ideas-0.6: src/Domain/Math/Examples/DWO1.hs
-----------------------------------------------------------------------------
-- Copyright 2010, Open Universiteit Nederland. This file is distributed
-- under the terms of the GNU General Public License. For more information,
-- see the file "LICENSE.txt", which is included in the distribution.
-----------------------------------------------------------------------------
-- |
-- Maintainer : bastiaan.heeren@ou.nl
-- Stability : provisional
-- Portability : portable (depends on ghc)
--
-- Example exercises from the Digital Mathematics Environment (DWO)
--
-----------------------------------------------------------------------------
module Domain.Math.Examples.DWO1
( calculateResults, fillInResult
, coverUpEquations, linearEquations
, quadraticEquations, findFactors
, modulusEquations, sqrtEquations, sqrtSubstEquations, brokenEquations
, simplerSqrt, simplerSqrt2, simplerSqrt3
) where
import Prelude hiding ((^))
import Domain.Math.Data.Relation
import Domain.Math.Expr
calculateResults :: [[Expr]]
calculateResults = [level1, level2, level3]
where
level1 =
[ -8*(-3)
, -3-9
, 55/(-5)
, -6*9
, -11- (-3)
, 6-(-9)
, -10+3
, 6+(-5)
]
level2 =
[ -3-(6*(-3))
, -12/3 - 3
, -4*(2+3)
, 2-6*6
, -27/(4-(-5))
, (-24/(-6)) - 3
, 8-(-77/(-11))
, 4/(-4+5)
]
level3 =
[ 4*(3-(6-2))
, (-16-9)/5 - 3
, 4- (4-13)/(-3)
, (3*(-3))-5-4
, -55/(3*(-5)+4)
, -4*(-2+ (-4)+7)
, -8 - (140/4*5)
, (13-(2-1)) / 3
]
fillInResult :: [[Equation Expr]]
fillInResult = [level1, level2, level3]
where
level1 =
let x = variable "x" in
[ x-2 :==: 2
, -4*x :==: -28
, -8*x :==: 72
, x+4 :==: 09
, 4+x :==: 2
, -10-x :==: -7
, x/(-8) :==: -3
, 11-x :==: 14
]
level2 =
let x = variable "x" in
[ -5-3*x :==: -23
, 21/x - 4 :==: 3
, -3*(x+3) :==: -27
, 2-5*x :==: 47
, 18/(7-x) :==: 6
, -77/x + 4 :==: -7
, -7-(x/(-5)) :==: -15
, -18/(-3+x) :==: 3
]
level3 =
let x = variable "x" in
[ -5*(5-(3-x)) :==: -20
, (-20-x)/(-5)-2 :==: 3
, 4-(x-14)/(-3) :==: 1
, 3*x - 3 - 7 :==: 8
, -42/(-2*x+2) :==: 7
, 3*(4+x+2) :==: 12
, -6-(-54/(-3*x)) :==: -12
, 14-(x-3)/4 :==: 3
]
coverUpEquations :: [[Equation Expr]]
coverUpEquations = [level1, level2]
where
level1 =
let x = variable "x" in
[ 38-7*x :==: 3
, sqrt (125/x) :==: 5
, 4*(12-x) + 7 :==: 35
, 5*x^2 :==: 80
, 5*(5-x) :==: 35
, 32/sqrt x :==: 8
, (21/x)-8 :==: -1
, 180/x^2 :==: 5
, 3*(x-8)^2 :==: 12
, (8-x)/3 + 7 :==: 9
]
level2 =
let x = variable "x" in
[ sqrt (x+9)/2 :==: 3
, (4*x-18)^2 :==: 4
, 3*(13-2*x)^2 - 20 :==: 55
, 5*((x/3) - 8)^2 :==: 20
, (6/sqrt (x-7))^3 :==: 8
, 8-(15/sqrt (31-x)) :==: 5
, sqrt (4*(x^2-21)) :==: 4
, 3 + (44/sqrt (87 + x)) :==: 7
, 13-(56 / (21 + (70/(3+x)))) :==: 12
, 12/(2+(24/(8+(28/(2+9/x))))) :==: 3
]
linearEquations :: [[Equation Expr]]
linearEquations = [level1, level2, level3, level4, level5]
where
level1 :: [Equation Expr]
level1 =
let x = variable "x" in
[ 5*x + 3 :==: 18
, 11*x - 12 :==: 21
, 19 - 3*x :==: -5
, -12 + 5*x :==: 33
, 15 - 9*x :==: 6
, 4*x + 18 :==: 0
, 11*x - 12 :==: -34
, -2*x - 3 :==: -4
, 6*x - 12 :==: 2
, -4*x - 13 :==: -11
]
level2 :: [Equation Expr]
level2 =
let x = variable "x" in
[ 6*x-2 :==: 2*x+14
, 3+6*x :==: 3*x+24
, 5*x+7 :==: 2*x - 10
, 2*x-8 :==: 18 - x
, 4*x - 6 :==: 7*x - 14
, -1 -5*x :==: 3*x - 20
, 4*x - 7 :==: -5*x - 24
, 4*x - 18 :==: 14 + 11*x
, 17 :==: 4 - 10*x
, -5*x + 6 :==: 2 - 3*x
]
level3 :: [Equation Expr]
level3 =
let x = variable "x" in
[ 4*(x-1) :==: 11*x - 12
, 4*(x-4) :==: 5*(2*x+1)
, 2*(5-3*x) :==: 6-x
, 4*x - (x-2) :==: 12 + 5*(x-1)
, -3*(x-2) :==: 3*(x+4) - 7
, 3*(4*x-1) + 3 :==: 7*x - 14
, 4*(4*x - 1) - 2 :==: -3*x + 3*(2*x -5)
, 2*x - (3*x + 5) :==: 10 + 5*(x-1)
, -5*(x+1) :==: 9*(x+4)-5
, 18 - 2*(4*x + 2) :==: 7*x - 4*(4*x -2)
]
level4 :: [Equation Expr]
level4 =
let x = variable "x" in
[ (1/2)*x - 4 :==: 2*x + 2+(1/2)
, (1/4)*x + (1/2) :==: (5/2)*x + 2
, (1/4)*x - (3/4) :==: 2*x + (1/2)
, -(1/2)*x + (3/4) :==: (5/2)*x + 3
, -(1/2)*x + 1+(1/2) :==: 2*x - 5
, -(1/3)*x + (3/4) :==: (1/4)*x + (1/6)
, (3/4)*x - (1/3) :==: (2/3)*x - (3/4)
, (2/5)*x - (1/4) :==: (1/2)*x + (3/4)
, (2/3)*x - 2 :==: (1/5)*x - (3/5)
, (-1+(2/5))*x + 3+(1/2) :==: (3/5)*x + (9/10)
]
level5 :: [Equation Expr]
level5 =
let x = variable "x" in
[ (1/4)*(x-3) :==: (1/2)*x - 4
, (x+3)/2 :==: 5*((1/2)*x + 1 + (1/2))
, (1/2)*(7-(2/3)*x) :==: 2 + (1/9)*x
, (3/4)*x - (x-1) :==: 3 + (2+(1/2))*(x-1)
, -(5/4)*(x-7) :==: (3/4)*(x+2) - (4+(1/2))
, 3*((1/5)*x - 1) + 5 :==: 7*x - 14
, ((5*x - 1) / 6) - 2 :==: -4*x + (3*x - 6)/2
, 2*x - ((2*x+2)/5) :==: 12 + (x-1)/6
, (-3*(x+2))/6 :==: 9*((2/3)*x + (1/3)) - (5/3)
, 1 - ((4*x + 2)/3) :==: 3*x - ((5*x - 1) / 4)
]
quadraticEquations :: [[Equation Expr]]
quadraticEquations = [level1, level2, level3, level4, level5, level6]
where
level1 =
let x = variable "x" in
[ x^2 :==: 2
, x^2+3 :==: 52
, x^2-7 :==: 0
, 9*x^2 - 6 :==: 75
, 32 - 2*x^2 :==: 14
, 2*(x^2 - 3) :==: 12
, (1/4)*x^2 + 12 :==: 16
, (x-1)^2 :==: 100
, 14 - 2*x^2 :==: 6
, (1/4)*(17-x^2) :==: 2
]
level2 =
let x = variable "x" in
[ (x-7)^2 + 3 :==: 11
, (6-2*x)^2 :==: 81
, (1/2)*(x+9)^2 :==: 4
, (3-x^2)/10 :==: 2
, 5*x^2 + 3*x :==: 3*x + 2
, 11 - (2*x + 1)^2 :==: 5
, (6*x - 3)^2 + 6 :==: 12
, (7+2*x)^2 :==: 7
, 4 - (x^2 / 10) :==: 6
, 12 - (2*x + 3)^2 :==: 6
]
level3 =
let x = variable "x" in
[ x^2 :==: 5*x
, x^2 - 6*x :==: 0
, 6*x + x^2 :==: 0
, x*(x+4) :==: 0
, x*(2*x-4) :==: 0
, 3*x^2 :==: 6*x
, 3*x :==: 2*x^2
, x*(1-6*x) :==: 0
, (x+5)*(x-8) :==: 0
, (3*x-1)*(x+3) :==: 0
]
level4 =
let x = variable "x" in
[ x^2-2*x :==: 3
, x^2+12*x+20 :==: 0
, x^2-x :==: 30
, x*(x+2) :==: 8
, x*(x-3) :==: 4
, 2*x+15 :==: x^2
, 4*x :==: 12 - x^2
, x^2 :==: 15 - 8*x
, x^2-9*x+18 :==: 0
, x^2+14*x+24 :==: 0
]
level5 =
let x = variable "x" in
[ (3*x+5)^2+(x-5)^2 :==: 40
, 4*(10-x^2) :==: -2*x*(2*x + 10)
, x*(x+12) :==: 2*x^2
, 3*(x-2)*(x+6) :==: 12*x
, 8*x^2+4*x-24 :==: (x+3)*(x-8)
, 3*x^2 - 11 :==: (3+2*x)^2
, 2*x*(x-3)-3 :==: (x+2)*(x+6)
, 12*(x^2-3*x)+8 :==: 56
, 4*x^2-6*x :==: x^2+9
, (x+1)*(x-5) :==: (x-2)*(x-3)
]
level6 =
let x = variable "x" in
[ x^2+4*x-4 :==: 0
, x^2-6*x :==: 4
, x^2-12*x+34 :==: 0
, 2*x^2+4*x-8 :==: 0
, (x-4)*(x-1) :==: 11
, (x-(7/2))^2 :==: 2*(x+4)
, x^2-3*x :==: 3*(x-2)
, (4-x)*(1-x) :==: 3*x
, 2*x^2 :==: x*(x+2)+7
, (1-x)^2 :==: x+2
]
-- Ontbinden in factoren (VWO B, hoofdstuk 1)
findFactors :: [[Expr]]
findFactors =
let x = variable "x" in
[ -- (buiten haakjes brengen)
[ 4*x^2 -4*x
, 36*x^2+30*x
, -6*x^2-18*x
, 14*x^2-10*x
] --(product-som methode)
, [ x^2+11*x+24
, x^2-8*x+15
, x^2-x-2
, x^2-11*x+28
]
]
modulusEquations :: [[Equation Expr]]
modulusEquations =
let x = variable "x" in
[ [ abs (2*x + 3) :==: 2
, abs (5 - 2*x) :==: 1
, abs (4*x + 7) :==: 3
, abs (11 - 3*x) :==: 15
]
, [ abs (x^2 + 3) :==: 9
, abs (x^2 - 7) :==: 2
, abs (3-x^2) :==: 6
, abs (9-x^2) :==: 7
]
, [ abs (2*x^2+3) :==: 19
, abs (3*x^2-2) :==: 1
, abs (6-2*x^2) :==: 2
, abs (3-(1/2)*x^2) :==: 15
]
, [ abs (4*x^3-72) :==: 36
, abs (0.2*x^4 - 112) :==: 13
, abs (2*x^4-4.25) :==: 3.75
, abs (1.75 - 0.5*x^3) :==: 2.25
]
]
sqrtEquations :: [[Equation Expr]]
sqrtEquations =
let x = variable "x" in
[ [ 5 - 2*sqrt x :==: 1
, 7 - 3 * sqrt x :==: 5
, 4 - 2*sqrt x :==: -3
, 6 - 3*sqrt x :==: 2
]
, [ 2*sqrt x :==: x
, 2*sqrt x :==: 3*x
, x-3*sqrt x :==: 0
, 3*x - 5*sqrt x :==: 0
]
, [ x :==: sqrt (2*x + 3)
, x :==: sqrt (3*x+10)
, x :==: sqrt (4*x + 21)
, x :==: sqrt (3*x + 4)
]
, [ 5*x :==: sqrt (50*x + 75)
, 2*x :==: sqrt (24*x + 28)
, 3*x :==: sqrt (27*x - 18)
, 2*x :==: sqrt (28*x - 40)
, 3*x :==: sqrt (3*x + 42)
, 5*x :==: sqrt (49*x + 2)
, 3*x :==: sqrt (10*x -1)
, 5*x :==: sqrt (30*x - 5)
]
, [ x - sqrt x :==: 6
, x - 4*sqrt x :==: 12
, x - sqrt x :==: 12
, x - sqrt x :==: 2
, 2*x + sqrt x :==: 3
, 3*x + 4*sqrt x :==: 20
, 2*x + sqrt x :==: 15
, 2*x - 3*sqrt x :==: 27
]
]
sqrtSubstEquations :: [[Equation Expr]]
sqrtSubstEquations =
let x = variable "x" in
[ [ 8*x^3 + 1 :==: 9*x * sqrt x
, 27*x^3 :==: 28*x*sqrt x - 1
, x^3 + 3 :==: 4*x*sqrt x
, x^3 :==: 10*x*sqrt x - 16
]
, [ x^3 :==: 6*x*sqrt x + 16
, x^3 - 24*x*sqrt x :==: 81
, x^3 + x*sqrt x :==: 20
, x^3 - 15 :==: 2*x*sqrt x
]
, [ x^5 +32 :==: 33*x^2*sqrt x
, 243*x^5 - 244*x^2*sqrt x + 1 :==: 0
, 32*x^5 + 31*x^2*sqrt x :==: 1
, x^5 :==: 242*x^2*sqrt x + 243
]
, [ x^5 + 8 :==: 6*x^2*sqrt x
, x^5 :==: 9*x^2 * sqrt x - 18
, x^5 :==: 5*x^2*sqrt x + 24
, x^5 + 4*x^2 * sqrt x :==: 12
]
]
brokenEquations :: [[Equation Expr]]
brokenEquations =
let x = variable "x" in
[ [ (2*x^2 - 10) / (x^2+3) :==: 0
, (7*x^2 - 21) / (2*x^2 - 5) :==: 0
, (3*x^2 - 6) / (4*x^2+1) :==: 0
, (4*x^2 - 24) / (6*x^2 - 2) :==: 0
, x^2 / (x+4) :==: (3*x+4) / (x+4)
, (x^2 + 2) / (x-2) :==: (x+8) / (x-2)
, (x^2 + 6*x - 6) / (x^2 - 1) :==: (4*x + 9) / (x^2 - 1)
, (x^2 + 6) / (x^2 - 2) :==: (7*x) / (x^2 - 2)
]
, [ (x^2 + 6*x) / (x^2 - 1) :==: (3*x + 4) / (x^2 - 1)
, (x^2 + 6) / (x - 3) :==: (5*x) / (x - 3)
, (x^2 + 4*x) / (x^2 - 4) :==: (3*x + 6) / (x^2 - 4)
, (x^2 + 2*x - 4) / (x-5) :==: (4*x + 11) / (x - 5)
, (5*x+2) / (2*x - 1) :==: (5*x + 2) / (3*x + 5)
, (x^2-9) / (4*x - 1) :==: (x^2 - 9) / (2*x + 7)
, (3*x - 2) / (2*x^2) :==: (3*x - 2) / (x^2 + 4)
, (2*x + 1) / (x^2+3*x) :==: (2*x + 1) / (5*x + 8)
]
, [ (x^2 - 1) / (2*x + 2) :==: (x^2-1) / (x+8)
, (x^2 - 4) / (3*x - 6) :==: (x^2-4) / (2*x+1)
, (x^2 + 5*x) / (2*x^2) :==: (x^2 + 5*x) / (x^2 + 4)
, (x^2 - 3*x) / (2*x - 6) :==: (x^2 - 3*x) / (4*x + 2)
, x/(x+1) :==: 1+3/4
, (x+2)/(3*x) :==: 1+1/3
, (2*x+3) / (x-1) :==: 3+1/2
, (x-3)/(1-x) :==: 1+2/5
]
, [ (x+4)/(x+3) :==: (x+1)/(x+2)
, (2*x+3)/(x-1) :==: (2*x-1) / (x-2)
, (3*x+6)/(3*x-1) :==: (x+4) / (x+1)
, (x+2)/(2*x+5) :==: (x+4)/(2*x-3)
, (x+5)/(2*x) +2 :==: 5
, (3*x+4)/(x+2) - 3 :==: 2
, (x^2)/(5*x+6) +4 :==: 5
, (x^2)/(2*x-3) + 3 :==: 7
]
, [ (x-2) / (x-3) :==: x/2
, (x+9) / (x-5) :==: 2/x
, (x+2) / (x+4) :==: 2/(x+1)
, (-3) / (x-5) :==: (x+3)/(x-1)
, (x+1)/(x+2) :==: (7*x+1) / (2*x-4)
, (2*x-7)/(5-x) :==: (x+1) / (3*x-7)
, (x+1)/(x-1) :==: (3*x-7)/(x-2)
, (3*x-7)/(x-2) :==: (7-x) / (3*x-3)
]
]
infix 9 *|
-- short-hand notation for a product with a square-root
(*|) :: Expr -> Expr -> Expr
a *| b = a * sqrt b
simplerSqrt :: [[Expr]]
simplerSqrt =
let a = Var "a" in
[ [ 9*|5 * 7*|3, 3*|2 * 2*|5, 5*|2 * 6*|7, 4*|6 * 2*|7, (6*a)*|3 * 9*|2
, 5*|5 * (2*a)*|7, a*|6 * 7*|5, 8*|7 * a*|3
]
, [ sqrt 15 / 6*|3, 5*|30/sqrt 5, 4*|10 / 5*|2, 5*|21 / 2*|7, (6*a)*|35 / 3*|5
, (5*a)*|14 / 9*|2, a*|6 / 7*|3, (a*3)*|42 / 7*|7
]
, [ 5/2*|2, 2/5*|3, 3/2*|5, 8/7*|6, (2*a)/3*|7, (6*a)/7*|10, (5*a)/3*|11
, (6*a)/5*|13
]
, [ sqrt (2/3), sqrt (5+1/3), sqrt (1+1/2), sqrt (3+4/7), sqrt (5*a^2)
, sqrt (7*a^2), sqrt (3*a^2), sqrt (6*a^2)
]
, [ sqrt ((2/9)*a^2), sqrt ((5/16)*a^2), sqrt ((3/25)*a^2), sqrt ((7/16)*a^2)
, ((1/3)*|2)^2, ((1/2)*|3)^2, ((2/7)*|5)^2, ((2/3)*|7)^2
]
]
simplerSqrt2 :: [[Expr]]
simplerSqrt2 =
let a = Var "a" in
[ [ (((1/7)*a)*|2)^2, (((3/5)*a)*|3)^2, (((1/3)*a)*|5)^2, (((4/7)*a)*|6)^2
, sqrt 8 + sqrt 2, sqrt 2 + sqrt 18, sqrt 12 - sqrt 3, sqrt 7 - sqrt 28
]
, [ sqrt 12 + sqrt 48, sqrt 18 - sqrt 8, sqrt 45 - sqrt 20, sqrt 80 + sqrt 45
, sqrt (50*a^2) - sqrt (32*a^2), sqrt (75*a^2) - sqrt (12*a^2)
, sqrt (27*a^2) + sqrt (3*a^2), sqrt (24*a^2) + sqrt (96*a^2)
]
, [ sqrt 27 + 1 / sqrt 3, sqrt 24 + 5/sqrt 6, sqrt 72- 7/ sqrt 2
, sqrt 98 - 5/sqrt 2, sqrt 24 + sqrt (1+1/2), sqrt 40 - sqrt (2+1/2)
, sqrt 75 - sqrt (1+1/3)
, sqrt (1+2/3) + sqrt 60
]
]
simplerSqrt3 :: [[Expr]]
simplerSqrt3 =
let a = Var "a" in
[ [ (2*|7 + 7*|3)^2, (sqrt 2 + 6*|3)^2, (4*|3 + 3*|2)^2, (2*|5 + sqrt 7)^2
, (3*|6 - 4*|5)^2, (5*|3 - sqrt 2)^2, (4*|6 - 2*|7)^2, (sqrt 5 - 2*|3)^2
]
, [ (2*|3 - 2)^2, (5*|2 - 1)^2, (3+4*|3)^2, (2+3*|6)^2, (4*|2+3)*(4*|2 - 3)
, (sqrt 7 + sqrt 3)*(sqrt 7 - sqrt 3), (2*|2 - sqrt 5)*(2*|2 + sqrt 5)
, (6-3*|3)*(6+3*|3)
]
, [ (a-sqrt 3)^2, (2*|6 + a)^2, (2*a + a*|5)^2, (a*|3 - (2*a)*|2)^2
, (a-sqrt 7)*(a+sqrt 7), (3*a + 2*|3)*(3*a - 2*|3)
, (2*a + a*|2)*(2*a - a*|2), ((3*a)*|5 - a)*((3*a)*|5 + a)
]
, [ 4/(sqrt 2 + 2), 3/(sqrt 5 + 1), 2/(sqrt 3 - 3), 5/(sqrt 6 - 2)
, 6/(sqrt 7 + sqrt 5), 4/(2*|3 + sqrt 6), 5/(3*|2 - sqrt 3)
, 2/(sqrt 11-sqrt 2)
]
, [ 2*|3 / (sqrt 5 + sqrt 2), 6*|5 / (sqrt 7 + sqrt 3)
, 4*|3 / (sqrt 5 - sqrt 3), 8*|7 / (sqrt 6 - sqrt 5)
]
]