Transposition of formulae. Changing the subject of an equation. 

rebel rebelmaths

Testing the understanding of the formal definition of $A\subseteq B$.

Fourth part of the Core Foundation Maths pre-arrival self-assessment material:

Question 1:  Algebra IV: Properties of indices (1) - Multiplication/Division                                 

Question 2: Algebra IV: Properties of indices (2) - Fractions                                 

Question 3: Algebra IV - Properties of Indices (4) - Further

Question 4: Numbers V: standard index form (conversions and operations)                                 

This is the question for week 3 of the MA100 course at the LSE. It looks at material from chapters 5 and 6. The following describes how two polynomials were defined in the question. This may be helpful for anyone who needs to edit this question.

In part a we have a polynomial. We wanted it to have two stationary points. To create the polynomial we first created the two stationary points as variables, called StationaryPoint1 and StationaryPoint2 which we will simply write as s1 ans s2 here. s2 was defined to be larger than s1. This means that the derivative of our polynomial must be of the form a(x-s1)(x-s2) for some constant a. The constant "a" is a variable called PolynomialScalarMult, and it is defined to be a multiple of 6 so that when we integrate the derivative a(x-s1)(x-s2) we only have integer coefficients. Its possible values include positive and negative values, so that the first stationary point is not always a max (and the second always a min). Finally, we have a variable called ConstantTerm which is the constant term that we take when we integrate the derivative derivative a(x-s1)(x-s2). Hence, we can now create a randomised polynomial with integers coefficients, for which the stationary points are s1 and s2; namely (the integral of a(x-s1)(x-s2)) plus ConstantTerm.

In part e we created a more complicated polynomial. It is defined as -2x^3 + 3(s1 + s2)x^2 -(6*s1*s2) x + YIntercept on the domain [0,35]. One can easily calculate that the stationary points of this polynomials are s1 and s2. Furthermore, they are chosen so that both are in the domain and so that s1 is smaller than s2. This means that s1 is a min and s2 is a max. Hence, the maximum point of the function will occur either at 0 or s2 (The function is descreasing after s2). Furthermore, one can see that when we evaluate the function at s2 we get (s2)^2 (s2 -3*s1) + YIntercept. In particular, this is larger than YIntercept if s2 > 3 *s1, and smaller otherwise. Possible values of s2 include values which are larger than 3*s1 and values which are smaller than 3*s1. Hence, the max of the function maybe be at 0 or at s2, dependent on s2. This gives the question a good amount of randomisation.

A graph of a straight line $f$ is given. Questions include determining values of $f$, of $f$ inverse, and determing the equation of the line.

This is a set of questions designed to help you practice adding, subtracting, multiplying and dividing fractions.

All of these can be done without a calculator.

Differentiate $\displaystyle \ln((ax+b)^{m})$

Provided with information on a sample with sample mean and standard deviation, but no information on the population variance, use the t test to either accept or reject a given null hypothesis.

Transposition of formulae. Changing the subject of an equation. 

rebel rebelmaths

The student is asked to factorise a quadratic $x^2 + ax + b$. A custom marking script uses pattern matching to ensure that the student's answer is of the form $(x+a)(x+b)$, $(x+a)^2$, or $x(x+a)$.

To find the script, look in the Scripts tab of part a.

This question shows how to ask for a number in scientific notation, by asking for the significand and exponent separately and using a custom marking algorithm in the gap-fill part to put the two pieces together.

Answers not in standard form, i.e. with a significand not in $[1,10)$, are accepted but given partial marks.

6 questions on standard statistical distributions.

Binomial, Poisson, Normal, Uniform, Exponential.

The student is asked to factorise a quadratic $x^2 + ax + b$. A custom marking script uses pattern matching to ensure that the student's answer is of the form $(x+a)(x+b)$, $(x+a)^2$, or $x(x+a)$.

To find the script, look in the Scripts tab of part a.

Putting algebraic fractions into their simplest forms

Straightforward question: student must find the general solution to a second order constant coefficient ODE. Uses custom marking algorithm to check that both roots appear and that the solution is in the correct form (e.g. two arbitrary constants are present). Arbitrary constants can be any non space-separated string of characters. The algorithm also allows for the use of $e^x$ rather than $\exp(x)$.

Unit tests are also included, to check whether the algorithm accurately marks when the solution is correct; when it's correct but deviates from the 'answer'; when one or more roots is incorrect; or when the roots are correct but constants of integration have been forgotten.

A collection of questions (frequently updated) to demonstrate the usage of the Logic extension.

Current questions:

• Make syllogisms (either valid, invalid or valid under an additional assumption);
• Write statements in Polish and reverse Polish notation, find the truth table, determine satisfiability;
• Test whether a collection of statements $\Gamma$ models a statement $\phi$;
• Write the Disjunctive and Conjunctive Normal Forms for a statement.

Needs the Logic Extension!

6 questions on standard statistical distributions.

Binomial, Poisson, Normal, Uniform, Exponential.

A multiple linear regression model of the form:

\[Y=\beta_0+\beta_1X_1+ \beta_2X_2+\beta_3X_3+\beta_4X_4+\epsilon \]

is fitted to some data in Minitab which generates a table showing estimates of the parameters with associated $p$-values. Determine which variable to exclude first.

A multiple linear regression model of the form:

\[Y=\beta_0+\beta_1X_1+ \beta_2X_2+\beta_3X_3+\beta_4X_4+\epsilon \]

is fitted to some data in Minitab which generates a table showing estimates of the parameters with associated $p$-values. Determine which variable to exclude first.

6 questions on standard statistical distributions.

Binomial, Poisson, Normal, Uniform, Exponential.

Solve for $x$: $\displaystyle ax ^ 2 + bx + c=0$.

No description given

Rearranging a formula

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

A few simple functions are provided of the form ax, x+b and cx+d. Values of the functions, inverses and compositions are asked for. Most are numerical but the last few questions are algebraic.

A graph of a straight line $f$ is given. Questions include determining values of $f$, of $f$ inverse, and determining the equation of the line.

Multiply two numbers in standard form, then divide two numbers in standard form.

Needs marking algorithm to allow equal values in standard form to gain equal marks

Calculation of the length and alternative form of the parameteric representation of a curve.

The student is asked to factorise a quadratic $x^2 + ax + b$. A custom marking script uses pattern matching to ensure that the student's answer is of the form $(x+a)(x+b)$, $(x+a)^2$, or $x(x+a)$.

To find the script, look in the Scripts tab of part a.

An example of using the GeoGebra extension to ask the student to create a geometric construction, with marking and steps.