Pure Year 1 Flashcards
What is the cosine formula- both variations?
a^2 = b^2 + c^2 - 2bc x cos(A) Cos(A) = b^2 + c^2 - a^2 / 2bc
What is the sine formula- both variations?
sin (A) / a = sin (B) / b - angles
a / sin (A) = b / sin (B) - sides
What is pythagorasβ theorem?
a^2 + b^2 = c^2
What is a rational fraction?
A fraction with a surd in the denominator
What is 8^2/3?
4
Bottom number outside
Top number inside
How do you rationalise a surd?
Multiply by either the denominator itself if it is on its own or by itself & the opposite to either + or -
What is the quadratic formula?
x = -b +- root (b^2 - 4ac) / 2a
What is the discriminant and when does it show the number of roots a quadratic has?
b^2 - 4ac = 0 - 1 root (tangent if root repeated)
b^2 - 4ac > 0 - 2 roots
b2 - 4ac < 0 - no roots
How do you find points of intersection?
Equate the 2 equations
How do you write an inequality when the equation is greater than (>) ?
2 separate inequalities with the smaller one as less than ()
Because above the graph- opposite directions of solutions
How do you write an inequality when the equation is less than (
As one inequality because below the graph
What does the reciprocal graph y = 1 / x look like and where are itβs asymptotes?
Line in top right quadrant & bottom left quadrant with asymptotes at x = 0 & y = 0
What does the reciprocal graph y = -2 / x look like and where are itβs asymptotes?
Line in top left quadrant & bottom right quadrant with asymptotes at x = 0 & y = 0
What does the graph y = 2 / x^2 look like & where are itβs asymptotes?
Line in top left quadrant & top right quadrant with asymptotes at x = 0 & y = 0
What does the graph y = -5 / x^2 look like & where are itβs asymptotes?
Line in bottom left quadrant & bottom right quadrant with asymptotes at x = 0 & y = 0
In a function such as (a / b ) what direction does a & b control?
a controls left/right movements
b controls up/down movements
When a graph is translated, are its asymptotes translated with it?
When a graph is translated its asymptotes are translated with it
What does the translation y = -f(x) mean?
Reflection in the x-axis
What does the translation y = f(-x) mean?
Reflection in the y-axis
How do you calculate the gradient of the line between 2 points?
gradient = change in y / change in x gradient = y2 - y1 / x2 - x1
What is the product of the gradients of 2 perpendicular lines?
-1
What is unique about the gradients of 2 lines which are perpendicular to one another?
Their gradients are negative reciprocals of one another
What is unique about the gradients of 2 lines which are parallel to one another?
Their gradients are the same
How do you find the distance between 2 points?
Find the net coordinates by minusing them from one another and then use pythagorasβ theorem
Root {(x2 - x1)^2 + (y2 - y1)^2}
How do you calculate the mid point of 2 points?
Add each of the x & y coordinates separately and divide each one by 2 separately to form the midpoints coordinates
(x1 + x2/2, y1 + y2/2)
What is a perpendicular bisector and what is its gradient?
A line that passes directly in the middle of another line or 2 points and forms a 90 degree angle with that line or the line the 2 points form
Its the gradient of the negative reciprocal of the line it cuts or the line formed by the 2 points
What is the general equation of a circle?
x2 + y2 = r2
How do you find the gradient and radius of a circle when its equation is given in its expanded form? (x2 + y2 + 2fx + 2gy + c = 0)
Complete the square
What is unique about the tangent of a circle and its radius?
They touch only once and when they do they form a 90 degree angle
What is unique about the perpendicular bisector of a chord of a circle?
It will go through the centre of the circle
Define what the circumcircle of a triangle is
A circumcircle is a circle drawn using the 3 vertices of any triangle
Define what the circumcentre of a triangle is
It is the centre of the circumcircle and is the point where the perpendicular bisector of each side of the triangle intersect
What is different about the circumcircle of a right angled triangle?
The hypotenuse of the right angled triangle is the diameter of the circle because the angle in a semi circle is always a right angle
How do you find the centre of a circle when you have 3 points on its circumference
Join the points together to form 3 lines and find the perpendicular bisector of 2 of the lines. Find the midpoints of of the chords and plug into the perpendicular bisector equation to find C. Equate the 2 perpendicular bisector equations and the intersection found is the midpoint.
Define a theorem
Statement that has been proven
Define a conjecture
Statement that has yet to be proven
What is proof by deduction?
Use of algebra to find proof
What is proof by exhaustion?
Work out using all possibilities- several examples
What is disproof by counter example?
When you prove a statement wrong with one example
What is 0 factorial (0!) equal to?
1
How do you find the rth entry in the nth row of Pascalβs triangle?
n-1 C r-1
What is a natural number (like x β¬ N)?
All positive integers- therefore x is a positive integer (positive whole number)
What is a real number (like x β¬ R)?
All numbers positive or negative, decimal or integer- therefore x is any number
How do you find the area of a triangle?
- 5 x a x b x Sin (C)
0. 5 x base x height
What is a unit circle?
A circle with a radius of 1 unit
How do you find the value of a sin, cos, tan 960?
Use the cast diagram but first take away 360 until you are left with a value which is between 0 and 360 so that it minimises time takes and chances of error of going around and around the CAST diagram. Therefore you would use the CAST diagram to find the value of 240 degrees which is equal to the value of 960 degrees.
What is the CAST diagram and how do you use it?
Four quadrants are split into CAST starting with C in the bottom right
C- cos only positive
A- all positive (sin, cos, tan)
S- sin only positive
T- tan only positive
You always you go anti-clockwise for positive angles and clockwise for negative angles
What are the trigonometric identities?
sin2x + cos2x = 1
tan x = sin x / cos x
How do your prove trigonometric identities?
Draw triangle from positive x-axis of a circle and label adjacent side as X, opposite side as Y and hypotenuse as 1. Then use the SohCahToa to find values in terms of X & Y SO sin x = y and cos x = x and tan x = y/x. As the equation of a circle is x2 + y2 = r2, sin2x + cos2x = 1 & tan = y/x = tan x = sin x / cos x
What are the inverse functions of sin, cos & tan called?
arcsin, arcos, arctan
What is the resultant vector?
The sum of 2 or more vectors
What is a unit vector?
Vector of length 1
How do you find the magnitude of a vector?
Add the i and j and square root the answer- pythagoras
How can you write a vector?
Column/position vector form (x/y)
xi + yj
How do you find the gradient of any point on a curve?
Find the gradient of the tangent to that point on the curve
What is 1^-368
1
1^x where x is any number = 1
Find the turning point of the equation π β π(π β π)^π = π
The turning point (maximum) will be at (4, 5)
Notice that the turning point is a maximum because the π₯^2 term is β
Solve the equation π^4 + ππ^2 + π = π
The equation could be rewritten as (π^2)^π + ππ^2 + π = π
So you can let y=π₯^2 and you have: π¦^2 +3π¦+2=0
Now you can solve by factorising
Solve the equation π^6 + ππ^3 + π = π
The equation could be rewritten as (π^3)^2+ ππ^3 + π = π
So you can let y=π₯^3 and you have: π¦^2 + 5π¦ + 6=0
Now you can solve by factorising
What MUST you remember with linear inequalities?
If you βοΈ or β by a β number you need to reverse the inequality sign
What MUST you remember with logs and inequalities?
Logs between 0 and 1 with any base number are β and therefore when β or βοΈ by them you MUST reverse the sign
Sketch the inequality: π < π β π
In this case you want the area below the line because π¦ is less than (
What is the equation of a straight line/tangent?
π¦ β π¦1 = π(π₯ β π₯1)
What are 3 properties of circles that you need to remember?
1) The angle in a semi-circle is a right angle
2) If you draw a line from the centre of the circle, perpendicular to a chord, then the line will bisect the chord (halfway point of the chord)
3) Any tangent to a circle is perpendicular to the radius at the point where it touches the circle
What is the binomial probability equation and what does it mean?
π·(πΏ) = ππͺππ^(x)q^(nβx) Where: π = number of trials π₯ = number of successes π = probability of success π = probability of failure (i.e. (1 β π))
How would you solve the following:
If you throw a die 8 times what is the probability of throwing three 6s?
π = 8 π₯ = 3 (we want 3 6s) π = 1/6 (probability of throwing a 6) π = 5/6 (probability of NOT throwing a 6) π(3 6π ) = 8πΆ3 (1/6)^3 (5/6)^5 = 0.104
Differentiate π = π^2 + ππ from first principles
π¦ = π₯^2 + 4π₯
Lim as h->0 = [(π₯+h)^2 +4(π₯+h)βπ₯^2 - 4π₯] / [h]
β¦ in first principles formula f(x+h) = (π₯+h)^2 +4(π₯+h)
AND f(x) = π₯^2 - 4π₯
β¦ as h β 0 we have
= [2xh + (h)^2 + 4h] / [h]
= 2π₯ + h + 4 = 2x + 4
What does the second derivative tell you?
Tells you where the stationary point(s) is/are
If dπ¦ > 0 and itβs a minimum (positive is minimum)
If dπ¦ < 0 and itβs a maximum (negative is maximum)
What MUST you remember when integrating?
INCLUDE C
What is the 1 exception to including the C when integrating?
When definite integrals with limits
What must you remember with integration with limits?
β answer = area above π₯-axis
β answer = area below π₯-axis
IGNORE β when considering area
If there is mixture (above and below)- find each area separately and add areas (ignoring β sign)
What does |π| mean?
Magnitude of a- note ALSO means modulus- ALWAYS POSITIVE- magnitude scalar quantity β¦ ALWAYS positive too
How can you write vectors?
1) Component form
2) Magnitude, direction form
What does component form look like?
1) Use of i and j
What does magnitude direction form look like?
(Magnitude of vector, angle of vector made with horizontal)
E.g. (4, 40) β¦ = Turn 40Β° from horizontal and draw line with length of 4
How do you find the resultant vector?
Add the vectors involved
What must you remember with position vectors?
ββββββ ββββββ π¨πΆ = βπΆπ¨
ββββββ ββββββ ββββββπ¨π© = πΆπ© β πΆπ¨
βββββββ ββββββ πΆπ΄ (midpoint) = πΆπ¨ + 1/π π¨π©
(IMAGINE ARROWS ABOVE ALL)
How do you find the distance between 2 points?
Pythagoras
|π¨π©|=β(ππ βππ)^π +(ππ βππ)^π
Find the distance between the points π¨ and π© given the vectors |ππ΄| = 3π β π and |ππ΅| = 5π + 2π
|π΄π΅|=β(π2 βπ1)^2 +(π2 βπ1)^2 βββββ |π΄π΅|=β(5β3)^2 +(2ββ1)^2 βββββ |π΄π΅| = β(2)^2 + (3)^2 βββββ |π΄π΅| = β13
What does the a^x graph look like?
- upwards sloping graph- same shape as e^x
- always crosses the π¦-axis at1(π^0 =1)
- the π₯-axis is an asymptote as you can never get a y- value of 0 (π^x β 0)
SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
What does the e^x graph look like?
- upwards sloping graph- same shape as a^x as you have just replaced π with e
SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
What do you do if you canβt remember what a graph looks like or what its shape is?
- Sub in values into its equation to plot a few points
- NOTE- math error typically means asymptote
- SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
What is the answer to: 2^[log(base 2) (5)]and why?
5- because 2 and 2^log(base 2) cancel each other out
What is the answer to: log(base 2) (2^5) and why?
5- because log(base 2) and 2 cancel out
What is the answer to: e^ln(7) and why?
7- because e and ln cancel out
What is the answer to: ln(π^7) and why?
7- because e and ln cancel out
What does the graph of ln(x) look like?
- increasing graph but a decreasing gradient
- always crosses the π₯-axis at 1 (ln1 = 0)
- the π¦-axis is an asymptote as you cannot get an answers for ln(0) (try it on your calculator, you will get an error - you canβt raise e to any power and get the answer 0)
SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
What are the laws of logs?
1) π₯π¨π (π) + π₯π¨π (π) = π₯π¨π (ππ)
2) π₯π¨π (π) β π₯π¨π (π²) = π₯π¨π (π/π)
3) π₯π¨π (ππ) = ππ₯π¨π (π)
4) π₯π¨π (π) = π
5) log(less than 1) = negative β
What MUST you remember about logs that are less than 1?
They are negative β
β¦ with inequalities MAKES SURE YOU SWITCH THE SIGH WHEN DIVIDING β OR MULTIPLYING βοΈ BY THEM
Solve the equation: π^(πβπ) = π
log 3^(π₯β5) = log 2 (π₯ β 5)log3 = log 2 (π₯ β 5) = (log2)/(log3) (π₯ β 5) = 0.6309 π₯ = 0.6309 + 5 π = π. ππππ
How do you plot π = ππ^π and β¦ find k and n?
log(π¦) = log(ππ₯^π) log(π¦) = log(π) + log(π₯^π) log(π¦) = log(π) + πlog(π₯) log(π¦) = π log(π₯) + log(π) Y = MX + C ... gradient = π ... intercept = log k
How do you plot π=ππ^π and β¦ find a and b?
log(π¦) = log(ππ^π₯) log(π¦) = log(π) + log(π^π₯) log(π¦) = log(π) + π₯log(π) log(π¦) = π₯log(π) + log(π) Y = MX + C ... gradient = logπ ... intercept = logπ
