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Core Pure Year 1
AS Further Maths- Edexcel
| Question | Answer |
|---|---|
| √(-a²) as a complex number | ai |
| To add/ subtract or multiply/divide complex numbers you... | Keep real and imaginary parts separate |
| i² | -1 |
| z=a+bi / z*= | a-bi |
| Any complex roots comes with... | Its conjugate as a root |
| Give the co-ordinates of z=3+4i on an argand diagram | (3,4) |
| What is the modulus of a complex number? | Distance from origin on an argand diagram |
| What is the argument of a complex number? | Angle between positive Real axis and line connecting complex point to origin (on an Argand Diagram). Measured anti-clockwise |
| What is the modulus-argument from of a complex number? | z=|z|(cos(arg(z))+i sin(arg(z))) |
| | z1z2 | = | |z1| |z2| |
| arg ( z1z2) = | arg (z1) + arg (z2) |
| | z1/ z2 | = | | z1 | / | z2 | |
| arg (z1/ z2) = | arg (z1) - arg (z2) |
| What does the loci of | z-3-4i | = 5 look like? | Circle radius 5, centre (3,4) |
| What does the loci of arg (z-1+4i) = -π / 4 look like? | Half line starting at (1,-4) at an angle of -π / 4 |
| What does the loci of | z-a | = | z-b | look like? | Perpendicular bisector of line connecting points a and b |
| How do you work out the region of a complex equation? | Work out loci and determine which side you need to shade, based on <, > etc... |
| ∑(1 -> n) 1= | n |
| ∑(1 -> n) r= | 1/2 n (n+1) |
| ∑ 3r + 5 = | 3∑r + 5∑1 |
| ∑(1 -> n) r²= | 1/6 n (n+1) (2n+1) |
| ∑(1 -> n) r³= | 1/4 n² (n+1)² |
| Give the equations you can form using: ax^2 + bx + c has roots α,β | α+β = -b / a αβ = c / a |
| Give the equations you can form using: ax^3 + bx^2 + cx + d has roots α,β,γ | ∑α = -b / a ∑αβ = c / a αβγ = -d / a |
| Give the equations you can form using: ax^4 + bx^3 + cx^2 + dx + e has roots α,β,γ,δ | ∑α = -b / a ∑αβ = c / a ∑αβγ = -d / a αβγδ = e / a |
| If youre given information about the roots and asked to work out a different relation between them, you... | Rewrite the relation you need in terms of the given information |
| If w=2x + 1 what do you sub into the equation | (w-1) / 2 for all x |
| How do you calculate the volume of a revolution around the x-axis? | π∫(between a & b) y² dx |
| How do you calculate the volume of a revolution around the y-axis? | π∫(between a & b) x² dy |
| What do you need to look out for when working out the volume of a more interesting revolution | Cones, cylinders and oportunities to add/ subtract volumes between curves |
| What is a square matrix | When no. rows = no. columns |
| What is a zero matrix | When all elements are 0 |
| What is an identity matrix | Matrix with central diagonal 1- rest 0 |
| How do you add / subtract matrices | Add / subtract corresponding elements |
| Do (2x2 matrix ) 1 0 2 3 x (2x2 matrix) 3 5 5 2 | (2x2 matrix) 3 5 21 16 |
| What does a det of 0 mean | Matrix is singular (no inverse) |
| How do you calculate det of (2x2 matrix) a b c d | ad - bc |
| What is the inverse of (2x2 matrix) a b c d | 1/detM (2x2 matrix) d -b -c a |
| (AB)^-1 = | B^-1 A^-1 |
| What are the steps to invert a 3x3 matrix | Find det A. Form the matrix of minors (M). Form the matrix of co factors (keep change keep change...). Transpose the matrix of minors. Multiply by 1/detA |
| How do you solve a system of equations using a matrix? | (matrix of multipliers of x y z) (x,y,z) = (matrix of numbers each line equates to). Then times the right by the inverse of the left |
| How do you know if 3 planes meet at a point? | System of equations determinate is NOT zero (one solution) |
| How do you know if 3 planes form a sheaf? | System of equations determinate is zero and the 3 equations are consistent (Form 2 equations after removing any variable (x,y,z) and check to see if they are multiples) (infinite solutions) |
| How do you know if 3 planes form a prism? | System of equations determinate is zero and the 3 equations are not consistent (Form 2 equations after removing any variable (x,y,z) and check to see if they are not multiples) (no solutions) |
| What are the 2 quick ways to check what a system of equations form? | If two+ are parallel they are inconsistent (no solutions). If they all are equal (consistent w/ infinite solutions) |
| How do you calculate det of (3x3 matrix) a b c d e f g h i ? | a|(2x2 matrix) a f h i| - b|(2x2 matrix) d f g i| + c|(2x2 matrix) d e g h| |
| Linear transformations always map what onto itself? | The origin |
| What represents a linear transformation? | A matrix |
| T: (x,y) -> ( ? , ? ) | ax+by, cx+dy |
| What can T: (x,y) -> (ax+by, cx+dy) be represented as? | (2x2 matrix) a b c d |
| What is an invariant point? | Point which doesnt change location after a transformation |
| What is an invariant line? | A line which its transformation maps onto itself |
| What matrix represents a reflection in the y-axis? | (2x2 matrix) -1 0 0 1 |
| What matrix represents a reflection in the x-axis? | (2x2 matrix) 1 0 0 -1 |
| What matrix represents a reflection along the line x=y? | (2x2 matrix) 0 1 1 0 |
| What matrix represents a reflection along the line y=-x? | (2x2 matrix) 0 -1 -1 0 |
| What matrix represent a rotation about the origin, anti clockwise of angle θ? | (2x2 matrix) cosθ -sinθ sinθ cosθ |
| What matrix represent a scale factor of 2 in the x axis and 5 in the y axis? | (2x2 matrix) 2 0 0 5 |
| How do you work out the area of an image (after a transformation) ? | Area of object (before transformation) x | detM | (M is the transformation) |
| What does matrix PQ represent | Transformation Q and then P |
| What is the matrix for a reflection in the x=0 plane? | (3x3 matrix) -1 0 0 0 1 0 0 0 1 |
| What is the matrix for a rotation θ on the x axis | (3x3 matrix) 1 0 0 0 cosθ -sinθ 0 sinθ cosθ |
| What is the matrix for a rotation θ on the y axis | (3x3 matrix) cosθ 0 sinθ 0 1 0 -sinθ 0 cosθ |
| What is the matrix for a rotation θ on the z axis | (3x3 matrix) cosθ -sinθ 0 sinθ cosθ 0 0 0 1 |
| If a transformation M is given as BA, how would you work out B? | MA^-1 = B |
| What are the 4 steps of induction? | Basis n=1, Assumption n=k, Inductive n=k+1, Conclusion (for all n) |
| How do you induct when looking at series? | Split the sum of k+1 into sum of k + (k+1) subbed in |
| How do you induct when looking at division? | f(k+1) - f(k) or f(k+1) - multiplier f(k) or make f(k+1) involve f(k) as a term |
| How do you induct when looking at sequences? | Sub in Uk into Uk+1 |
| How do you induct when looking at matricies? | (matrix)^k+1 = (matrix)^k (matrix) |
| What is the vector equation of a stright line? | r=a+λb (a is point on line, b is direction vector) |
| What is r=(a,b,c)+λ(d,e,f) in cartesian form? | (x-a)/d = (y-b)/e = (z-c)/f (= λ) |
| What is the vector equation of a plane? | r=a+λb+μc (a is point on plane, b and c are non-parallel non-zero vectors on the plane) |
| What is the cartesian equation of a plane with normal (a,b,c)? | ax+by+cz=d (where d is a constant) |
| What is the equation for angles between two lines? | cosθ= (a.b) / |a||b| |
| How can you check vectors a and b are perpendicular? | If a.b = 0 |
| a.a= | |a|² |
| (a,b,c) . (d,e,f) = | ad+be+cf |
| What is the scalar product form of a plane? | r.n=a.n (n is normal to the plane and a is a fixed point on the plane) |
| How do you work out the angle between a line and a plane? | 90° - (angle between normal and line) |
| How do you work out the angle between two planes? | 180° - (angle between the normals) |
| What are the two reasons for 3d lines never meeting? | They are parallel. They are skew. |
| How do you work out the shortest distance between 2 3d lines? | Find the general point equations for both lines. Find the general vector for the direction of lines connecting the two original lines. Dot that vector with the direction of both lines (=0) and solve for λ and μ. Plug into direction vector and find length. |
| How do you work out the shortest distance between a 3d line and a given point? | Find general equation for points on the line. Find general direction vector between general point and the given point. Dot with direction of line (=0). Solve for λ. Plug into general direction vector and work out length. |
| How do you work out shortest distance between a plane and a point? | ((point . normal) -d ) / |normal| |
| How do you work out the co-ordinates of a point reflected in a plane? | Work out equation of the shortest line connecting the point to the plane (r= point + λ normal). Sub in r into the scalar equation of the plane. Solve for λ. Insert 2λ into the original line equation. |
| How do you work out the equation of a line reflected in a plane? | Work out the point the original line intersects with the plane. Work out the equation of any point on the line after its been reflected. Use those two points to form an equation. |
Created by:
XanderMoore
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