Input as a single fraction with the numerator as a quadratic and all terms expanded in the numerator.
", "showStrings": false, "partialCredit": 0, "strings": ["^"]}, "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "std", "variableReplacements": [], "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\nNote: you do not need to expand the denominator, but you must enter the numerator as a polynomial in $x$.
\nInput the fraction here: [[0]]
\nClick on Show steps for more information. You will lose one mark if you do so.
", "steps": [{"prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / (b*d)}\\]
and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nNote that in your answer you do not need to expand the denominator.
", "scripts": {}, "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "stepsPenalty": 1}], "statement": "Add the following two fractions together and express as a single fraction over a common denominator.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator", "MAS1601", "mas1601"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t \t\t5/08/2012:
\n \t\t \t\t \t\tAdded tags.
\n \t\t \t\t \t\tAdded description.
\n \t\t \t\t \t\tChanged to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t \t\t \t\t12/08/2012:
\n \t\t \t\t \t\tBack to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\t \t\tUsed the except feature of ranges to get non-degenerate examples.
\n \t\t \t\t \t\tChecked calculation.OK.
\n \t\t \t\t \t\tImproved display in content areas.
\n \t\t \t\t \n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{ax+b}{x + c} \\pm \\frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "let(\n qa,a*a2+a1*c,\n qb,b*c+a1*b2+b1*a2+a*d,\n qc,b1*d+b2*b,\n roots,[-b/a1,-d/a2],\n \n not (((-qb+sqrt(qb*qb+4*qa*qc))/(2*qa) in roots) or ((-qb-sqrt(qb*qb+4*qa*qc))/(2*qa) in roots))\n)", "maxRuns": "300"}, "advice": "The formula for {nb} fractions is :
\n\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / b*d}\\]
\nand for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nHence we have:
\\[\\begin{eqnarray*}\\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]
Input as a single fraction.
\nAlso make sure that the numerator of your answer is input in the $(ax+b)$ with no brackets other than the ones shown.
", "showStrings": false, "partialCredit": 0, "strings": [")-", ")+"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std1", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\nExpress \\[\\simplify{{a} /((x+{p}) ({a1}x + {b})) + ({c} /( (x+{p})({a2}x + {d})))}\\] as a single fraction.
\nInput the fraction here: [[0]].
\nClick on Show steps if you need help. You will lose one mark of you do so.
\n ", "steps": [{"type": "information", "prompt": "\nNote that the denominators both have the factor $\\simplify{x+{p}}$ hence we see that a common denominator is $\\simplify{(x+{p})({a1}x+{b})({a2}x+{d})}$ as both denominators, $\\simplify{(x+{p})({a1}x+{b})}$ and $\\simplify{(x+{p})({a2}x+{d})}$, divide into it.
\nNote that in your answer you do not need to expand the denominator.
\n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Add the following two fractions together and express as a single fraction over a common denominator.
\nMake sure that your answer has a numerator which you input in form $(ax+b)$ with no brackets other than the ones shown.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator", "MAS1601", "mas1601"], "rulesets": {"std1": ["std", "collectNumbers"], "std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t5/08/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tAdded description.
\n \t\t \t\tChanged to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t \t\t12/08/2012:
\n \t\t \t\tBack to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\tUsed the except feature of ranges to get non-degenerate examples.
\n \t\t \t\tChecked calculation.OK.
\n \t\t \t\tImproved display in content areas.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{a}{(x+r)(px + b)} + \\frac{c}{(x+r)(qx + d)}$ as an algebraic single fraction over a common denominator. The question asks for a solution which has denominator $(x+r)(px+b)(qx+d)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\nShow steps tells us that a good choice for the denominator of the algebraic fraction we are looking for is $\\simplify{(x+{p})({a1}x+{b})({a2}x+{d})}$.
\nHence we have:
\\[\\begin{eqnarray*} \\simplify{{a} /((x+{p}) ({a1}x + {b})) + ({c} /( (x+{p})({a2}x + {d})))} &=& \\simplify[std]{({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / ((x+{p})({a1}*x + {b}) * ({a2}*x + {d})) }\\\\&=& \\simplify{({a*a2 + c*a1} * x + {a * d + c * b}) / ((x+{p})({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]
You must write your answer in the form p/q for integers p and q
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "You must write your answer in the form p/q for integers p and q
", "showStrings": false, "partialCredit": 0, "strings": ["+", ".", "(", ")", "1-", "2-", "3-", "4-", "5-", "6-", "7-", "8-", "9-"]}, "showpreview": true, "maxlength": {"length": 7, "message": "answer too long
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\\[\\simplify[std]{{a} / {g} + ({s1*b} / {f})}\\]
Input your answer here: [[0]]
No decimal numbers allowed.
\nDo not include brackets in your answer.
\nYou can get help by clicking on Steps. If you do so you will lose 1/2 mark.
", "steps": [{"type": "information", "prompt": "The rule for {action1} fractions is \\[\\simplify{a/b+ {s1}*(c/d)=(a*d+{s1}*b*c)/(b*d)}.\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n{dosomething} the following fractions and reduce the\n \n resulting fraction to lowest form.
Input your answer as a fraction and not\n \n as a decimal.
Putting something here so Loughborough doesn't break.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Add/subtract fractions and reduce to lowest form.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The rule for {action1} fractions is \\[\\simplify{a/b+ {s1}*(c/d)=(a*d+{s1}*b*c)/(b*d)}.\\]
In this case we have:
\\[\\simplify[std,!unitFactor]{{a} / {g} + ({s1*b} / {f}) = ({a} * {f} + {g} * {s1*b}) / ({g} * {f}) ={a*f+s1*g*b}/{g*f}}.\\]
Note that this fraction is in its lowest form as there are no common factors in the denominator and the numerator.
write in the form $a(x+b)^2+c$
", "showStrings": false, "partialCredit": 0, "strings": ["(", ")", "^"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "write in the form $a(x+b)^2+c$ without using decimals
", "showStrings": false, "partialCredit": 0, "strings": [".", "x*x", "x x", "x(", "x (", ")x", ") x"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\\[q(x)=\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}\\]
Now write $q(x) = a(x+b)^2+c\\;\\;$ for fractions or integers $a$, $b$, $c$.
$q(x)=\\;$ [[0]]
You can get more information on completing the square by clicking on Show steps.
\nYou will lose 1 mark if you do so.
\nRemember: Input all numbers as fractions or integers and not as decimals.
\n ", "steps": [{"type": "information", "prompt": "\n \n \nGiven the quadratic $\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}$ we complete the square by:
\n \n \n \n1. Writing the quadratic as \\[\\var{n5}\\left(\\simplify{x^2+({-n1}/{n5})x+ {c*d}/{n5}}\\right)\\]
\n \n \n \n2. Then complete the square for the quadratic \\[\\simplify{x^2+({-n1}/{n5})x+ {c*d}/{n5}}\\]
\n \n \n \n3. Remember to multiply by {n5} the expression found from the second stage.
\n \n \n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{n1-n4}/{2*a*b}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "input numbers as fractions or integers not as a decimals
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all,fractionNumbers", "marks": 1, "vsetrangepoints": 5}, {"answer": "{n1+n4}/{2*a*b}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "input numbers as fractions or integers not as a decimals
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all,fractionNumbers", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\nNow find the roots of the quadratic equation $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$.
\nThe least root is $x=\\;$ [[0]]. The greatest root is $x=\\;$ [[1]].
\nInput numbers as fractions or integers not as a decimals
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\nComplete the square for the quadratic expression $q(x)$ by writing it in the form \\[a(x+b)^2+c\\] for numbers $a,\\;b$ and $c$.
\nHence find both roots of the equation $q(x)=0$.
\n ", "tags": ["algebra", "algebraic manipulation", "checked2015", "completing the square", "functions", "MAS1601", "mas1601", "quadratic equations", "quadratic expressions", "quadratics", "roots of a quadratic", "solving a quadratic", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tImproved spacing, added some full stops!
\n \t\tChanged \"Using that\" to \"Hence\" in statement.
\n \t\t3/07/2012:
\n \t\tAdded tags.
\n \t\t4/07/2012:
In part a - When submitted answer c=7-(31^2)/48 is accepted however question asks for answer as a fraction or an integer but this is a combination of both.
\n \t\t9/07/2012:
\n \t\tAdded reminder about using integers or fractions in last part.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Complete the square for a quadratic polynomial $q(x)$ by writing it in the form $a(x+b)^2+c$. Find both roots of the equation $q(x)=0$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Completing the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$.
\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&=&\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right)\\\\ &=&\\var{n5}\\left(\\simplify{x+({-n1}/{2*n5})}\\right)^2 -\\simplify{ {n2^2}/{4*(n5)}} \\end{eqnarray} \\]
So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:
\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}& -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}&=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]
So we get the two {rep} solutions:
\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{{abs(n2)}/{2*a*b}} \\Rightarrow &x& = \\simplify{({abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{-({abs(n2)}/{2*a*b})} \\Rightarrow &x& = \\simplify{({n1-abs(n2)}/{2*a*b})} \\end{eqnarray}\\]
please input in the form $(x+a)^2+b$
", "showStrings": false, "partialCredit": 0, "strings": ["(", ")", "^"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input your answer in the form $(x+a)^2+b$.
", "showStrings": false, "partialCredit": 0, "strings": ["x^2", "x*x", "x x", "x(", "x*("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$\\simplify{x^2+{2*a}x+ {a^2+b}} = \\phantom{{}}$ [[0]].
", "steps": [{"type": "information", "prompt": "Given the quadratic $q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}$ we complete the square by:
\n1. Halving the coefficient of $x$ gives $\\var{a}$
\n2. Work out $\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}$.
This gives the first two terms of $q(x)$.
3. But the constant term $\\simplify[all]{{a^2}}$ in $p(x)$ is not the same as in $q(x)$ – so we need to adjust by adding on a suitable constant to $p(x)$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Put the following quadratic expression in the form $(x+a)^2+b$ for suitable numbers $a$ and $b$.
\nNote that you have to input your answer in the form $(x+a)^2+b$ and the numbers $a,\\;b$ must be input exactly.
", "tags": ["MAS1601", "Steps", "algebra", "algebraic manipulation", "checked2015", "complete the square", "completing the square", "mas1601", "quadratics", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "14/7/2015
\nAdded module tag
\n5/08/2012:
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Given the quadratic $q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}$ we complete the square by:
\n1. Halving the coefficient of $x$ gives $\\var{a}$
\n2. Work out $\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}$.
This gives the first two terms of $q(x)$.
3. But the constant term $\\simplify[all]{{a^2}}$ in $p(x)$ is not the same as in $q(x)$, so we need to adjust by adding on $\\simplify[std,!fractionNumbers]{{a^2+b}-{a^2}={b}}$ to $p(x)$.
Hence we get \\[q(x) = \\simplify[all]{p(x)+{b} = (x+{a})^2+{b}}=\\simplify[all]{ (x+{a})^2+{b}}\\]
Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.
"}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "(({m} * (x ^ 2)) + ({n} * x) + {t})", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input numbers as integers not decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{t*n+be-t*s}", "maxValue": "{t*n+be-t*s}", "variableReplacements": [], "marks": 2}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "\n \n \n$q(x)=\\;\\;$[[0]]
\n \n \n \nInput all numbers as integers and not as decimals.
\n \n \n \n$r=\\;\\;$[[1]]
\n \n \n ", "variableReplacements": [], "marks": 0}], "statement": "\nDivide $\\displaystyle{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}$ by $\\simplify[std]{{r}x+{s}}$ so that:
\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=q(x)+\\frac{r}{\\simplify[std]{{r}x+{s}}}\\]
where $q(x)$ is the quotient polynomial and $r$ is the remainder ($r$ is a constant).
\nThe coefficients of $q(x)$ are integers, do not input as decimals.
\n ", "tags": ["algebra", "algebraic manipulation", "checked2015", "dividing polynomials", "division of polynomials", "polynomial division", "quotient polynomial", "remainder polynomial"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "We have:
\n\\[\\begin{eqnarray*} \\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}&=&\\simplify[std]{(x+{s})x^2+{n}x^2+{n*s+t}x+{t*n+be}}\\\\&=&\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+{t}x+{t*n+be}}\\\\ &=&\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+(x+{s})*{t}+{t*n+be-s*t}}\\\\ &=&\\simplify[std]{(x+{s})(x^2+{n}x+{t})+{t*n+be-s*t}} \\end{eqnarray*} \\]
\nHence
\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=\\simplify[std]{x^2+{n}x+{t}+{t*n+be-s*t}/({r}x+{s})}\\]
The gradient $m =$ [[0]] (input your answer to 3 decimal places).
\nThe equation of the chord is $y=ax+b$ where:
\n$a= \\;$[[1]] and $b=\\; $[[2]]
\nEnter both values $a$ and $b$ correct to 3 decimal places.
\n ", "steps": [{"answer": "", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "prompt": "Given two points $(a,f(a))$ and $(a+h,f(a+h))$ on the graph of the function $y=f(x)$.
Then the chord is the straight line between these two points and has the equation \\[y-f(a)=m(x-a)\\] where $m$ is the gradient of the chord.
The gradient is given by dividing the change in $y$ by the change in $x$.
Hence for this example \\[m = \\frac{f(a+h)-f(a)}{h} = \\frac{f(\\var{a+h})-f(\\var{a})}{\\var{h}}\\]
Let $f(x)=\\simplify[std]{(x+{b})^{n}}$. What are the gradient and equation of the chord between $(\\var{a},f(\\var{a}))$ and $(\\simplify[std]{{a}+{h}},f(\\simplify[std]{{a}+{h}}))$?
\nYou can get help by clicking on Show steps. If you do so you will lose 1 mark.
\n ", "tags": ["calculus", "Calculus", "checked2015", "equation of a chord", "equation of a straight line", "function", "functions", "gradient of chord", "mas1601", "MAS1601", "Newton quotient", "steps", "Steps", "straight line"], "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tDecided to include Steps as a tag. Perhaps the presence of Steps can be searched for in another way?
\n \t\t03/07/2012:
Added tags.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Given $f(x)=(x+b)^n$. Find the gradient and equation of the chord between $(a,f(a))$ and $(a+h,f(a+h))$ for randomised values of $a$, $b$ and $h$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Given two points $(a,f(a))$ and $(a+h,f(a+h))$ on the graph of the function $y=f(x)$.
Then the chord is the straight line between these two points and has the equation \\[y-f(a)=m(x-a)\\] where $m$ is the gradient of the chord.
The gradient is given by dividing the change in $y$ by the change in $x$.
Hence for this example \\[m = \\frac{f(a+h)-f(a)}{h} = \\frac{f(\\var{a+h})-f(\\var{a})}{\\var{h}} = \\var{d1} = \\var{val}\\] to 3 decimal places.
Hence the equation of the chord is of the form $y=\\var{d1}x+b$ for some constant $b$.
But we know that when $x=\\var{a}$ then $y=f(\\var{a}) = \\var{a+b}^\\var{n}=\\var{(a+b)^n}$
So \\[b=\\var{(a+b)^n}-\\var{d1}\\times\\var{a} = \\var{d}=\\var{val1}\\] to 3 decimal places
Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$y=\\;\\phantom{{}}$[[0]]
", "steps": [{"type": "information", "prompt": "The equation of the line is of the form $y=mx+c$.
\nYou are given the gradient $m$ and you can calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Find the equation of the straight line which:
\n\n
\n
Input your answer in the form $mx+c$ for suitable values of $m$ and $c$.
\nInput $m$ and $c$ as fractions or integers as appropriate and not as decimals.
\nClick on Show steps if you need help, you will lose 1 mark if you do so.
", "tags": ["checked2015", "diagram", "equation of a straight line", "gradient of a line", "mas1601", "MAS1601", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
\nImproved display in content areas. Corrected some minor typos.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the equation of a straight line which has a given gradient $m$ and passes through the given point $(a,b)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The equation of the line is of the form $y=mx+c$.
\nYou are given the gradient $\\displaystyle m= \\simplify{{b-d}/{a-c}}$ and we can calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
\nUsing this we get:
\\[ \\begin{eqnarray} \\var{b}&=&\\simplify[std]{({b-d}/{a-c}){a}+c} \\Rightarrow\\\\ c&=&\\simplify[std]{{b}-({b-d}/{a-c}){a}={(b*c-a*d)}/{(c-a)}} \\end{eqnarray} \\]
Hence the equation of the line is
\\[y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}\\]
Factorise $\\displaystyle{ax ^ 2 + bx + c}$ into linear factors.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Factorise the following quadratic expression $q(x)$ into linear factors i.e. input $q(x)$ in the form $(ax+b)(cx+d)$or of the form $a(x+b)(cx+d)$ for suitable integers $a$, $b$, $c$ and $d$ .
", "advice": "Direct Factorisation.
\nIf you can spot a direct factorisation then this is the quickest way to do this question.
\nFor this example we have the factorisation
\n\\[\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}\\]
\nFactorisation by finding the roots.
\nFor if $x=r$ and $x=s$ and are the roots then $q(x)=a(x-r)(x-s)$ where $a$ is the coefficient of $x^2$.
\nThere are several methods of finding the roots – here are the main methods.
\nFinding the roots of a quadratic using the standard formula.
\nWe can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
\nThe two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their common value is $-\\frac{b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\nFor this question the discriminant of $\\simplify{{a*b}x^2+{-b*c-a*d}x+{c*d}}$ is $\\Delta = \\simplify{{-(b*c+a*d)}^2-4*{a*b}*{c*d}={disc}}$
\n{rdis}.
\nSo the {rep} roots are:
\n\\[\\begin{eqnarray} x = \\frac{\\var{n1} + \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{\\var{n1} + \\var{n4} }{\\var{n3}} &=& \\simplify{{n1 + n4}/ {n3}}\\\\ x = \\frac{\\var{n1} - \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{(\\var{n1} - \\var{n4}) }{\\var{n3}} &=& \\simplify{{n1 - n4}/ {n3}} \\end{eqnarray}\\]
So we see that:
\\[q(x)=\\simplify{{a*b}}\\left(\\simplify{x-{n1 + n4}/ {n3}}\\right)\\left(\\simplify{x-{n1 - n4}/ {n3}}\\right)=\\simplify{({b} * x + { -d}) * ({a} * x + { -c})}\\]
Completing the square.
\nFirst we complete the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$
\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&=&\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right) \\end{eqnarray} \\]
So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:
\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}& -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}&=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]
So we get the two {rep} solutions:
\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{{abs(n2)}/{2*a*b}} \\Rightarrow &x& = \\simplify{({abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{-({abs(n2)}/{2*a*b})} \\Rightarrow &x& = \\simplify{({n1-abs(n2)}/{2*a*b})} \\end{eqnarray}\\]
Finding these roots then gives the factorisation as before.
\\[q(x)=\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}\\]
$q(x)=\\;$ [[0]]
You can get more information on factorising a quadratic by clicking on Show steps. You will lose 1 mark if you do so.
", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "
Factorisation by finding the roots
\nIf you cannot spot a direct factorisation of a quadratic $q(x)$ then finding the roots of the equation $q(x)=0$ can help you.
\nFor if $x=r$ and $x=s$ and are the roots then $q(x)=a(x-r)(x-s)$ for some constant $a$.
\nFinding the roots of a quadratic using the standard formula
We can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
The two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their value is $\\frac{-b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "((({a} * x) + {( - c)}) * (({b} * x) + {( - d)}))", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.0001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "factorise the expression into two factors
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\n\\[\\log_a(x^{\\var{a1}}y^{\\var{b1}})=\\alpha\\log_a(x)+\\beta\\log_a(y)\\]
\n$\\alpha=\\;\\;$[[0]], $\\beta=\\;\\;$[[1]]
\n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "((x ^ {f}) * (({c} * x) + {d}))", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\\[\\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})=\\log_a(q(x))\\]
\n$q(x)=\\;\\;$[[0]]
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Answer the following questions on logarithms.
", "tags": ["checked2015", "log laws", "logarithm laws", "logarithmic expressions", "logarithms", "logs", "MAS1601", "mas1601", "rules for logarithms", "simplifying logarithms"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "2/06/2012:
\nAdded tags.
\nChanged statement to make question clearer.
\n19/07/2012:
\nAdded description.
\n25/07/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n17/08/2012:
\nMade copy to include in Simplify Algebraic Expressions exam.
", "licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\tExpress $\\log_a(x^{c}y^{d})$ in terms of $\\log_a(x)$ and $\\log_a(y)$. Find $q(x)$ such that $\\frac{f}{g}\\log_a(x)+\\log_a(rx+s)-\\log_a(x^{1/t})=\\log_a(q(x))$
\n \t\t\n \t\t"}, "advice": "
The rules for combining logs are
\n\\[\\begin{eqnarray*} \\log_a(bc)&=&\\log_a(b)+\\log_a(c)\\\\ \\\\ \\log_a\\left(\\frac{b}{c}\\right)&=&\\log_a(b)-\\log_a(c)\\\\ \\\\ \\log_a(b^r)&=&r\\log_a(b) \\end{eqnarray*} \\]
\na)
Using these rules gives:
\\[ \\begin{eqnarray*} \\log_a(x^{\\var{a1}}y^{\\var{b1}})&=&\\log_a(x^{\\var{a1}})+\\log_a(y^{\\var{b1}})\\\\ &=&\\var{a1}\\log_a(x)+\\var{b1}\\log_a(y) \\end{eqnarray*} \\]
b)
\\[\\begin{eqnarray*} \\simplify[std]{{a2}/{b2}}\\log_a(x)+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})&=&\\log_a(x^\\frac{\\var{a2}}{\\var{b2}})+\\log_a(\\simplify{{c}*x+{d}})-\\log_a(\\simplify{x^(1/{b2})})\\\\ \\\\ &=&\\log_a\\left(\\simplify[std]{(x^({a2}/{b2})*({c}x+{d}))/(x^(1/{b2}))}\\right)\\\\ &=&\\log_a\\left(\\simplify{x^{f}*({c}x+{d})}\\right) \\end{eqnarray*} \\]
You must write your answer in the form p/q for integers p and q
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "You must write your answer in the form p/q for integers p and q
", "showStrings": false, "partialCredit": 0, "strings": ["+", ".", "(", ")", "1-", "2-", "3-", "4-", "5-", "6-", "7-", "8-", "9-"]}, "showpreview": true, "maxlength": {"length": 7, "message": "answer too long
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[\\simplify{{g} / ({a} + {s1} * ({b * g} / {f}))}\\]
Input your answer here: [[0]]
Your answer must be of the form a/b for suitable integers a and b. No decimal numbers allowed.
\nDo not include brackets in your answer.
", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Write the following expression as a single fraction in its lowest form:
", "tags": ["checked2015", "Fractions", "fractions", "lowest form", "mas1601", "MAS1601", "simplifying fractions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded description.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\frac{a} {b + \\frac{c}{d}}$ as a single fraction in the form $\\displaystyle \\frac{p}{q}$ for integers $p$ and $q$.
"}, "advice": "We have:
\\[\\simplify[std]{{g} / ({a} + {s1} * ({b * g} / {f})) = {g} / (({a} * {f} + {s1} * {b * g}) / {f}) ={g} / (({a * f + s1 * b * g}) / {f})= ({f}*{g}) / ({a * f + s1 * b * g}) = ({g * f} / {(a * f + s1 * b * g)})}\\]
Here we use the result that dividing by a fraction $\\frac{a}{b}$ is the same as multiplying by $\\frac{b}{a}$.
The resulting fraction is in lowest form i.e. the top and bottom do not have a common factor.
\\[2\\log_{\\var{a}}(\\simplify{x+{b}})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\var{d}\\]
\n$x=\\;$ [[0]].
\nIf you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\nInput all numbers as fractions or integers and not as decimals.
", "marks": 0, "scripts": {}, "gaps": [{"answer": "{sol1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"prompt": "Three rules for logs should be used:
\n1. $n\\log_a(m)=\\log_a(m^n)$
\n2. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$
\n3. $\\log_a(p)=r \\Rightarrow p=a^r$
\nSo use rule 1 followed by rules 2 and 3 to get an equation for $x$.
", "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "stepsPenalty": 1}], "statement": "Solve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
", "tags": ["algebra", "algebraic manipulation", "checked2015", "combining logarithms", "logarithm laws", "logarithms", "MAS1601", "mas1601", "simplifying logarithms", "solving", "solving equations", "Solving equations", "steps", "Steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
\nImproved display in content areas.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle 2\\log_{a}(x+b)- \\log_{a}(x+c)=d$.
\nMake sure that your choice is a solution by substituting back into the equation.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "We use the following three rules for logs :
\n1. $n\\log_a(m)=\\log_a(m^n)$
\n2. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$
\n3. $\\log_a(p)=r \\Rightarrow p=a^r$
\nUsing rule 1 we get
\\[2\\log_{\\var{a}}(\\simplify{x+{b}})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}((\\simplify{x+{b}})^2)- \\log_{\\var{a}}(\\simplify{(x+{c})})\\]
Using rule 2 gives
\\[\\log_{\\var{a}}(\\simplify{(x+{b})^2})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)\\]
So the equation to solve becomes:
\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)=\\var{d}\\]
and using rule 3 this gives:
\\[ \\begin{eqnarray} \\simplify{(x+{b})^2/(x+{c})}&=&{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ \\simplify{(x+{b})^2}&=&{\\var{a}}^{\\var{d}}(\\simplify{x+{c}})=\\simplify{{a^d}(x+{c})}\\Rightarrow\\\\ \\simplify{x^2+{2*b-a^(d)}x+{b^2-a^(d)*c}}&=&0 \\end{eqnarray} \\]
Solving this quadratic we get two solutions:
$x=\\var{sol1}$ and $x=\\var{sol2}$
\nWe should check that these solutions gives positive values for $\\simplify{x+{b}}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.
\nThe value $x=\\var{sol1}$ gives:
\nSubstituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{2*a^d}})$ so OK.
\nSubstituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{4*a^d}})$ so OK.
\nHence $x=\\var{sol1}$ is a solution to our original equation.
\nThe value $x=\\var{sol2}$ gives:
\nSubstituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{-a^d}})$ so NOT OK.
\nSubstituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{a^d}})$ so OK.
\nHence $x=\\var{sol2}$ is NOT a solution to our original equation as $\\log_{\\var{a}}(\\simplify{x+{b}})$ is not defined for this value of $x$.
\nSo there is only one solution $x=\\var{sol1}$.
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Input your answer as a fraction or an integer. Do NOT input the answer as a decimal.
$x\\;=$[[0]]
", "marks": 0}], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..12)", "name": "b", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b=td,td+1,td)", "name": "d", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..12)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a=tc,tc+1,tc)", "name": "c", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s3", "description": ""}, "tc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..20)", "name": "tc", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "td": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..20)", "name": "td", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "td", "tc"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Solve the following linear equation for $x$.
", "tags": ["checked2015", "equations", "linear equation", "mas1601", "MAS1601", "solving equations", "Solving equations", "solving linear equations"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded more tags.
\nAdded description.
\nChecked calculation. OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve $\\displaystyle ax + b = cx + d$ for $x$.
"}, "advice": "Given the equation \\[\\simplify[std]{{a}x+{b}={c}x+{d}}\\] we first collect together all the constant terms, and collect together all the terms in $x$.
\nThe equation can then be written as:
\\[\\simplify[std]{({a}-{c})x=({d}+{-b})}\\] i.e.
\\[\\simplify{{a-c}x={d-b}}\\]
which gives \\[x =\\simplify[std]{{(d-b)}/{(a-c)}}\\] as the solution.
Input numbers as fractions or integers not as a decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{n1+n4}/{2*a*b}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input numbers as fractions or integers not as a decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Solve for $x$: \\[\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}=0\\]
The least root is $x=\\;$ [[0]]. The greatest root is $x=\\;$ [[1]]
You can get more information on solving a quadratic by clicking on Show steps. You will lose 1 mark if you do so.
\nEnter the least root first. If the roots are equal, enter the root in both input boxes.
\nEnter the roots as fractions or integers, not as decimals.
", "steps": [{"type": "information", "prompt": "Finding the roots by factorisation.
\nFinding a factorisation of a quadratic $q(x)=a(x-r)(x-s)$ where $a$ is the coefficient of $x^2$ gives the roots $x=r$, $x=s$ immendiately.
\nIf you cannot find a factorisation then there are several other methods you can use.
\nUsing the formula for the roots.
\nYou can find the roots by using the formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
\nThe two roots are:
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their value is $\\displaystyle \\frac{-b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\n", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "stepsPenalty": 1}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "
Find the roots of the following quadratic equation.
", "tags": ["algebra", "checked2015", "factorisation", "find roots of a quadratic equation", "mas1601", "MAS1601", "quadratic formula", "quadratics", "roots of a quadratic equation", "solving a quadratic equation", "solving a quadratic equation ", "solving equations", "Solving equations", "Steps", "steps"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded tags.
\nAdded description.
\nImproved display in various content areas.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle ax ^ 2 + bx + c=0$.
"}, "advice": "Direct Factorisation
\nIf you can spot a direct factorisation then this is the quickest way to do this question.
\nFor this example we have the factorisation
\n\\[\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}\\]
\nHence we find the roots:
\\[\\begin{eqnarray} x&=& \\simplify{{n1-n4}/{2*a*b}}\\\\ x&=& \\simplify{{n1+n4}/{2*a*b}} \\end{eqnarray} \\]
Other Methods.
\nThere are several methods of finding the roots – here are the main methods.
\nFinding the roots of a quadratic using the standard formula.
\nWe can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$
\nThe two roots are
\n\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]
there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$
1. $\\Delta \\gt 0$. The roots are real and distinct
\n2. $\\Delta=0$. The roots are real and equal. Their common value is $\\displaystyle -\\frac{b}{2a}$
\n3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.
\nFor this question the discriminant of $\\simplify{{a*b}x^2+{-b*c-a*d}x+{c*d}}$ is $\\Delta = \\simplify[std]{{-n1}^2-4*{a*b*c*d}}=\\var{disc}$
\n{rdis}.
\nSo the {rep} roots are:
\n\\[\\begin{eqnarray} x = \\frac{\\var{n1} - \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{\\var{n1} - \\var{n4} }{\\var{n3}} &=& \\simplify{{n1 - n4}/ {n3}}\\\\ x = \\frac{\\var{n1} + \\sqrt{\\var{disc}}}{\\var{n3}} &=& \\frac{\\var{n1} + \\var{n4} }{\\var{n3}} &=& \\simplify{{n1 + n4}/ {n3}} \\end{eqnarray}\\]
\nCompleting the square.
\nFirst we complete the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$
\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&=&\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &=&\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right) \\end{eqnarray} \\]
So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:
\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}& -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&\\phantom{{}}&=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]
So we get the two {rep} solutions:
\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{-{abs(n2)}/{2*a*b}} \\Rightarrow &x& = \\simplify{({-abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&=&\\simplify{({abs(n2)}/{2*a*b})} \\Rightarrow &x& = \\simplify{({n1+abs(n2)}/{2*a*b})} \\end{eqnarray}\\]
Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "Rearrange the equation by adding {-c} to both sides to get:
\\[\\simplify[std]{{t} / ({a} * x + {b}) = {d} + { -c} = {d -c}}\\]
This gives \\[\\simplify{({a} * x + {b}) / {t} = 1 / {d -c}}\\] (this is because if $\\displaystyle \\frac{a}{b}=c$ then $\\displaystyle \\frac{b}{a}=\\frac{1}{c}$ on turning the fraction round the other way)
\nand so \\[\\simplify{({a} * x + {b}) = {t} / {d -c}}\\] on multiplying both sides by {t}.
\nSolve this equation for $x$.
", "scripts": {}, "marks": 0}], "prompt": "\\[\\simplify{{t} / ({a} * x + {b}) + {c} = {d}}\\]
\n$x=\\;$ [[0]]
\nIf you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\nInput all numbers as fractions or integers and not as decimals.
", "stepsPenalty": 1}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Solve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "MAS1601", "mas1601", "rearranging equations", "solving", "solving equations", "Solving equations", "subject of an equation"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
\nImproved display in content areas.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle \\frac{a} {bx+c} + d= s$
"}, "advice": "Rearrange the equation by adding {-c} to both sides to get:
\\[\\simplify{{t} / ({a} * x + {b}) = {d} + { -c} = {d -c}}\\]
This gives \\[\\simplify{({a} * x + {b}) / {t} = 1 / {d -c}}\\] (this is because if $\\displaystyle \\frac{a}{b}=c$ then $\\displaystyle \\frac{b}{a}=\\frac{1}{c}$ on turning the fraction round the other way)
and so \\[\\simplify{({a} * x + {b}) = {t} / {d -c}}\\] on multiplying both sides by {t}.
Hence \\[\\simplify{{a} * x = {t} / {d -c} -{b} = ({a * an1} / {an2})}\\]
and so \\[\\simplify{x={an1}/{an2}}\\] is the solution on dividing both sides by {a}.
Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "{c*a1-a*c1}/{b*a1-a*b1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1} \\end{eqnarray} \\]
\n$x=\\phantom{{}}$[[0]], $y=\\phantom{{}}$[[1]]
\nInput your answers as fractions or integers, not as decimals.
", "marks": 0}], "statement": "Solve the following simultaneous equations for $x$ and $y$. Input your answers as fractions or integers, not as decimals.
", "tags": ["checked2015", "equations", "linear", "mas1601", "MAS1601", "pair of linear equations", "simultaneous", "simultaneous linear equations", "solve linear equations", "solving equations", "Solving equations"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded more tags.
\nAdded description.
\nChecked calculation. OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$ and $y$: \\[ \\begin{eqnarray} a_1x+b_1y&=&c_1\\\\ a_2x+b_2y&=&c_2 \\end{eqnarray} \\]
"}, "advice": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}&\\mbox{ ........(1)}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1}&\\mbox{ ........(2)} \\end{eqnarray} \\]
To get a solution for $x$ multiply equation (1) by {this} and equation (2) by {that}
This gives:
\\[ \\begin{eqnarray} \\simplify[std]{{a*this}x+{b*this}y}&=&\\var{this*c}&\\mbox{ ........(3)}\\\\ \\simplify[std]{{a1*that}x+{b1*that}y}&=&\\var{that*c1}&\\mbox{ ........(4)} \\end{eqnarray} \\]
Now {aort} (4) {fromorto} equation (3) to get
\\[\\simplify[std]{({a*this}+{s6*a1*that})x={this*c}+{s6*that*c1}}\\]
And so we get the solution for $x$:
\\[x = \\simplify{{c*b1-b*c1}/{b1*a-a1*b}}\\]
Substituting this value into any of the equations (1) and (2) gives:
\\[y = \\simplify{{c*a1-a*c1}/{b*a1-a*b1}}\\]
You can check that these solutions are correct by seeing if they satisfy both equations (1) and (2) by substituting these values into the equations.
Refresher questions on topics in algebra that students beginning a maths undergraduate course should be familiar with.
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