// Numbas version: exam_results_page_options {"question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Expansion of two brackets: Linear 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic manipulation", "expansion of brackets", "expansion of the product of two linear terms"], "advice": "\n

1. Using the method given by Show steps we have:

\n

\$\\simplify[std]{ {a}x*({c}x+{d})}=\\simplify[std]{{a*c}x^2+{a*d}x}\$

\n

2.

\n

\$\\simplify[std]{ ({a1}x+{b1})*({c1}x)}=\\simplify[std]{{a1*c1}x^2+{b1*c1}x}\$

\n

\n

\n ", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

$\\simplify[std]{({a}x)({c}x+{d})}=\\;$[[0]].

\n

$\\simplify[std]{({a1}x+{b1})({c1}x)}=\\;$[[1]].

\n

Your answers should be quadratics in $x$ and should not include any brackets.

\n

You can click on Show steps to get more information, but you will lose one mark if you do so.

\n ", "gaps": [{"notallowed": {"message": "

Do not include brackets in your answer. Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.

", "showstrings": false, "strings": ["("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a*c}x^2+{a*d}x", "type": "jme", "maxlength": {"length": 13.0, "message": "

Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.

", "partialcredit": 0.0}}, {"notallowed": {"message": "

Do not include brackets in your answer. Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.

", "showstrings": false, "strings": ["("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a1*c1}*x^2+{b1*c1}*x", "type": "jme", "maxlength": {"length": 13.0, "message": "

Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.

", "partialcredit": 0.0}}], "steps": [{"prompt": "

\$ax(cx+d)=acx^2+adx\$

", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "

Expand the following to give quadratics in $x$.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-5..5 except 0)", "name": "a"}, "c": {"definition": "random(-5..5 except 0)", "name": "c"}, "b": {"definition": 0.0, "name": "b"}, "d": {"definition": "random(-9..9 except [0,c])", "name": "d"}, "a1": {"definition": "random(-5..5 except [0,a])", "name": "a1"}, "b1": {"definition": "random(-9..9 except [0,c])", "name": "b1"}, "c1": {"definition": "random(-5..5 except 0)", "name": "c1"}}, "metadata": {"notes": "\n \t\t \t\t

15/08/2012:

\n \t\t \t\t

\n \t\t \t\t

\n \t\t \n \t\t", "description": "

Expand $ax(cx+d)$ and expand $(rx+s)(px)$

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Expansion of two brackets: Linear 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic manipulation", "expansion of brackets", "expansion of the product of two linear terms"], "advice": "\n

Using the method given by Show steps we have:

\n

\$\\begin{eqnarray*}\\simplify[std]{ ({a}x+{b})({c}x+{d})}&=&\\simplify[std]{{a}x*({c}x+{d})+{b}({c}x+{d})}\\\\&=&\\simplify[std]{{a*c}x^2+{a*d}x+{b*c}x+{b*d}}\\\\&=&\\simplify[std]{{a*c}x^2+{(a*d+b*c)}x+{b*d}}\\end{eqnarray*}\$

\n

\n ", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

$\\simplify[std]{({a}x+{b})({c}x+{d})}=\\;$[[0]].

\n

Your answer should be a quadratic in $x$ and should not include any brackets.

\n

You can click on Show steps for more information, but you will lose one mark if you do so.

\n ", "gaps": [{"notallowed": {"message": "

Do not include brackets in your answer. Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.

", "showstrings": false, "strings": ["("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "maxlength": {"length": 17.0, "message": "

Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.

", "partialcredit": 0.0}, "answer": "{a*c}x^2+{b*c+a*d}x+{b*d}", "marks": 2.0, "type": "jme", "musthave": {"message": "

Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.

", "showstrings": false, "strings": ["x^2"], "partialcredit": 0.0}}], "steps": [{"prompt": "\n

There are many ways to expand an expression such as $(ax+b)(cx+d)$.

\n

One way:

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\$\\begin{eqnarray*} (ax+b)(cx+d)&=&ax(cx+d)+b(cx+d)\\\\&=&acx^2+adx+bcx+bd\\\\&=&acx^2+(ad+bc)x+bd\\end{eqnarray*}\$

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "

Expand the following to give a quadratic in $x$.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-5..5 except 0)", "name": "a"}, "c": {"definition": "random(-5..5 except 0)", "name": "c"}, "b": {"definition": "random(-9..9 except [0,a])", "name": "b"}, "d": {"definition": "random(-9..9 except [0,c])", "name": "d"}}, "metadata": {"notes": "\n \t\t

15/08/2012:

\n \t\t

\n \t\t

\n \t\t", "description": "

Expand $(ax+b)(cx+d)$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Expansion of two brackets: Quadratic and Quadratic ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic manipulation", "expansion of brackets", "expansion of two quadratic terms"], "advice": "\n

Using the method given by Show steps:

\n

\$\\begin{eqnarray*}\\simplify[std]{ ({a}z^2+{b}z+{c})({m}z^2+{p}z+{q})}&=&\\simplify[std]{{a}z^2*({m}z^2+{p}z+{q})+{b}*z*({m}z^2+{p}z+{q})+{c}({m}z^2+{p}z+{q})}\\\\&=&\\simplify[std]{{a*m}z^4+{a*p}z^3+{a*q}z^2+{b*m}z^3+{b*p}z^2+{b*q}z+{c*m}z^2+{c*p}z+{c*q}}\\\\&=&\\simplify[std]{{a*m}z^4+{a*p+m*b}z^3+{(a*q+c*m+b*p)}z^2+{b*q+c*p}z+{c*q}}\\end{eqnarray*}\$

\n

\n ", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

$\\simplify[std]{({a}z^2+{b}z+{c})({m}*z^2+{p}z+{q})}=\\;$[[0]].

\n

Your answer should be a quartic (degree 4 polynomial) in $z$ and should not include any brackets.

\n

You can click on Show steps for more information, but you will lose one mark if you do so.

\n ", "gaps": [{"notallowed": {"message": "

Do not include brackets in your answer. Input your answer as a quartic in $z$, in the form $az^4+bz^3+cz^2+dz+f$ for appropriate integers $a,\\;b,\\;c,\\;d$ and $f$.

", "showstrings": false, "strings": ["(", "zz", "z*z"], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{a*m}z^4+{a*p+b*m}z^3+{a*q+p*b+c*m}z^2+{q*b+c*p}z+{c*q}", "type": "jme", "maxlength": {"length": 31.0, "message": "

Input our answer as a quartic polynomial with all terms cllected together in the form $az^4+bz^3+cz^2+dz+f$ for appropriate integers $a,\\;b,\\;c,\\;d$ and $f$.

", "partialcredit": 0.0}}], "steps": [{"prompt": "\n

One way to expand this is as follows:

\n

$(az^2+bz+c)(dz^2+pz+q)=az^2(dz^2+pz+q)+bz(dz^2+pz+q)+c(dz^2+pz+q)$ etc..

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "

Expand the following to give a quartic in $z$.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "c": {"definition": "random(2..5)", "name": "c"}, "b": {"definition": "random(-9..9 except [0,a])", "name": "b"}, "d": {"definition": "random(-9..9 except [0,c])", "name": "d"}, "m": {"definition": "random(1..4 except a)", "name": "m"}, "q": {"definition": "random(-3..3 except 0)", "name": "q"}, "p": {"definition": "random(-3..3 except 0)", "name": "p"}}, "metadata": {"notes": "\n \t\t

17/08/2012:

\n \t\t

\n \t\t

\n \t\t

Checked calculation.

\n \t\t", "description": "

Expand $(az^2+bz+c)(dz^2+pz+q)$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Jim's copy of Frank's copy of Solving system of equations using row operations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Jim Kelly", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2145/"}], "variable_groups": [], "rulesets": {}, "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(1..10#1)", "description": ""}, "t": {"name": "t", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a31}*{x1}+{a32}*{y1}+{a33}*{z1}", "description": ""}, "s": {"name": "s", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a21}*{x1}+{a22}*{y1}+{a23}*{z1}", "description": ""}, "a13": {"name": "a13", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(0..10#1)", "description": ""}, "c13": {"name": "c13", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a13}-{a12}*{b23}", "description": ""}, "c14": {"name": "c14", "group": "Ungrouped variables", "templateType": "anything", "definition": "{r}-{a12}*{b24}", "description": ""}, "k1": {"name": "k1", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(2..6#1)", "description": ""}, "y1": {"name": "y1", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(1..10#1)", "description": ""}, "a31": {"name": "a31", "group": "Ungrouped variables", "templateType": "anything", "definition": "k1*a11", "description": ""}, "b33": {"name": "b33", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a33}-{k1}*{a13}", "description": ""}, "a33": {"name": "a33", "group": "Ungrouped variables", "templateType": "anything", "definition": "{k1}*({a23}+{a13}-{k}*{a13})+1", "description": "

3x33 matrix with determinant = a11

"}, "r": {"name": "r", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a11}*{x1}+{a12}*{y1}+{a13}*{z1}", "description": ""}, "a22": {"name": "a22", "group": "Ungrouped variables", "templateType": "anything", "definition": "k*a12+1", "description": ""}, "b34": {"name": "b34", "group": "Ungrouped variables", "templateType": "anything", "definition": "{t}-{k1}*{r}", "description": ""}, "b32": {"name": "b32", "group": "Ungrouped variables", "templateType": "anything", "definition": "{k1}", "description": ""}, "a23": {"name": "a23", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(1..8#1)", "description": ""}, "d24": {"name": "d24", "group": "Ungrouped variables", "templateType": "anything", "definition": "{b24}-{b23}*{z1}", "description": ""}, "k3": {"name": "k3", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a23}-{k}*{a13}", "description": ""}, "k": {"name": "k", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(2..5#1)", "description": ""}, "b24": {"name": "b24", "group": "Ungrouped variables", "templateType": "anything", "definition": "{s}-{k}*{r}", "description": ""}, "a12": {"name": "a12", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(2..6#1)", "description": ""}, "d34": {"name": "d34", "group": "Ungrouped variables", "templateType": "anything", "definition": "{z1}", "description": ""}, "c33": {"name": "c33", "group": "Ungrouped variables", "templateType": "anything", "definition": "{b33}-{b32}*{b23}", "description": ""}, "z1": {"name": "z1", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(1..12#1)", "description": ""}, "d14": {"name": "d14", "group": "Ungrouped variables", "templateType": "anything", "definition": "{c14}-{c13}*{z1}", "description": ""}, "b23": {"name": "b23", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a23}-{k}*{a13}", "description": ""}, "a32": {"name": "a32", "group": "Ungrouped variables", "templateType": "anything", "definition": "k1*(a12+1)", "description": ""}, "a21": {"name": "a21", "group": "Ungrouped variables", "templateType": "anything", "definition": "k*a11", "description": ""}, "k2": {"name": "k2", "group": "Ungrouped variables", "templateType": "anything", "definition": "{a13}-{a12}*({a23}-{k}*{a13})", "description": ""}, "a11": {"name": "a11", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,2,4,5,10)", "description": ""}, "c34": {"name": "c34", "group": "Ungrouped variables", "templateType": "anything", "definition": "{b34}-{b32}*{b24}", "description": ""}}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

This question asks learners to use row operations to find the inverse of a 3x3 matrix.

"}, "ungrouped_variables": ["a11", "a12", "a13", "k", "a21", "a22", "a23", "k1", "a31", "a32", "a33", "k2", "k3", "x1", "y1", "z1", "r", "s", "t", "b23", "b24", "b32", "b33", "b34", "c13", "c14", "c33", "c34", "d14", "d24", "d34"], "statement": "

Solve the following syastem of equations using row operations.

\n

\$$\\var{a11}x+\\var{a12}y+\\var{a13}z=\\var{r}\$$

\n

\$$\\var{a21}x+\\var{a22}y+\\var{a23}z=\\var{s}\$$

\n

\$$\\var{a31}x+\\var{a32}y+\\var{a33}z+\\var{t}\$$

\n

\n

\n

First set up the augmented matrix:

\n

\$$\\left(\\begin{array}{rrr|c} \\var{a11}&\\var{a12}&\\var{a13}&\\var{r}\\\\\\var{a21}&\\var{a22}&\\var{a23}&\\var{s}\\\\\\var{a31}&\\var{a32}&\\var{a33}&\\var{t}\\\\\\end{array}\\right)\$$

\n

Use appropriate row operations to get zeroes below the diagonal in the first column:

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 {a11} 1 2 3 4 5 6 7 8 9 0
\n

\n

\$$\\left(\\begin{array}{rrr|c}\\var{a11}&\\var{a12}&\\var{a13}&\\var{r}\\\\0&1&[[1]]&[[2]]\\\\0&[[3]]&[[4]]&[[5]]\\\\\\end{array}\\right)\$$

\n

Use appropriate row operations to get zeroes above and below the diagonal in the second column:

\n

\$$\\left(\\begin{array}{rrr|c} \\var{a11}&0&[[6]]&[[7]]\\\\0&1&[[8]]&[[9]]\\\\0&0&[[10]]&[[11]]\\\\\\end{array}\\right)\$$

\n

Use appropriate row operations to get zeroes above and below the diagonal in the third column:

\n

\$$\\left(\\begin{array}{rrr|c} \\var{a11}&0&0&[[12]]\\\\0&1&0&[[13]]\\\\0&0&\\var{c34}&[[14]]\\\\\\end{array}\\right)\$$

\n

Hence

\n

\$$x=\$$ [[15]]

\n

\$$y=\$$ [[16]]

\n

\$$z=\$$ [[0]]

\n

The correct row operations in the first iteration are

\n

new row2 = old row2- \$$\\var{k}*\$$row1

\n

new row3 = old row3- \$$\\var{k1}*\$$row1

\n

\n

The correct row operations in the second iteration are

\n

new row1 = old row1 - \$$\\var{a12}*\$$row2

\n

new row3 = old row3 - \$$\\simplify{{a32}-{k1}*{a12}}*\$$row2

\n

\n

The correct row operations in the third iteration are

\n

new row1 = old row1 - \$$\\simplify{{a13}-{a12}*({a23}-{k}*{a13})}*\$$row3

\n

new row2 = old row2 \$$-(\\simplify{{a23}-{k}*{a13}})*\$$row3

\n

\n

", "type": "question"}]}], "navigation": {"showresultspage": "oncompletion", "reverse": true, "allowregen": true, "showfrontpage": true, "onleave": {"message": "", "action": "none"}, "browse": true, "preventleave": true}, "showQuestionGroupNames": false, "duration": 0, "name": "Jim's First Exam", "feedback": {"advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true, "feedbackmessages": [], "showtotalmark": true, "intro": ""}, "timing": {"timedwarning": {"message": "", "action": "none"}, "timeout": {"message": "", "action": "none"}, "allowPause": true}, "showstudentname": true, "percentPass": "80", "metadata": {"description": "

This is a practise exam.

", "licence": "All rights reserved"}, "type": "exam", "contributors": [{"name": "Jim Kelly", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2145/"}], "extensions": [], "custom_part_types": [], "resources": []}