KS5 Mathematics - User contributions [en-gb]

FP1 Series

2014-06-12T07:17:51Z

CBI: /* Preparation */

Part of [[FP1]]
== Objectives ==
<onlyinclude>
*Can use the <math>\sum</math> notation
*Use the formula for the sum of the first n natural numbers
*Use formulae for the sum of the first n square numbers and cubic numbers
*Use standard formulae to sum more complex series
</onlyinclude>

== Preparation ==
Exploring series: [http://nrich.maths.org/7290 nrich activity]

== Supporting activities ==
== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==

FP1 Complex Numbers

2014-06-12T07:16:32Z

CBI: /* Supporting activities */

Part of [[FP1]]
== Objectives ==
<onlyinclude>
*Can use real and imaginary numbers
*Know the definition of complex numbers in the form <math>a + ib</math>
*Can add subtract multiply and divide complex numbers
*Can represent complex numbers on the Argand diagram
*Know the definition of complex numbers in the form <math>r\cos{\theta}+ir\sin{\theta}</math>
*Know the meaning of, and can find, the discriminant, complex conjugate argument, real and imaginary part
*Can find complex solutions of quadratic equations with real coefficients
*Can solve problems using complex numbers
*Can solve some types of polynomials with real coefficients
</onlyinclude>
== Preparation ==
Introduction to complex numbers: [http://nrich.maths.org/9858 nrich article]

== Supporting activities ==
Squaring complex numbers: [http://nrich.maths.org/8109 nrich activity]

Exploring argand diagrams: [http://nrich.maths.org/9859 interactive activity]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==
[[File:COMPLEX_NUMBER_ARGAND_DIAGRAM.pdf]]

[[File:Complex_numbers_hw_overview.docx]]

[[File:Objectives_Complex_numbers_1_and_2.docx]]

[[File:Revision_complex_numbers.docx]]

FP1 Complex Numbers

2014-06-12T07:16:15Z

CBI: /* Supporting activities */

Part of [[FP1]]
== Objectives ==
<onlyinclude>
*Can use real and imaginary numbers
*Know the definition of complex numbers in the form <math>a + ib</math>
*Can add subtract multiply and divide complex numbers
*Can represent complex numbers on the Argand diagram
*Know the definition of complex numbers in the form <math>r\cos{\theta}+ir\sin{\theta}</math>
*Know the meaning of, and can find, the discriminant, complex conjugate argument, real and imaginary part
*Can find complex solutions of quadratic equations with real coefficients
*Can solve problems using complex numbers
*Can solve some types of polynomials with real coefficients
</onlyinclude>
== Preparation ==
Introduction to complex numbers: [http://nrich.maths.org/9858 nrich article]

== Supporting activities ==
Squaring complex numbers: [http://nrich.maths.org/8109 nrich activity]
Exploring argand diagrams: [http://nrich.maths.org/9859 interactive activity]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==
[[File:COMPLEX_NUMBER_ARGAND_DIAGRAM.pdf]]

[[File:Complex_numbers_hw_overview.docx]]

[[File:Objectives_Complex_numbers_1_and_2.docx]]

[[File:Revision_complex_numbers.docx]]

FP2 Complex Numbers

2014-06-12T07:15:53Z

CBI: /* Preparation */

Part of [[FP2]]
== Objectives ==
<onlyinclude>
* Euler's relation <math>e^{i\theta}=\cos{\theta}+i\sin{\theta}</math>
* De Moivre's theorem and its application to trigonometric identities and to roots of a complex number
* Loci and regions in the Argand diagram
* Transformations from the <math>z</math>-plane to the <math>w</math>-plane

</onlyinclude>
== Preparation ==

== Supporting activities ==
== Consolidation ==
== Homework ==
== Key questions ==
For example
*<math>\sum_{r=1}^{n}\dfrac{1}{r(r+1)}</math>
== Assessment ==
== Notes ==
These two identities are really useful:
* <math>2\cos{n\theta}\equiv z^n + \frac{1}{z^n} </math>
* <math>2i\sin{n\theta}\equiv z^n - \frac{1}{z^n} </math>
Proof of De Moivre by induction for positive integer powers is required, and for the negative case as a result.

Loci:
* <math>|z-z_1|=r</math> is a circle, centre <math>z_1</math>, radius <math>r</math>
* <math>|z-z_1|=|z-z_2|</math> is the perpendicular bisector of <math>z_1</math> and <math>z_2</math>
* <math>|z-z_1|=k|z-z_2|</math> is a circle. Convert to Cartesian to find centre and radius
* <math>arg(z-z_1)=\theta</math> is a half-line, centre <math>z_1</math>, angle <math>\theta</math> to the positive real direction
* <math>arg(\frac{z-z_1}{z-z_2})=\theta</math> is the circular arc formed by angles <math>\theta</math> subtended by the chord <math>z_1,z_2</math>
Transformation problems can often be solved by using the properties of modulus and argument, but breaking into <math>z=x+iy</math> and <math>w=u+iv</math> is sometimes necessary.

FP1 Matrices

2014-06-12T07:14:32Z

CBI: /* Supporting activities */

Part of [[FP1]]
== Objectives ==
<onlyinclude>
*Can find the dimension of a matrix
*Can add and subtract matrices of the same dimension
*Can multiply a matrix by a scalar
*Can multiply matrices together
*Can use matrices to describe linear transformations, rotations, reflections and enlargements
*Understands matrix product and can use this to describe combinations of transformations
*Can find the inverse of a 2x2 matrix and use inverse matrices to reverse the effect of a linear transformation
*Can find the determinant of a matrix and sue this to determine area scale factors
*Can use matrices to solve linear simultaneous equations
</onlyinclude>
== Preparation ==
== Supporting activities ==
Matrix meaning: [http://nrich.maths.org/6876 nrich activity]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==

FP1 Complex Numbers

2014-06-12T07:13:27Z

CBI: /* Supporting activities */

Part of [[FP1]]
== Objectives ==
<onlyinclude>
*Can use real and imaginary numbers
*Know the definition of complex numbers in the form <math>a + ib</math>
*Can add subtract multiply and divide complex numbers
*Can represent complex numbers on the Argand diagram
*Know the definition of complex numbers in the form <math>r\cos{\theta}+ir\sin{\theta}</math>
*Know the meaning of, and can find, the discriminant, complex conjugate argument, real and imaginary part
*Can find complex solutions of quadratic equations with real coefficients
*Can solve problems using complex numbers
*Can solve some types of polynomials with real coefficients
</onlyinclude>
== Preparation ==
Introduction to complex numbers: [http://nrich.maths.org/9858 nrich article]

== Supporting activities ==
Squaring complex numbers: [http://nrich.maths.org/8109 nrich activity]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==
[[File:COMPLEX_NUMBER_ARGAND_DIAGRAM.pdf]]

[[File:Complex_numbers_hw_overview.docx]]

[[File:Objectives_Complex_numbers_1_and_2.docx]]

[[File:Revision_complex_numbers.docx]]

FP1 Complex Numbers

2014-06-12T07:12:51Z

CBI: /* Preparation */

Part of [[FP1]]
== Objectives ==
<onlyinclude>
*Can use real and imaginary numbers
*Know the definition of complex numbers in the form <math>a + ib</math>
*Can add subtract multiply and divide complex numbers
*Can represent complex numbers on the Argand diagram
*Know the definition of complex numbers in the form <math>r\cos{\theta}+ir\sin{\theta}</math>
*Know the meaning of, and can find, the discriminant, complex conjugate argument, real and imaginary part
*Can find complex solutions of quadratic equations with real coefficients
*Can solve problems using complex numbers
*Can solve some types of polynomials with real coefficients
</onlyinclude>
== Preparation ==
Introduction to complex numbers: [http://nrich.maths.org/9858 nrich article]

== Supporting activities ==
== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==
[[File:COMPLEX_NUMBER_ARGAND_DIAGRAM.pdf]]

[[File:Complex_numbers_hw_overview.docx]]

[[File:Objectives_Complex_numbers_1_and_2.docx]]

[[File:Revision_complex_numbers.docx]]

FP2 Inequalities

2014-06-12T07:08:37Z

CBI: /* Preparation */

Part of [[FP2]]
== Objectives ==
<onlyinclude>
* The manipulation and solution of algebraic inequalities and inequations, including those involving the modulus sign.
</onlyinclude>

== Preparation ==
Proofs with inequalities: [http://nrich.maths.org/7051 nrich activity]

== Supporting activities ==
== Consolidation ==
== Homework ==
== Key questions ==
For example
*<math>\dfrac{1}{x-4}>\dfrac{x}{x-2}</math>
*<math>|x^2-1|>2(x+1)</math>
== Assessment ==
== Notes ==

FP2 Complex Numbers

2014-06-12T06:57:04Z

CBI: /* Preparation */

Part of [[FP2]]
== Objectives ==
<onlyinclude>
* Euler's relation <math>e^{i\theta}=\cos{\theta}+i\sin{\theta}</math>
* De Moivre's theorem and its application to trigonometric identities and to roots of a complex number
* Loci and regions in the Argand diagram
* Transformations from the <math>z</math>-plane to the <math>w</math>-plane

</onlyinclude>
== Preparation ==
Exploring argand diagrams: [http://nrich.maths.org/9859 interactive activity]

== Supporting activities ==
== Consolidation ==
== Homework ==
== Key questions ==
For example
*<math>\sum_{r=1}^{n}\dfrac{1}{r(r+1)}</math>
== Assessment ==
== Notes ==
These two identities are really useful:
* <math>2\cos{n\theta}\equiv z^n + \frac{1}{z^n} </math>
* <math>2i\sin{n\theta}\equiv z^n - \frac{1}{z^n} </math>
Proof of De Moivre by induction for positive integer powers is required, and for the negative case as a result.

Loci:
* <math>|z-z_1|=r</math> is a circle, centre <math>z_1</math>, radius <math>r</math>
* <math>|z-z_1|=|z-z_2|</math> is the perpendicular bisector of <math>z_1</math> and <math>z_2</math>
* <math>|z-z_1|=k|z-z_2|</math> is a circle. Convert to Cartesian to find centre and radius
* <math>arg(z-z_1)=\theta</math> is a half-line, centre <math>z_1</math>, angle <math>\theta</math> to the positive real direction
* <math>arg(\frac{z-z_1}{z-z_2})=\theta</math> is the circular arc formed by angles <math>\theta</math> subtended by the chord <math>z_1,z_2</math>
Transformation problems can often be solved by using the properties of modulus and argument, but breaking into <math>z=x+iy</math> and <math>w=u+iv</math> is sometimes necessary.

FP2 2nd Order Differential Equations

2014-06-12T06:54:36Z

CBI: /* Supporting activities */

Part of [[FP2]]
== Objectives ==
<onlyinclude>
* Solution of the linear second order differential equation <math>a\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}+b\frac{\mathrm{d}y}{\mathrm{d}x}+c=\mathrm{f}(x)</math>, where the coefficients are real and constant
* Solution of D.E.s reducible to the above by a given substitution
</onlyinclude>

== Preparation ==
== Supporting activities ==
Describing differential equations: [http://nrich.maths.org/5875 nrich activity]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==

FP2 Polar Graphs

2014-06-12T06:52:14Z

CBI: /* Preparation */

Part of [[FP2]]
== Objectives ==
<onlyinclude>
* Convert between polar and Cartesian coordinates
* Sketch polar graphs
* Find tangents parallel or perpendicular to the initial line using <math>\frac{\mathrm{d}y}{\mathrm{d}\theta}</math> or <math>\frac{\mathrm{d}x}{\mathrm{d}\theta}</math>
* Find the area within a polar graph using <math>\int_\alpha^\beta \frac{1}{2} r^2 \mathrm{d}\theta</math>
</onlyinclude>

== Preparation ==
Introduction to polar coordinates [http://nrich.maths.org/2755 nrich article]

== Supporting activities ==
Polar bearings: [http://nrich.maths.org/8055 nrich activity]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==
Should be able to sketch, at least, the following:

<math>\theta=\alpha,\,r=p\sec(\theta-\alpha),\,r=a,\,r=2a\cos\theta,\,r=k\theta,\,r=a(1\pm\cos\theta),\,r=a\cos(2\theta),\,r^2=a^2\cos(2\theta)</math>

<math>r</math> will always be positive.

FP2 Polar Graphs

2014-06-12T06:49:30Z

CBI: /* Supporting activities */

Part of [[FP2]]
== Objectives ==
<onlyinclude>
* Convert between polar and Cartesian coordinates
* Sketch polar graphs
* Find tangents parallel or perpendicular to the initial line using <math>\frac{\mathrm{d}y}{\mathrm{d}\theta}</math> or <math>\frac{\mathrm{d}x}{\mathrm{d}\theta}</math>
* Find the area within a polar graph using <math>\int_\alpha^\beta \frac{1}{2} r^2 \mathrm{d}\theta</math>
</onlyinclude>

== Preparation ==
== Supporting activities ==
Polar bearings: [http://nrich.maths.org/8055 nrich activity]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==
Should be able to sketch, at least, the following:

<math>\theta=\alpha,\,r=p\sec(\theta-\alpha),\,r=a,\,r=2a\cos\theta,\,r=k\theta,\,r=a(1\pm\cos\theta),\,r=a\cos(2\theta),\,r^2=a^2\cos(2\theta)</math>

<math>r</math> will always be positive.

FP2 Polar Graphs

2014-06-12T06:49:12Z

CBI: /* Supporting activities */

Part of [[FP2]]
== Objectives ==
<onlyinclude>
* Convert between polar and Cartesian coordinates
* Sketch polar graphs
* Find tangents parallel or perpendicular to the initial line using <math>\frac{\mathrm{d}y}{\mathrm{d}\theta}</math> or <math>\frac{\mathrm{d}x}{\mathrm{d}\theta}</math>
* Find the area within a polar graph using <math>\int_\alpha^\beta \frac{1}{2} r^2 \mathrm{d}\theta</math>
</onlyinclude>

== Preparation ==
== Supporting activities ==
Polar bearings: [http://nrich.maths.org/8055 link polar bearings]

== Consolidation ==
== Homework ==
== Key questions ==
== Assessment ==
== Notes ==
Should be able to sketch, at least, the following:

<math>\theta=\alpha,\,r=p\sec(\theta-\alpha),\,r=a,\,r=2a\cos\theta,\,r=k\theta,\,r=a(1\pm\cos\theta),\,r=a\cos(2\theta),\,r^2=a^2\cos(2\theta)</math>

<math>r</math> will always be positive.