single AA-size battery (R6P (SUM-3)) Battery Life: Approximately 17,000 hours continuous display of flashing cursor. Approximately 2 years when left with power turned off. 19.5 (H) i 78 (W) i 155 (D) mm 3/4 (H) 31/16 (W) 61/8 (D) 130 g (4.6 oz) including battery

Dimensions: Weight:

Power Consumption: 0.0002 W Operating Temperature: 0C to 40C (32F to 104F)
Important notice for fx-912MS owners
The names fx-912MS (Japan domestic model name) and fx-115MS (international model name) apply to the same model. Owners of the fx-912MS should refer to the description of fx-115MS. 39

CASIO COMPUTER CO., LTD.

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A supplementary reader

for CASIO fx-991MS/fx-570MS/fx-115MS/fx-100MS fx-95MS/fx-82MS/fx-350MS/fx-85MS
SA0204-010002B Printed in Japan

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About this book.

This book is a collection of easy-to-understand middle and high school practical problems that can be solved using a CASIO Scientific Calculator. Problems have been carefully selected to provide practice in performing calculations that are commonly encountered in our modern digital information age. As you will probably discover when you work through the problems in this book, the scientific calculator is a tool that makes it possible to perform very complex calculations that would be impossible to perform using manual calculation alone. We hope this lightening of the calculation load will provide you with more time to spend developing knowledge of theory and logic, and contribute towards making the study of mathematics more fun. It is important to remember that this book is not intended to replace the Users Manual that comes with your CASIO Scientific Calculator. Make sure you also carefully read the Users Manual and familiarize yourself with the features and functions of your calculator before and while using this book. When using this book, also remember that it was prepared for a global audience, so some of the units and contents contained herein may not apply in your particular country or region. We at CASIO sincerely hope you enjoy working through the problems in this book, and that by doing so you are better prepared to meet the challenges of the future digital information age.
The History of the CASIO Calculator
As the company that developed the worlds first relay-type calculator, CASIO boasts a proud history of continual innovation and development of new calculators that expand new horizons and meet the needs of students the world over. Ever since the 1972 introduction of the fx-1 with simple scientific function calculator capabilities, CASIO FX Series calculators have been meeting the needs of teachers, students and other calculator users for some 30 years. Today, the lineup of CASIO FX Series Scientific Function calculators covers a wide variety of needs, and are featured in mathematics texts the world over.

Contents

fx-82MS fx-100MS fx-570MS fx-85MS fx-95MS fx-115MS fx-991MS Page fx-350MS
Negative Value Calculations..1 Expression Values and Calculations..2 Proportion and Inverse Proportion.3 Circles and Sectors..5 Square Roots...7 Pythagorean Theorem..9 Permutation, Combination...10 Trigonometric Function Addition Theorem..12 Exponential Functions..13 Logarithmic Functions..14 Solving Quadratic Equations and Cubic Equations.15 Solving Simultaneous Equations.18 Problems Using CALC and SOLVE.22 Problems Involving Complex Numbers, with an Emphasis on Polar Form.25 Statistics Problems...28 Solving Differentials, with an Emphasis on the Derivative.31 Solving Integrations, with an Emphasis on Definite Integrals.32 Matrix Problems...34 Vector Problems..38

*Please note that some calculator models cannot be used for certain activities. *Operational procedures may differ depending on the calculator model you use.
Negative Value Calculations

Example

Operation

(3) (7) = ?

1. Enter the COMP Mode. F1 2. Perform the calculation (3) (7). D3-D7=

Result: 21

Expression Values and Calculations
Determine the value of xy when x = 5 and y = 3. 5
1. Enter the COMP Mode. F1 2. Assign 5 to variable X. 5AJx 3. Assign 3 to variable Y. D3AJy
2 4. Determine the value of X 5 Y. pxK,2C5py=

Result: 5

Proportion and Inverse Proportion
For y = 12/x, calculate the values of y for x = 1, 2, 3, 4, 5, 6, to determine how the value of y changes in proportion to a gradual increase in the value of x. Next, calculate the values of y for x = 1, 2, 3, 4, 5, 6, to determine how the value of y changes in proportion to a gradual decrease in the value of x.
1. Enter the COMP Mode. F1 2. Determine the value of y when x = 1. 1 a - 12 =

Result: 12

3. Determine the value of y when x = 2. 2 a - 12 =

Result: 6

4. Likewise, determine the values of y for x = 3, 4, 5, 6. 5. Press the [ key five times to display the value of y = 12 when x = 1.
6. Use the ] key to scroll through the values of y for x = 2, 3, 4, 5, 6.
The above reveals that the value of y decreases as the value of x increases.
7. Repeat steps 2 through 6 to calculate the values of y for x = 1 through 6.
Doing so will reveal that the value of y increases as the value of x decreases.

Circles and Sectors

Example 1
Determine, up to one decimal place, the circumference and the area of a circle with a radius of 8 cm.

Explanation Explanation

(Radius = r)
Circumference of a circle: l = 2r Area of a circle: S = r2
1. Enter the COMP Mode. F1 2. Press the F key a number of times until 1 (Fix1) is specified as the fixed number of decimal

places.

F 1(Fix) 1 ] 3. Calculate the circumference of the circle. 2Ax-8=

Result: 50.3

The circumference of the circle is approximately 50.3 cm.
4. Calculate the area. Ax-8K=

Result: 201.1

The area is approximately 201.1 cm2.

Example 2

Determine, up to one decimal place, the length of the arc and area of a sector that has a radius of 6 cm and a central angle of 240.
(Radius = r, central angle = a)
Length of the arc of a sector: l = 2r Area of a sector: S = r2
1. Enter the COMP Mode. F1 2. Press the F key a number of times until degrees (Deg) are specified as the angle unit. F 1(Deg)] 3. Press the F key a number of times until 1 (Fix1) is specified as the fixed number of decimal
F 1(Fix) 1 ] 4. Determine the length of the arc of the sector. 2 A x - 6 - R 240 \ 360 T =
The length of the arc of the sector is approximately 25.1 cm.
5. Calculate the area. A x - 6 K - R 240 \ 360 T =
The area is approximately 75.4 cm2.

Square Roots

Arrange the following from smallest to largest. 7, 2.7, 22/3, 18/ 7
Calculate the approximate square root values and then compare the results.
1. Enter the COMP Mode. F1 2. Determine the approximate value of 7. L7= Result: 7 2.645751311] 3. Determine the approximate value of L R 22 \ 3 T =

Result: 22/3 2.708012802

4. Convert 18/7 to its decimal equivalent. 18 \ 7 =
Result: 18/ 7 = 2.571428571
5. Using the [ key to scroll through and compare the results produced by steps 2 through 4
reveals that the proper ascending order arrangement of the values is: 18/ 7 < 7 < 2.7 < 22/3.
Determine the length of one side of a square whose area is equal to that of a circle with a radius of 10 cm.
First, determine the area of the circle. If the length of one side of a square is l, the area of the square is l 2. This means that the length of one side of a square with the same area as the above circle can be determined by calculating the square root of the area.

1. Enter the COMP Mode. F1 2. First, determine the area of the circle with a 10cm radius. A x - 10 K =

Result: S = 314.1592654

3. Calculate the square root of the area. L=
Result: l = S = 17.72453851
The above indicates that the length of one side of a square whose area is equal to that of a circle with a radius of 10 cm is 17.7 cm.

Pythagorean Theorem

Determine the length of the hypotenuse of a right triangle whose sides are 9 cm and 12 cm long.
(Two sides = a, b; Hypotenuse = c)

a2 + b2 = c2

1. Enter the COMP Mode. F1 2. Determine the sum of the squares of the two sides of the right triangle. 9 K + 12 K = ]
3. Calculate the square root. L=

Result: 225 = 15

The above shows that the length of the hypotenuse is 15 cm.

Permutation, Combination

What would be the maximum number of different flag signals possible using eight flags of eight different colors, when each signal consists of three flags?

Determine 8P3.

1. Enter the COMP Mode. F1 2. Determine P. 8Am3=

Result: 8P3 = 336

The above indicates there are 336 different signals possible.
How many different combinations are possible when selecting three 38 individuals from a class of 38?

Determine 38C3.

* The following shows operation using the fx-82MS/85MS/350MS/95MS.
1. Enter the COMP Mode. F1 2. Determine C. 38 n 3 =

Result: 38C3 = 8436

The above shows there are 8436 different combinations possible.
Trigonometric Function Addition Theorem
Determine the validity of: sin75 = sin (30 + 45) = sin30 cos45 + cos30 sin45.
sin Function Addition Theorem: sin( + ) = sin cos + cos sin
1. Enter the COMP Mode. F1 2. Press the F key a number of times until degrees (Deg) are specified as the angle unit. F 1(Deg)] 3. Determine value of sin75. S 75 =
Result: sin75 = 0.965925826
4. Determine value of sin30 cos45 + cos30 sin45. S 30 - W 45 + W 30 - S 45 =
Result: sin30 cos45 + cos30 sin45 = 0.965925826
5. Use the [ key to display the result obtained in step 3 to see if it matches the values obtained in

step 4.

Exponential Function
Determine the value of 64(-2/3).
When a 0: a(p) = 1/(ap) When a > 0: a(q/p) = p aq
1. Enter the COMP Mode. F1 2. Determine the value of 64. 64 W R D 2 \ 3 T = Result: 64(2/3) = 0.0625]
3. Convert to a fraction. C
1 The above indicates that 64(2/3) = 0.0625 = . 16

Logarithmic Function

Determine the value of log2

1. 128

This calculator has two logarithmic functions: log whose base is e (natural logarithm), and ln whose base is 10 (common logarithm). When the base is something other than that described above, the following formula can be used to convert the base: When a > 0, b > 0, M > 0 and a b 1, use logaM = (logbM)/(logba) for the calculation.
1. Enter the COMP Mode. F1
2. Determine log 128 = (log 128 )/(log 2). R 128 a \ R 2 =

Result: log2

= log /log 2 = 7 ] 128 128
Solving Quadratic Equations and Cubic Equations
(fx-95MS/fx-100MS/fx-115MS/fx-570MS/fx-991MS only)
Solve the equation 2x 5x + 2 = 0.
*The following shows operation using the fx-100MS/115MS/570MS/ 991MS.
1. Select the EQN Mode and then specify the degree of the equation.
In this example we want to solve a quadratic equation, so you would specify 2.

FFF1 r (Degree?) 2

2. Input values for a, b, and c in the quadratic equation ax + bx + c = 0. (a?) 2 =

(b?) D 5 = (c?) 2 =

3. One of the solutions obtained appears on the display.
4. Display the next solution. ]
The above produces the solutions x = 2 and x = 0.5. 15
Solve the equation x3 + 2 xx + 3 = 0 in the range of complex numbers.
In this example we want to solve a cubic equation, so you would specify 3.

FFF1 r (Degree?) 3

2. Input values for a, b, c, and d in the cubic equation ax + bx + cx + d = 0. (a?) 1 =
(b?) 2 = (c?) D 2 = (d?) 3 =
This causes the symbol RI to appear in the upper right corner of the display. This indicates that the displayed solution is a complex number, and the real part is 0.5. 16
5. Display the imaginary part of the solution. Ar
Each press of A r toggles the display between the real part and imaginary part.
6. Display the other solution. ] Ar
This solution is also a complex number, which is the conjugate of the previous solution. The above example shows us that the solutions for the equation are x = 3, x = 0.5 0.866025403i, and x = 0.5 + 0.866025403i.
Solving Simultaneous Equations
A group of 13 weights, some weighing 50 grams and some weighing 20 grams, weigh a combined total of 530 grams. How many of each type of weight are there?
If the number of 50g weights is x and the number of 20g weights is y, the relation
x + y = 13 50x + 20y = 530
? ? ? ? ? ? ? ? ? ? ? ? ?

can be

established and solved as simultaneous equations.

*The following shows operation using the fx-100MS/115MS/570MS/991MS.
1. Select the EQN Mode and then specify the number of unknowns.
In this example we have two unknowns, x and y, so we would specify 2.

(Unknowns?) 2

2. Input the values for the coefficients of the linear equations with two unknowns (a1?) 1 = (b1?) 1 = (c1?) 13 = (a2?) 50 = (b2?) 20 = (c2?) 530 = 3. One of the solutions obtained appears on the display.
{ a x + b y = c }. ax+by=c
The above tells us there are nine 50g weights and four 20g weights.
170 beans are divided among three individuals named A, B, and C. The ratio of beans given to A and B is 3:5, while the number of beans given to C is 22 less than the combined total given to A and B. How many beans each were given to A, B, and C?
Let us say A received x number of beans, B received y number of beans, and C received z number of beans. This means the total number of beans is x + y + z = 170. Also, since we know the ratio of beans received between A (x) and B (y) is 3:5, we can say 5x = 3y, which can be transformed to 5x 3y = 0. Finally, since C (z) received 22 beans less than A and B combined, we can say z = x + y 22.
Combining all of this leaves us with the set of simultaneous equations we should be able to solve and come up with the answer we need.
x + y + z = 170 5x 3y = 0 , which x + y z = 22
In this example we have three unknowns, x, y, and z so we would specify 3.

(Unknowns?) 3

2. Input the values for the coefficients of the linear equations with three unknowns
a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3.
(a1?) 1 = (b1?) 1 = (c1?) 1 = (d1?) 170 = (a2?) 5 = (b2?) D 3 = (c2?) 0 = (d2?) 0 = (a3?) 1 =
(b3?) 1 = (c3?) D 1 = (d3?) 22 =
3. Display the solutions.
The above shows us that A received 36 beans, B 60 beans, and C 74 beans.
Problems Using CALC and SOLVE
(fx-100MS/fx-115MS/fx-570MS/fx-991MS only)
Perform the following steps to sketch the general graph of the function y = x2 + x + 1. Substitute various values for x, and solve for y. Plot the points (x, y) on a plane. Connect adjacent points.
The CALC function comes in handy when substituting various different values for multiple instances of the same variable within a single function.

1. Select the COMP Mode. F1 2. Input the function and store it. pypupxK+px+1 C 3. Input values for x to display the values obtained for y.

(Input 1.)

(Input 2.)

< Note >

In this way, use CALC to solve for other values and then draw the graph to see what it looks like.
An object is dropped from a height of A (m) (free fall). If the speed of the object when it reaches the ground is B (m/s) and gravitational acceleration is C = 9.8 (m/s2), we can set up the following relation: (1/2) B2 = CA. Determine the speed of the object when it reaches the ground if it is dropped from a height of 30 meters. Convert the speed to kilometers per hour.
The SOLVE function lets you solve for the value of a given relation without changing its form. A unit conversion function (fx-570MS/fx-991MS only) lets C = 9.8 (m/s ) you convert from meters per second to kilometers per hour. gh = 1 VB = ? (m/s) g = C = 9.8, h = A = 30, V = B = ?

30m (A)

1. Select the COMP Mode. F1 2. Input the function and store it. R1\2T-pwKpupe-pq AI 3. Input values for the variables. Do not input anything for the variable for which you want to solve.
In this example we want to solve for B, so we input values for A and C.

9. 8 = 30 =

4. Display the variable for which you want to solve, and then run SOLVE. [[ A I (It will take some time before the solution appears.)
5. Now convert the speed to kilometers per hour.
The unit conversion number for converting meters per second to kilometers per hour is 20.

A c 20 =

The above shows us that the approximate speed of the object would be 24.25 meters per second, which converts to 87.3 kilometers per hour.
Problems Involving Complex Numbers, with an Emphasis on Polar Form
Rotate 3 2i 60 degrees around the origin of the complex plane, and then determine the point with a ratio of 2 with the origin as the center.

imaginary

Defining r as the distance of point (a, b) from the origin on the complex plane and as the angle formed with the positive part of the x-axis makes it possible to express complex number z = a + bi as z = r (cos + i sin ). This is called polar representation of complex number z. Using polar representation for z2 in the complex number multiplication z1 z2 gives us z2 = r2 (cos2 + i sin2). Now we can rotate zaround the origin of the complex plane, giving us a value with a ratio of r2 with the origin as the center.
1. Select the CMPLX Mode. F2 2. Specify the angle unit. F F F F 1(Deg) 3. Input the polar form of the complex number, with r = 2, and = 60. The values you input are
automatically converted to rectangular form on the display, but you can also display them in polar form.

2 A Q 60 =

4. Multiply by 3 2i. -R L3,2iT= Ar

The above obtains the complex number 5.196152423 + i.
Examine the relationship between 5 + 3i and 4 + i, which are two points on the complex plane.

5 + 3i

You can examine the relationship between two points on the complex plane (z1 and z2) by determining the absolute value (r) and the argument ( ) of z1/z2.
1. Select the CMPLX Mode. F2 2. Perform the calculation (5 + 3i ) (4 + i ). RD5+3iT\R4+iT= Ar 3. Convert to polar form and display the result. AY= Ar
The above indicates that 5 + 3i represents a 135 rotation of 4 + i around the origin, and that it is a point with a ratio of 1.414213562 (= 2 ) with the origin as the center.

Statistics Problems

The following are the measured 100-meter race times for five students. 12.5 11.6 10.8 12.8 11.4 (Unit: Seconds) Calculate the mean and the sample standard deviation for these results. Replace the times of the 12-second runners (runner 1 and runner 4) with 11.1 and 11.7 seconds (which are times run by other students), and calculate the mean and sample standard deviation again.
Sample standard deviation is a value that expresses the level of variation among data. When the number of data samples is n, the value of each data is xi (i = 1.n), and the mean is x : Sample standard deviation n-1 = (xi 2 nx 2)/(n1). Instead of using these complex formulas, you can obtain the same results using a simple key operation on the calculator.

12.5 11.6 10.8 12.8 11.4

*The following shows operation using the fx-82MS/85MS/350MS/95MS.
1. Enter the SD Mode, and clear the statistical memory area. F2 A B 1 (Scl) = 2. Input the data.
In the SD Mode, the | key operates as the S key. Input a value and then press the S key to register it. To input two sequential values that are identical, input the value and then press S twice. The display shows how many values have been input (number of samples = n).
12.5 S 11.6 S 10.8 S 12.8 S 11.4 S
3. Calculate the mean of the data. tAX1 =
Calculate the sample standard deviation of the data.

tA X 3 =

The above shows that the mean is 11.82 seconds, and the sample standard deviation is 0.82.
5. You can view values you have already input by using the ] and [ keys to scroll
them on the display. You can also edit a displayed value and recalculate, if you want. For the second part of this problem, first display the first value (12.5 seconds).
t] 6. Input the new value (11.1) and then press =. 11.1 = 7. Press the ] key a number of times to display the other data item to be edited (12.8 seconds for
runner 4 in this example), and then input the new data (11.7 seconds).
] ] ] ] ] ] 11.7 = 8. Calculate the mean and sample standard deviation of the data. tA X 1 =
The above shows that the mean is 11.32 seconds, and the sample standard deviation is 0.37.
The following table shows the mathematics and English test results for eight students. Math English 58 71
Use this data to determine the correlation coefficient of the test scores.
The correlation coefficient expresses the extent to which a change in one type of data tends to correspond to a M 58 M 61 change in another type of data. M 86 M 61 E 71 The correlation coefficient r for the two types of data x and y is obtained M 53 E 68 M 85 E 96 M 72 E 75 using the following expression. M 64 E 84 E 68

i=1 n i=1

(xi x)(yi y)

(xi x)2. i (yi y)2

n.xn.yn
_ _ (x , y: x, y mean; xn, yn: x, y standard deviation)
1. Enter the REG Mode and select liner regression (Lin).
First, clear statistical memory.
FF21 A B 1 (Scl) = ]] 2. Input the data.
Data items are input in pairs. Separate each item that makes up a pair with P , and press S to register the pair.
64 P 80 S 72 P 77 S 85 P 68 S 53 P 84 S 61 P 75 S 86 P 96 S 61 P 68 S 58 P 71 S
3. Calculate the correlation coefficient. A X r r 3 (r) =
The correlation coefficient of the above data is about 0.259.
Solving Differentials, with an Emphasis on the Derivative
Determine the equation for a tangent line at (1, 3) on y = x2 3x + 5.
Derivative f ' (a) at x = a on y = f (x) is the slope of the tangent line at x = a. Also, the tangent line passes through (a, f (a)), which means that its equation is y = f '(a) (x a) + f (a).

f (a) a

tangent line
1. Select the COMP Mode. F1 2. Determine derivative f ' (1) at x = 1 on y = x 3x + 5. AJp xK,3px+5P1T=

3. Determine f '(1) = 1.

4. The form of an equation for a tangent line is y = x + c, which can be transformed to c = y + x.
Since this tangent line passes through (1, 3), substitute these values.

5. This produces c = 4.

Based on the above, the equation for the tangent line is y = x + 4.
Solving Integrations, with an Emphasis on Definite Integrals
Determine the area enclosed by y = x2 + 2, the x-axis, and x = 1, x = 2.

The area enclosed by y = f (x), the x-axis, and x = a, x = b is generally b determined using the definite integral a f (x)dx.

f (x) ab x

1. Select the COMP Mode. F1 2. Calculate the value of the definite integral values for y = x + 2 from x = 1 to x = 2. dp xK+2P1P2T=
3. This produces a definite integral value of 4.333333333.
The above indicates that the area being calculated is 4.333333333 (= 13/3).
Determine the area enclosed by y = 4 x2 and the x-axis.
First, solve y = 4 x2 and find the x-intercepts. Use this to determine the integration partition, whose definite integral value is the area.

b f (x)

1. Select the EQN Mode, and then solve y = 4 x. FFF1r (Degree?) 2
(a?) D 1 = (b?) 0 = (c?) 4 =
The solutions are 2 and 2.
3. Select the COMP Mode and then calculate integration values for y = 4 x F1 d4,pxKPD2P2T= 4. This produces a definite integral value of 10.66666667.

from x = 2 to x = 2.

This above tells us that the area being calculated is 10.66666667 (= 32/3).

Matrix Problems

( fx-570MS/fx-991MS only)
Determine X2 Y2 for two matrices named X and Y when X + Y = and X Y = 7 1.
Matrix multiplication normally does not follow the commutative law XY = YX. Consequently, (X + Y) (X Y) = X2 XY + YX Y2 must be used.
1. Select the MAT Mode. FFF2 2. Specify the matrix name, and the number of rows and columns, and then input its elements.
For this problem, we need to input X + Y into Matrix A and X Y into Matrix B. The following shows input of the elements of Matrix A. Specify Matrix A.

A j 1 (Dim) 1 (A)

Specify 2 for the number of rows (m).
Specify 2 for the number of columns (n).
Matrix A is displayed as First, input 3 for A11.

A11 A12. A21 A22

Next, input 2 for A12, 3 for A21, and 1 for A22.
The display returns to the initial element (A11) of the matrix after input of values for all elements is complete.
Use the same procedure to input Matrix B elements.
A j 1 (Dim) 2 (B) 2 = 2 = 1=D4=7=1=t
3. Determine X and Y from X + Y and X Y.
First, use (X + Y) + (X Y) = 2X to determine X.
A j 3 (Mat) 1 (A) + A j 3 (Mat) 2 (B) =
1 C 2 - A j 3 (Mat) 4 (Ans) =
This produces the result X =
You can use [, ], e, and r to view the other elements of the matrix.
4. Input result obtained for X into Matrix C. A j 1 (Dim) 3 (C) 2 = 2 = 2=D1=5=0=t 5. Now determine Y from (X + Y) X = Y. A j 3 (Mat) 1 (A) , A j 3 (Mat) 3 (C) =
This produces the result Y =
6. Now perform the calculation Matrix C Matrix C Matrix Ans Matrix Ans (= X Y ). A j 3 (Mat) 3 (C) K , A j 3 (Mat) 4 (Ans) K =

The above steps show us that X2 Y2 =
Use a matrix to solve the simultaneous equations 2x 3y = 18 3x + 2y = 1.
x a b k ax + by = k Placing simultaneous equations cx + dy = l inside matrices A = c d , X = y , and B = l makes it possible to express them as AX = B. This means that if inverse matrix A-1 exists for Matrix A, then X = A-1B.
1. Select the MAT Mode. FFF2 2. Input
for Matrix A and 18 for Matrix B. 2
A j 1 (Dim) 1 (A) 2 = 2 = 2=D3=3=2=t A j 1 (Dim) 2 (B) 2 = 1 = 18 = 1 = t 3. Determine in inverse for Matrix A. A j 3 (Mat) 1 (A) a = 4. Multiply the inverse matrix by Matrix B (from the left). A j 3 (Mat) 2 (B) =
The above procedure tells us that x = 3 and y = 4.

Vector Problems

Express c = (12, 5) in the form c = s a + t b when a = (2, 1), b = (3, 4).
b = (3,4) a = (2,1) sa tb
Though this is a vector problem, it can be solved by solving the following simultaneous equations for each component.

2s 3t = 12 s + 4t = 5

c = (12, 5) = sa + t b
1. Select the EQN Mode and specify 2 as the number of unknowns. FFF1 (Unknowns?) 2 2. Input the function and store it. (a1?) 2 = (b1?) D 3 = (c1?) 12 = (a2?) 1 = (b2?) 4 = (c2?) D 5 = 3. This shows that the solutions of the simultaneous equations are 3 and 2.
The above tells us that c =3a 2 b.
What type of triangle would be produced by the vertices of the three points A = (5, 1, 2), B = (1, 0, 1), and C = (2, 4, 2)?
When determining the properties of a triangle, Explanation Explanation the length of a side is determined using the absolute value of its vector, while the angle formed by two sides is determined using the inner product of their vectors. For this problem, first lets determine the lengths of the three sides of triangle ABC.
C (2, 4, 2) B (1, 0, 1) y A (5, 1, 2)
1. Select the VCT Mode. FFF3
2. Specify the vector name and dimension, and then input its elements.
The following shows input of the elements of Vector A. Specify Vector A.

A z 1 (Dim) 1 (A)

Specify three dimensions.
3-dimensional Vector A is displayed as (A1 A2 A3). First, input 5 for A1.
Next, input 1 into A2 and 2 into A3.
The display returns to the initial element (A1) after input of values for all elements is complete.
Use the same procedure to input elements for Vector B and Vector C.
A z 1 (Dim) 2 (B) 3 = 1=0=1=t A z 1 (Dim) 3 (C) 3 = 2=4=2=t
3. Calculate A =. B b a A z 3 (Vct)
2 (B) , A z 3 (Vct) 1 (A) =
4. Calculate the absolute value of A. B

The value of A B is stored in Ans memory by step 3, so you can use Vct Ans as shown below.
A A A z 3 (Vct) 4 (Ans) =
C A 5. Use the same procedure to obtain absolute values for B and C. A z 3 (Vct) 3 (C) , A z 3 (Vct) 2 (B)
A z 3 (Vct) 1 (A) , A z 3 (Vct) 3 (C) =
The above tells us that the sides of triangle ABC are of equal length, which means it is a regular triangle.