Pure maths 2

finding x and y-intercepts of (2x - 1)(x + 3)(x - 2)

(-1)(3)(-2) = + 6 = y x = -3, or 2, or 1/2

sin^2x is the same as

(sin x)^2

y = Sin2x and y = 2Sinx effect on Sinx graph

-Sin 2x = stretch parallel to x of factor 1/2 (so graph has been compressed to the left) -2Sinx = stretch parallel to y of factor 2 (so graph has been stretched upwards and downwards but x intercepts not affected)

y = -SinX vs y = Sin(-X)

-SinX = reflection in x-axis (flipped) Sin(-X) = reflection in y-axis (flipped) cos(-x) has no change

how to find line of symmetry of graph

-always x translation vector (opposite of inside of completed square)

two terms containing t⁷ are 126a⁵t⁷ - 84a⁶t⁷√t find the coefficient of t⁷

-cannot be the second term because t is not combined: - 84a⁶t⁷√t = -84a⁶√t¹⁵ which does not contain t⁷ -so the coefficient of t⁷ is 126a⁵ from the expression 126a⁵t⁷

how many teams of 4 can be produced from a pool of 12 enginners?

-combination is used because ABCD is the same group as DCBA 12C4 = 495

find equation of y = x^3 translated by (-1, 4)

-complete square y = (x )^3 -add translation y = (x + 1)^3 + 4 -expand and simplify y = x^3 + 3x^2 + 3x + 5

(x - 1) and (x + 2) are both factors of 2x^3 - x^2 + px + q. Find P and Q:

-convert from bracket form to intercepts (x - 1) -> 1 (x + 2) -> -2 -sub intercepts into x of polynomial to generate simultaneous equations p + 1 = -q 2p + 20 = q -solve simultaneous equations p = -7 q = 6

How many ways can you arrange the letters of Mississippi

-count the number of letters 11 -count the number of times each letter is repeated i = 4 s = 4 p = 2 M = 1 -do total!/each repeat! 11!/(4! x 4! x 2! x 1!) = 34650

find the values of x for which (x - 2)³ = x³ - 8

-expand using binomial expansion x³ - 6x² + 12x - 8 -set equal to other side and simplify x³ - 6x² + 12x - 8 = x³ - 8 -6x² + 12x = 0 -factorise and solve -6x(x - 2) x = 0 or 2

y = f(ax), y = f(x/a), y = af(x)

-f(ax) = stretch of scale factor 1/a, parallel to x-axis -f(x/a) = stretch of scale factor a, parallel to x-axis -af(x) = stretch of scale factor a, parallel to y-axis

how to find factors of a polynomial (or factorise)

-find factors of last digit in equation -substitute factors into equation until f(x) = 0. This one is an intercept -set intercept into bracket form -divide polynomial by bracket -factorise leftover equation to find remaining brackets

A cricket team consisting of 6 batters, 4 bowlers and 1 wicket-keeper is to be selected from a group of 18 cricketers comprising 9 batters, 7 bowlers and 2 wicket-keepers. How many different teams can be selected?

-find number of ways each title can be selected batters: 9C6 = 84 bowlers: 7C4 = 35 wicket-keepers: 2C1 = 2 -find number of ways of combining them 84 x 35 x 2 = 5880

show how graph of y = x^2 -4x -1 can be obtained from y = x^2

-find translation -> -complete square (x - 2)^2 - 5 -use completed square to find translation (2, -5)

affect on curve of increasing k and x in inverse proportion

-increasing k stretches the curve outwards more -increasing x tightens curve inwards (only if multiplied. If an integer is added to x then it will translate the graph opposite than expected)

transformations: y = f(x + 2) and y = f(x) + 2

-inside brackets: change opposite and affects x-axis -graph translated by vector (-2, 0) -outside brackets: change is expected and affects y-axis -graph translated by vector (0, 2)

given that arcsin x = arcos y, prove that x² + y² = 1

-let arcsin x = θ -then x = sin θ -and arcos y = θ -so y = cos θ sin²θ + cos²θ = 1 so x² + y² = 1

what does the graph of √x look like?

-line curving up and to the right until almost flat -only in positive quadrant √x² = straight line coming down and bouncing off axes to go up (graph of y = x but no negative y and +- x) √x^3 = positive half of x^2 graph

solve the equation 3tan x = -1 for -180<x<360

-make tan x the subject tanx = -1/3 -find reverse of tan and use graph to find all values tan^-1 (-1/3) = -18.4 -18.4 + 180 = 161.6 -if the equation involves a square function (sin^2 0), then replace the functions with x and solve as a quadratic

when to use combinations or permutations

-permutation - associated with arranging things in different orders -order matters (eg: ABC can be rearranged to CBA) -combinations - associated with combining things -order does not matter (eg: ABC and CBA are the same combination)

given that sinx = 3/4 and x is obtuse, find cosx and tanx

-rearrange rules and substitute in known values cos^2 x = 1 - Sin^2 x cos^2 x = 1 - (3/4)^2 cos x = +- (root7)/4 -use graph to confirm whether answer is positive or negative (on cos graph, obtuse angles have negative values) so cos x = - (root 7)/4 -use calculated value to find tan x by rearranging and substituting in rule tanx = sinx / cosx tanx = (3/4)/((root7)/4) tanx = -3/(root 7)

tan 3x = 0, solve for 0<x<360

-replace 3x with y, and solve for y 3x = y -> tan y = 0 -> tan^-1(0) = 0 -find new range 3(0<x<360) = 0<y<1080 -add 180 onto 0 to find repeats until the maximum is reached 0, 180, 360, 540, 720, 900, 1080 (values of y) -since 3x = y, find the values of x by dividing by 3 0, 60, 120, 180, 240, 300, 360 -same method can be applied to x + 10 for example

pascal's triangle

-represents multipliers used to find coefficients in binomial expansions ((x + y)ⁿ is a binomial) -the numbers 1 across from 1s are used to find the correct row (equal to n) -the triangular numbers can be found next to the Ns eg: find the multipliers of (x + y)³ -use pascal's triangle to find the 3rd row -multipliers are 1 3 3 1

it can be shown that (1 - x/2)^8 is approximately 1 - 4x + 7x² - 7x³. Use this answer to find an approximate value of 0.9^8

-set inside of bracket equal to decimal 1 - x/2 = 0.9 1 - 0.9 = x/2 0.1 = x/2 x = 0.2 -sub x in position of expanded polynomial and simplify -does not have to be whole polynomial, estimate will be very close

the curve y = 2ˣ is stretched by a factor of 1/4 parallel to the x-axis. What is the equation of the new curve?

-since factors affecting x are not as expected: a stretch of factor 1/4 is made by doing f(4x) -f(4x) -> y = 2⁴ˣ

it can be shown that the binomial expansion of (2 + x)⁵ is 32 + 80x + 80x² + 40x³ + 10x⁴ + x⁵. Hence find the coefficient of y³ in the expansion of (2 + 3y + y²)⁵

-since the brackets are similar, 3y + y² = x -sub 3y + y² in x spot in expanded binomial A y³ will be created with the parts 80x² and 40x³ -because (3y + y²)² = 9y² + 6y³ + y⁴ (from 80x²) -and (3y + y²)³ = 27y³ ... -add the expanded values with y³ into the brackets and combine 80(6y³) + 40(27y³) = 480y³ + 1080y³ = 1560y³

a graph of -x^z has intercepts at x = -1 twice and x =2, as well as y = 4. Find the equation

-starting off the equation using x intercepts: (x + 1)^2 (x - 2) -the graph is negative so a negative coefficient is needed -(x + 1)^2 (x - 2) -to achieve y = 4, we need to multiply the current value by 2 without affecting the x intercepts, so change the coefficient -2(x + 1)^2 (x - 2) -expand: -2x^3 +6x +4

find the value of the term in the expansion of (x - 1/x)⁸ which is independent of x

-the term with no x is when the powers of x and -1/x are equal because they will multiply and cancel out so (x)⁴(-1/x)⁴ 8C4(x)⁴(-1/x)⁴ = 70

Find the equation of the graph with roots (1, 0) and (9, 0) that crosses the point (3, 36)

-use a multiplier k and find it by subbing in x and y of point y = k(x - 1)(x - 9) 36 = k(3 - 1)(3 - 9) 36 = k(-12) k = -3 -expand the brackets to form the equation y = -3(x - 1)(x - 9) y = -3x² + 30x - 27

Expand and simplify (2x + 3)³

-use nCr to find multipliers: 1 3 3 1 -write out sequence with powers and multipliers 1(2x)³(3)⁰ + 3(2x)²(3)¹ + 3(2x)¹(3)² + 1(2x)⁰(3)³ -expand and simplify (8x³) + 9(4x²) + 27(2x) + 27 8x³ + 36x² + 54x + 27

Find the coefficient of x⁵ in the expansion of (3x + 4)⁵

-use nCr to find x⁵ term in 7th row of pascal's triangle -since it starts at x⁷, x⁵ is the second term in the sequence (7 - 5 = 2) 7C2 = 21 -write out term with multipliers and powers to solve 21(3x)⁵(4)² 21(243x⁵)(16) 81648x⁵ -coefficient of x⁵ = 81648

A committee of four is to be selected from five boys and four girls. The members are selected at random. What is the probability that the committee will be made up of more boys than girls?

-write down possible ways 4 boys 0 girls 3 boys 1 girl -find combinations for each scenario from their groups 4 boys 0 girls: 5C4 = 5 3 boys 1 girl: 5C3 = 10 and 4C1 = 4, 10 x 4 = 40 -add possible ways' combinations 40 + 5 = 45

0! and how to find probability

1 probability = 1/total combinations or permutations eg: probability of picking ABC from alphabet in that order 26P3 = 15600 so probability = 1/15600 eg: probability of picking ABC from alphabet in any order 26C3 = 2600 so probability = 1/2600 eg: probability of winning lottery from 6 numbers from group of 59 59C6 = 45057474 so probability = 1/45057474

f(x) = (x + 1)^2(2 - x) what are its intercepts? And find f(2x)

1 x 1 x 2 = 2 (y intercept) - don't modify brackets to find y -for f(2x), sub 2x into x spot of f(x) and expand f(2x) = (2x + 1)^2(2 - 2x) f(2x) = -8x^3 + 6x + 2

(1 = 2tan 2x)/2

1/2 = tan 2x y = 2x -> 1/2 = tan y -dividing does not affect coefficients of theta

area of triangle

1/2absinC

(x - 2) to the power of 1, 2, 3, 4 representation on graph

1: crosses through root 2: touches root but bounces off (turning point) 3: changes direction on root (goes flat on point) 4: changes direction on root (goes flat on point)

2Cosx - Sinx = 0

2cosx = sinx 2 = sinx/cosx 2 = tanx

translate 3(x - 1)(x + 2)(x + 5) by (3, 0)

3(x - 4)(x - 1)(x + 2) -take 3 from each bracket

find the point of intersection of the graphs y = 4cos(x) and y = 2sin(x) in the form (arctan x, surd)

4cos(x) = 2sin(x) tan(x) = 2 -since tan(x) = 2 when 4cosx and 2sinx intercept: -draw a triangle with 2 as the opposite of angle and 1 as the adjacent to angle -hypotenuse = √2² + 1² = √5 cos x = 1/√5 4cosx = 4/√5 final coordinate of intersection: (arctan 2, 4/√5)

practice dividing polynomials

6x^3 + 28x^2 - 7x + 15 / x + 5

5 - 5sin^2(x)

=5(1 - sin^2 x) =5(cos^2 x)

CAST diagram

CAST anticlockwise from bottom right quadrant -starts at 0 (right), 90(up), 180(left), 270(down) -used to find if sin cos or tan will be negative or positive depending on type of angle

prove the identity: cos^2 0 - sin^2 0 = 2cos^2 0 -1

LHS = cos ^2 0 - sin^2 0 = cos^2 0 - (1 - cos^2 0) = cos^2 0 = -1 + cos^2 0 = 2Cos^2 0 -1 =RHS

y = SinX + a vs y = Sin(X + a)

SinX + a = shift parallel to y-axis with scale factor a Sin(X + a) = shift parallel to x-axis with scale factor of -a

sine rule + relationship with A and a

a/sinA = b/sinB = c/sinC A = angle opposite side 'a'

y = aSinX vs y = Sin(aX)

aSinX = stretch parallel to y, with scale factor a Sin(aX) = stretch parallel to x, with scale factor 1/a

Cosine rule

a² = b² + c² - 2bcCosA -keep answers as surds and fractions where possible, do not use decimals

discontinuous curve vs continuous

discontinuous = curve with different branches (eg: 1/x) continuous = curve with no breaks (eg: x^2)

y = f(-x) vs y = -f(x)

f(-x) = graph reflected in y-axis -f(x) = graph reflected in x-axis

powers of binomials into the sequence

for (x + y)ⁿ: -the power of x starts at n and decreases to 0 -the power of y starts at 0 and increases to n when powers are unknown: y-power = number of terms into sequence - 1 (eg: 4th term = y³) x-power = n - y power (powers must always add up to n)

degree of polynomial

greatest power in polynomial

nCr function

n = power of binomial r = number into sequence (starts at 0) n!/r!(n - r)! eg: 5C0 = 5!/0!(5 - 0)! = 5!/5! = 1 -so multiplier of first term in (x + y)⁵ sequence is 1 -found on calculator by doing MATH -> OPTN -> PROB

factorial (n!)

n! = n(n - 1)(n - 2)(n - 3)... etc eg: 8! = 8x7x6x5x4x3x2x1

what is the value of permutation + how to use it

n!/(n - r)! for nPr n = total number r = number selected -eg: how many ways can i arrange 2 books from a set of 4 books? 4p2 = 210 xPx = x! -eg: how many different ways can we arrange 5 books on a shelf? 5p5 or 5! = 120

f:x -> x^2

same as f(x)

f(x) = ..., g(x) =... find f(x) = g(x)

set equations equal to each other and solve for x (where curves intersect)

rules to find other angle if 0<x<360, and repeats

sin B = sin(180 - A) cos B = cos(360 - A) tan B = tan(180 + A) -for repeats of cos and sin, + 360 to both answers -for repeats of tan, + 180 to both answers -always check values on graph line up correctly

trigonometry rules

sin^2 0 + cos^2 0 = 1 tan0 = sin0/cos0 0 = theta

if x is obtuse and sinx = 0.53, find cosx and tanx

sin^2 x + cos^2 x = 1 (0.53)^2 + cos^2 x = 1 cos^2 x = 0.72 cos x = +- 0.85 -using cast diagram, obtuse angle means cos is negative so cosx = -0.85 sinx/cosx = tanx 0.53/-0.85 = 0.63 = tanx

2sinxcosx + sinx = 0

sinx(2 cosx + 1) sinx = 0 and cosx = -1/2 x = 0, 120, 240, 360

graph shape of y = 1/x, y = 1/x^2, y = 1/x^3 etc

y = 1/x: one half curve in the top right quadrant, opposite in opposite quadrant y = 1/x^2: one half curve in top two quadrants -asymptotes at origin, because x = 0 or y = 0 is undefined -if the power of x is odd, the curves are in the opposite quadrants and become tighter the higher the power -if the power of x is even, the curves are in the top quadrants, the curves become tighter the higher the power

sin^2(3x) + cos^2(3x) = 1

y = 3x sin^2(y) + cos^2(y) = 1 -so the identities are not dependent on coefficients of theta

inverse proportion

y decreases as x increases y = k/x

direct proportion

y increases as x increases y = kx