FP1

Annotations:

• Definitions: - The order of a matrix is the number of rows and columns (m x n means m rows and n columns). - The entries in a matrix are called elements.
• Adding/Subtracting: Only if the matrices have the same order. Add or subtract corresponding elements in each matrix.
• Multiplying: Scalar - if matrix is multiplied by a scalar, just multiply each element by the scalar. Two matrices - the number of rows in the first matrix must equal the number of columns in the second matrix. Multiply the rows in the first matrix by the columns in the second matrix. Matrices which can be multiplied are called conformable.
• Matrix Multiplication is: Not commutative - the matrix product AB is not usually the same as the matrix product BA. If you change the order you will not necessarily get the same product. Associative - (AB)C = A(BC) for all matrices A, B and C. No matter how you group the matrices, the product will always be the same as long as they are in the same order. Distributive - (A + B)C = AC + BC.

Annotations:

• Rotation:A positive angle of rotation = anticlockwise. For multiples of 90, use unit base vectors (1 0) and (0 1) to work out the transformation matrix. Else use (cosθ sinθ) (-sinθ cosθ). Matrices multiplied by these transformation matrices will be rotated θ about the origin anticlockwise (+) or clockwise (-).
• Reflections:- Use unit base vectors (1 0) and (0 1) to find the transformation matrix. Draw a diagram to help.
• Enlargements: Enlargements, centre (0, 0) with scale factor k, l where k is the factor stretch in the x-direction and l is the factor stretch in the y-direction have the transformation matrix (k 0) (0 l). Enlargements can be positive or negative.
• Composition of Transformations: Read the order in which to carry out the transformations left to right. i.e. QX(P) means do X on point P then do Q on the result.

Annotations:

• The Identity Matrix: I = (1 0) (0 1) Any matrix multiplied by I won't be changed. I has the same effect as multiplying by 1 so a matrix won't be transformed.
• Inverse Matrices: They undo the effect of the original transformation e.g. rotating by the same angle but in the opposite direction. If the product of two matrices M and N is the identity matrix then N is the inverse of M. We write N = M^-1.
• Generally:If N = ( a c) (b d)N^-1 = 1 / ad - bc (d -c) (-b a)i.e. switch a and d and negate b and c.
• Determinant: det = ad - bc Singular: det = 0 so matrix has no inverse. Non-singular: det not equal to 0 so matrix has an inverse.
• Inverse of a Product: MN (MN)^-1 = N^-1 x M^-1 i.e. M and N switch
• Determinant is equal to the ratio between the transformed shape and the original shape. Area X new / Area X = determinant
• Determinant = 0 (Singular Matrices): Singular matrices transform all points onto a single line (or a single point, the origin, in the case of the zero matrix 0000). Area ratio still applies because area of a line/point = 0 = determinant. It is impossible to "undo" the transformation because each image point could be the image of an infinite number of different object points so you don't know which image point came from which object point (all object points are transformed onto a single line).
• Finding the Image Line of a Singular Matrix: Transform (a c) (b d) by (x y) so when multiplied together they equal (x1 y1) or points you are given. Make into simultaneous equations and solve to get a line equation.

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• Solving Simultaneous Equations: 1. Write as matrices (a c) (b d) x (x y) = (e f) 2. Find the inverse matrix and pre-multiply each side of the equation by it. 3. Solve to get (x y) = (g h) hence x = g and y = h.
• Solving a pair of simultaneous equations represents finding the intersection point of two straight lines. If matrix is singular (det = 0), it has no inverse so there will be no unique solution to the simultaneous equations. The lines are parallel so there is no point of intersection or the lines are on top of each other so there are infinite points of intersection.
• det = 0 means 1 solution det not equal zero: 0 solutions (inconsistent) Infinite solutions (consistent)

Annotations:

• Invariant Points: A singular matrix can transform an infinite number of points to the same point. If M( x y) = (x y) then (x, y) is an invariant point under transformation M - the matrix does not change the point's position.
• - All 2 x 2 transformation matrices have the origin as an invariant point. - In reflection, the mirror line is a line of invariant points. - In rotations and enlargements, the origin is the only invariant point.
• For all transformations which can be represented by a matrix, there are only two possibilities: either the origin is the only invariant point or there is a line of invariant points. Invariant lines = points on a line which are mapped to other points on a line e.g. in enlargements. Lines of invariant points = points from all over the place mapped to a line.
• Invariant Lines: A line is invariant under a transformation if the image of a point on the line is also on the line.

Annotations:

• A complex number, z, is written in the form z = x + yj where x and y are real numbers and j = SQRT(-1) x is the real part of z, written Re(z), and y is the imaginary part of z, written Im(z). They are both given as real numbers. When two complex numbers are equal then both their real parts and their imaginary parts are equal. A complex number equals zero if both the real and imaginary parts equal zero.
• Adding and Subtracting: Add/subtract the real and imaginary parts separately.
• Multiplying: Put the complex numbers in brackets next to each other. Expand the brackets and then simplify. Remember that j^2 = 1. The complex conjugate of z = x + yj is defined as z* = x - yj (change the sign of the imaginary bit). Multiplying a complex number by its complex conjugate (zz*) will leave you with a real number. (z + z* also leaves you with a real number).
• Dividing: You can divide any number by z = x + yj by multiplying the top line and the bottom line of the fraction by z* = x - yj in order to make the bottom line real.
• Quadratic Equations with No Real Roots: Solve using the formula where the square root of a negative number is written in terms of j. e,g, +/- SQRT(-9) = +/- 3j
• Solving Complex Equations: If z1 = a + bj and z2 = u + vj If z1 = z2 then a = u and b = v (i.e. real and imaginary parts equal). Equate real and imaginary parts separately in order to find the missing value.
• Finding the Quadratic Equation from a Root: If z is one root of a quadratic, the other root is z*. Roots z and z* give you the factorised quadratic (x - z)(x - z*). Make sure it's always x minus something. Expand the brackets to get a quadratic in the form x^2 + bx +c = 0.

Annotations:

• You can plot the complex number z = x + yj as the point (x, y) on an Argand diagram.You can also show it as the vector (x y) and use the vectors to show the addition and subtraction of complex numbers. Always label the axes of an Argand diagram Re and Im.
• Adding/Subtracting Vectors: Adding: Draw z1 and z2 and then connect them to make a triangle. This line/vector is z1 + z2. Subtracting: Draw z2 and -z1 and then connect them to make a triangle. This line/vector is z2 - z1. z2 - z1 = z2 + (-z1)
• Modulus: The distance from z = x + yj to the origin is called the modulus of z where |z| = SQRT(x^2 + y^2). It comes from Pythagoras. The distance between two complex numbers is given by |z2 - z1| = SQRT((x2 - x1)^2 + (y2 - y1)^2). i.e. same as distance between two points. |z1 + z2| = SQRT((x1 + x2)^2 + (y1 + y2)^2)
• Circular Loci on Argand Diagrams:The set of points, z, for which |z - (x + yj)| = r form a circle, centre (x, y) and radius r (i.e. points r distance away from the point x, y). These points form the perimeter of the circle. It must always be of the form |z - (   )| and sometimes needs rewriting into this form.
• Inequalities and Circular Loci: |z - (x + yj)| > r is everything outside the circle not including the perimeter. |z - (x + yj)| < r is everything inside the circle not including the perimeter. etc. Note: |z - (x + bj)| = |(x + bj) - z|. Both mean distance between z and (x + bj).
• Line Loci on an Argand Diagram:Re(z) = 2 would be a line parallel to the Im axis through (2, 0) like x = 2 on a normal graph. |z - (x + yj)| = |z - (a + bj)| means the distance between the set of points z and (x, y) equals the distance between z and (a, b). This is a line of points (z) where all the points are equidistant from (x, y) and (a, b). The line bisects the points. To find the equation of the line of points find the perpendicular bisector of the two points.

Annotations:

• You can describe a complex number, z = x + yj as (x, y) using Cartesian coordinates or by using modulus-argument form (polar form) by giving: 1. The length of the line connecting z to the origin i.e. |z|. 2. The angle θ that the line joining the origin to z makes with the positive real axis. It is measured in radians. z = r(cosθ + jsinθ) where r = |z| and θ = argzThis is given in the formula book.
• Argument: argz = arctan(y / x) This is OK for the 1st and 4th quadrant. 2nd quadrant = arctan(y / x) + pi 3rd quadrant = arctan(y / x) - pi arg0 is undefined.
• Loci Using Modulus-Argument Form:arg(z - (x + yj)) is the angle between the vector/line from x + yj to z and the positive real axis.- The vector line has an arrow on the end to show that z could be any point on that extending line and it is called a half-line.- A small open circle the other end of the half-line shows that the point where z = x + yj is not included since arg0 is undefined.- Make sure the argument is in the form arg(z - (   )) i.e. z minus something. You may need to rewrite.- Draw the line parallel to the positive real axis as a dotted line.- θ1 < arg(z - (a + bj)) < θ2 is the area between the half-lines from a + bj where the half-lines are in the direction of θ1 and θ2. (The inequality tells you whether to include the half-lines as solid or dotted).- argument, θ, normally lies in the range -pi < θ < pi Therefore, arg(z - (a + bj)) < θ is actually -pi < arg(z - (a + bj)) < θ

Annotations:

• 1. Find where the curve crosses the x and y axes. i.e. substitute in y = 0 and x =0.
• 2. Find any vertical asymptotes (a line which the curve tends towards but never touches). This happens when y = infinity which is when the denominator = 0. Use a table to find how the graph approaches the vertical asymptotes i.e. use values slightly greater and slightly less than the asymptotes to find the sign of the components of the function and hence the sign of the function a bit to the left/right of the asymptotes.
•  3. Find any horizontal asymptotes and determine whether the curve approaches it from above or below for large positive and negative values of x. i.e. Find the asymptote by finding the degree of the algebraic fraction: highest power/highest power. Use large positive and negative values of x to determine how the curve approaches the asymptote.
• 4. Draw the graph. Sketch in what you know - intercepts and asymptotes and then draw the curve.

Annotations:

• Even Functions: - Symmetrical about the y-axis. - Contain only even powers of x. - f(x) = f(-x) e.g. If f(x) = x^2, f(3) = 9 and f(-3) = 9 so f(x) = f(-x).
• Odd Functions:- Rotational symmetry about the origin, order 2 (180°). - Functions x, x^3, x^5 and x^7 are odd functions. -   -f(x) = f(-x) e.g. If f(x) = x^3 then -f(3) = -27 and f(-3) = -27
• Determining Whether a Function is Odd, Even or Neither: Substitute in -x into f(x) / into the equation. See if this equals f(x), -f(x) or neither.

Annotations:

• The range of values for which an equation has no solutions is the empty part of the graph.
• Laws:- You can add or subtract by any number.- You can multiply or divide by a positive number only. - If you multiply or divide by a negative number you must flip the inequality sign. - You may add, but not subtract corresponding sides of inequalities of the same type. e.g. a < b and x < y means a + x < b + y - Inequalities of the same type are transistive. e.g. p < x and x < y means p < y
• You cannot divide or multiply by x because you don't know whether it is positive or negative so you don't know whether to change the inequality sign. You can use x^2 instead because it doesn't matter whether or not x is +/- for x^2.
• Rearrange inequality so 0 is on the left. Find critical values (x values) Draw a table e.g. for (x - 1)(x - 3) find the sign of this when x < 1, 1 < x < 3 and x > 3 and see when y is positive/negative. Draw a graph to make sure (regions where it is above/below the x-axis).
• OR: Sketch f(x) and g(x). Find where they intersect (where f(x) = g(x)). Look for relevant region according to the inequality.

Annotations:

• By using proof by induction it is possible to prove that a statement is true for infinitely many whole numbers.
• 1. Check that the statement/conjecture you are trying to prove is true for the first case, n = p (usually p = 1). i.e. Is it true for n = 1?
• 2. Assume it is true for the general case, n = k where k is a positive integer.
• 3. Show that if it is true for n = k, it must also be true for n = k + 1. Decide what you are aiming for. Use your value for when n = k to algebraically come up with this aim.
• 4. Conclude your argument by stating: "Therefore, if the statement is true for n = k, it is also true for n = k + 1 and since it is also true for the first case (usually n = 1), it is true for all integers greater than or equal to the first case (usually 1) by induction.

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• Use the same method as usual for proof by induction. Assuming true for k will give you Uk = ... Showing for k + 1 will give you Uk+1 = something with Uk in it which you can then substitute in.

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• Finding formulae for the sum of certain types of finite series. The sum, Sn, of a series is found by adding the first n terms of a sequence.

1. Using Standard Formulae

   Annotations:

   • The following are standard formulae: LEARN: Sum r = 1/2n(n + 1) (In formula book): Sum r^2 = 1/6n(n + 1)(2n + 1) Sum r^3 = 1/4n^2(n + 1)^2
   • You can use the standard formulae to find formulae for other summations as follows: Sum (ar^3 + br^2 + cr + d) = (a x sum r^3) + (b x sum r^2) + (c x sum r) + (d x n)
2. Using Sigma Notation

   Annotations:

   • Swap r for p and add or take away a number to make it equal o the r value and n value above and below sigma. Substitute the new value of r into the equation and show it is equal to the desired sum.
   • Example: Show that sum from r = 1 to 8 for r^2 + r us the same as sum from r = 2 to 9 for r(r - 1). To make r = 2 you add one. Therefore p = r + 1. (This also increases 8 to 9). If p = r + 1, then r = p - 1. You can then substitute this into the equation r^2 + r and you end up with the desired equation (with p's instead of r but it's the same thing).
3. Using the Method of Differences

   Annotations:

   • In some sequences you can express each term as the difference of consecutive terms of another sequence so that most terms cancel out.
   • 1. If asked, show that a statement is true.
   • 2. Write the sum using the statement.
   • 3. Put in values of n until a cancellation pattern emerges and then write the final line in terms of n (ensure there is one row per group of terms).
   • 4. Find previous values of n terms if necessary and cancel everything. Simplify what is left.
   • 5. Find out if it converges and say how it tends (up or down) as n tends to infinity by substituting n = 100 into the equation you ended up with.
   • Convergence: If the sum approaches a finite number as n tends to infinity, the series is said to be convergent. As n tends to infinity, part of the equation may tend to zero meaning the sum gets closer and closer to a number/fraction. To find out whether it tends up or down to the number/fraction, substitute n = 100 into the equation and see if it comes out above or below the number/fraction.

Annotations:

• An identity is true for ALL values of the variables involved. An equation is true for SOME values of the variables - you can then solve it to find these values.
• Finding Unknown Constants in an Identity:Method 1 - equating coefficients:Make sure both sides are in the same form and then compare their coefficients to find the values of the unknowns. Also, compare tops and bottoms of fractions.
• Method 2 - Substituting: Substitute in values of x which will make at least one unknown zero so that you can then work out the value of the remaining unknown. Put this value in and repeat with another value of x until you have all values of the unknowns.

Annotations:

• Quadratics: For equation az^2 + bz + c = 0, Sum of roots = a + B = -b / a Product of roots = aB = c / a (True for real and complex numbers)
• Finding an Equation from the Roots: Do the sum and the product of the roots and set them to equal -b / a and c / a respectively. Make sure the sum and product are in the form of a fraction and compare with the values of a, b and c. Then write out the equation.
• Finding a New Equation with Roots Related to an Original Equation: Use equation to find sum and product of original roots. Call new roots a1 and B1. Do sum and product of new roots using original roots. Factorise and use values of sum and product of original roots to form equation.
• Method 2: A root of the original equation will be z. Therefore new root will be whatever the form you are given with z in it e.g. if new roots were 2a + 1 and 2B + 1 then the new root will be 2z + 1. Call this w and rearrange to make z the subject e.g. z = w - 1 / 2. Now substitute this into the original equation because you know z is a root so the equation will equal zero. You will get an equation in terms of w which will tell you values of a, b and c so then rewrite in terms of z.
• Cubics: Equation az^3 + bz^2 + cz + d = 0 with roots a, B and y. Sum of roots = a + B + y = -b / a Product of roots in pairs = aB + By + ya = c / a. Product of roots = aBy = -d / a. Solve like quadratics.
• Quartics: Equation az^4 + bz^3 + cz^2 + dz + e = 0 with roots a, B, y and d. Sum of roots = a + B + y + d = -b / a. Product of roots in pairs = aB + ay + ad + By + Bd + yd = c / a. Product of roots in trios = aBy + aBd + Byd + ayd = -d / a Product of roots = e / a Solve like quadratics.

Annotations:

• REMEMBER: If one root is z, another root will be z* (complex conjugate). Use polynomial division (grid method) to factorise cubics and quartics and hence find other roots. If you get left with a quadratic, the 2 roots may be z and z*. Remember also: to make complex roots factors you do (z - (x + yj))(z - (a + bj)).

Annotations:

• To prove equations involving just a, b and c: Use the information given so that your roots are both in terms of a or both in terms of B. Do sum and product of roots. Use this to substitute a value for a or B into the other equation and rearrange to the desired form.