Search - Numbas at mathcentre.ac.uk

48 results.

Graphing: quadratic
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Question in Graphing and Polynomials by Ben Brawn

No description given
Graphing: nth degree polynomial
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Question in Graphing and Polynomials by Ben Brawn

Understanding the general facts about polynomials of degree n.
Graphing: linear
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Question in Graphing and Polynomials by Ben Brawn

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Graphing: quartic
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Question in Graphing and Polynomials by Ben Brawn

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Graphing: cubic
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Question in Graphing and Polynomials by Ben Brawn

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Ugur's copy of Find eigenvalues, characteristic polynomial and a normalised eigenvector of a 3x3 matrix
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Question in Ugur's workspace by Ugur Efem and 1 other

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$ .
Characteristic poly, eigenvalues and eigenvectors 3x3, digonailsability (non-randomised)
Ready to use
Question in Ugur's workspace by Ugur Efem and 1 other

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".
Polynomials: Expand and Simplify 1a
Ready to use
Question in MASH Bath: Question Bank by Ben McGovern and 1 other

Finding linear combinations of two quadtratic expressions of the form $ax^2+bx+c$ .
Cubics: Solving a Cubic Equation
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Question in MASH Bath: Question Bank by Ben McGovern and 1 other

Solving a cubic equation of the form $ax^3+bx^2+cx+d=0$ .
Algebraic division and partial fractions, practice question 3
Needs to be tested
Question in Nahid's workspace by Nahid Chowdhury and 2 others

Find the remainder when dividing two polynomials, by algebraic long division.
Roots of a cubic real polynomial
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.
Find characteristic polynomial, eigenvalues and eigenvectors of a 2x2 matrix
Ready to use
Question in Julia Goedecke's contributions by Julia Goedecke and 1 other

Example of an explore mode question. Student is given a 2x2 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".
Characteristic poly, eigenvalues and eigenvectors 3x3 non-randomised
Ready to use
Question in Julia Goedecke's contributions by Julia Goedecke

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".
Quotient rule
Draft
Question in Bill's workspace by Bill Foster

Other method. Find $p,\;q$ such that $\displaystyle \frac{ax+b}{cx+d}= p+ \frac{q}{cx+d}$ . Find the derivative of $\displaystyle \frac{ax+b}{cx+d}$ .
Dividing Polynomials
Draft
Question in Bill's workspace by Bill Foster

Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.
Dividing polynomials 3
Draft
Question in Bill's workspace by Bill Foster

Divide $f(x)=x ^ 4 + ax ^ 3 + bx^2 + cx+d$ by $g(x)=x^2+p$ so that:
$\displaystyle \frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}$
Dividing polynomials 2
Draft
Question in Bill's workspace by Bill Foster

Algebraic manipulation/simplification.

Simplify $\displaystyle \frac{ax^4+bx^2+c}{a_1x^4+b_1x^2+c_1}$ by cancelling a a common degree 2 factor.
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $(ax+b)^m(cx+d)^n$ using the product rule. The answer will be of the form $(ax+b)^{m-1}(cx+d)^{n-1}g(x)$ for a polynomial $g(x)$ . Find $g(x)$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $x ^ m(ax+b)^n$ using the product rule. The answer will be of the form $x^{m-1}(ax+b)^{n-1}g(x)$ for a polynomial $g(x)$ . Find $g(x)$ .
Product rule
Draft
Question in Bill's workspace by Bill Foster

Differentiate $f(x) = x^m(a x+b)^n$ .
Complete the square. (Video)
Draft
Question in Bill's workspace by Bill Foster

Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$ .

Hence, or otherwise, find roots of $ax^2+bx+c=0$ .

Includes a video which shows how to solve a quadratic by completing the square.
Find eigenvalues, characteristic polynomial and a normalised eigenvector of a 3x3 matrix
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$ .
Integrate algebraic fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$ . Enter the constant of integration as $C$ .
Indefinite integral of polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{nx^3+mx^2+px +m}{x^2+1} \;dx$
Integral of improper polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$ . Solution involves $\arctan$ .
Integral of improper polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$ . Enter the constant of integration as $C$ .
Indefinite integral of improper polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$ . Solution involves $\arctan$ .
Indefinite integral of improper polynomial fraction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find $\displaystyle\int \frac{ax^3-ax+b}{1-x^2}\;dx$ . Input constant of integration as $C$ .
Integration of fraction with power in denominator
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$ for a polynomial $g(x)$ . Find $g(x)$ .
How to enter algebraic expressions - Getting Started
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Inputting algebraic expressions into Numbas.

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