Pure 1 Flashcards by Aditya Jadhav

a^m x a^n =

a^m+n

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a^m / a^n =

a^m-n

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(a^m)^n =

a^mn

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(ab)^n =

a^n b^n

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root(ab) =

root(a) x root(b)

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root(a/b) =

root(a) / root(b)

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quadratic formula

x = (-b ± root(b^2 - 4ac)) / 2a

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What is the domain

The set of possible inputs for a function

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What is the range

The set of possible outputs for a function

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What are the roots of a function?

The values of x for which f(x) = 0

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if the completed square was: f(x) = (x+p)^2 + q what would the turning point be?

(-p, q)

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What is the discriminant?

b^2 - 4ac

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How many roots does b^2 - 4ac > 0 have?

Two distinct real roots

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How many roots does b^2 - 4ac = 0 have?

One repeated root

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How many roots does b^2 - 4ac < 0 have?

No real roots

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What is x is greater than y or x is smaller than or equal to b in set notation?

{x: x is greater than y} u {x: x is smaller than or equal to b}

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What is x is greater than y and x is smaller than or equal to b?

{x: y is smaller than x which is smaller than or equal to b}

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Is the circled filled in or empty for x is greater than or equal to y?

Filled in

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Is the circled filled in or empty for x is greater than y?

Empty

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Steps to solve a quadratic inequality

.Solve like a normal quadratic
.Draw a graph
.If solving for ax^2 + bx + c is greater than 0, then the answer is the x values above the x axis
.If solving for ax^2 + bx + c is smaller than 0, then the answer is the x values below the x axis

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Is y is greater than f(x) dotted or solid?

Dotted

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Is y greater than or equal to f(x) dotted or solid?

Solid

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and

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if x^3 is positive where does the graph start?

Bottom left

if x^3 is negative where does the graph start?

Top left

f(x) + a

Vertical by a

f(x + a)

Horizontal by -a

a X f(x)

Stretch by factor of a vertically

f(a X x)

Stretch by a scale factor of 1/a horizontally

-f(x)

Reflection of f(x) in x axis

f(-x)

Reflection of f(x) in the y axis

formula for m

(y2 - y1) / (x2 - x1)

formula for finding equation of a straight line

y - y1 = m(x - x1)

What do parallel lines have that are the same?

Gradient

How are the gradients of two perpendicular lines found?

Their gradients will multiply to make -1

How to find length of straight line between two points?

Pythagoras’ formula

How to find mid point of two points

((x1 + x2) / 2, (y1 + y2) / 2)

Where does a perpendicular bisector pass through?

The mid point

The equation of a circle with centre (0,0) and radius = r is?

x^2 + y^2 = r^2

The equation of a circle with centre (a,b) and radius = r is?

(x - a)^2 + (y - b)^2 = r^2

What is a tangents relationship with the radius?

Perpendicular

What will the perpendicular bisector of a chord do?

Go through the centre of a circle

The angle in a semicircle is always a…

Right angle

What does it mean to prove a mathematical statement true by exhaustion?

Breaking the statement into smaller cases and proving each case separately

What does it mean to prove a mathematical statement not true by counter-example?

A counter-example is one example that does not work for the statement, you do not need to give more than one example, as one is sufficient to disprove a statement

What does it mean to prove a mathematical statement true by deduction?

Starting from known factors or definitions, then using logical steps to reach the desired conclusion

What is the general formula for binomial expansion? Using (a+b)^n as an example

a^n + nC1a^(n-1)b + nC2a^(n-2)b^2 + nC3a^(n-3)b^3 + … + b^n

Cosine rule for a missing side

a^2 = b^2 + c^2 - 2bc cos(A)

Cosine rule for a missing angle

cos(A) = (b^2 + c^2 - a^2) / 2bc

Sine rule for finding missing side

a / sin(A) = b / sin(B) = c / sin(C)

Sine rule for finding missing angle

sin(A) / a = sin(B) / b = sin(C) / c

Trigonometric formula for area of a triangle

area = 1/2 ab sin(C)

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

sin(x) / cos(x)

tan(x)

A vector has both magnitude and direction? T or F

If PQ–> = RS–> then

The line segments PQ and RS are equal in length and are parallel

If AB–> = -BA–> then

AB is equal in length, parallel and in the opposite direction to BA

AB–> + BC–> =

AC–>

What is lamda?

A non-zero scalar

What is a unit vector? How are they usually denoted?

A vector of length 1, usually denoted by i and j, for x and y respectively

How would you find the magnitude of: a = xi + yj

[a] = sqrt(x^2 + y^2)

What are position vectors?

Vectors giving the position of a point, relative to a fixed origin

Equation for differentiating from first principles

f’(x) = lim of h→0 of (f(x+h) - f(x)) / (h)

How do you find the derivative usually?

Times by the power and then minus one from the power

if f’‘(a) > 0 then

The point is a local minimum

if f’‘(a) < 0 then

The point is a local maximum

Generally how do you integrate?

Add one to the power and then divide the coefficient by the value of the new power

If not integrating to find an area on a graph, what must you remember to do?

Constant of integration = +c

How must normal integration be set out?

Integration squiggle, equation wanting to integrate, dx

Where do you put the limits on definite integrals?

At the top and bottom of the squiggly line

Which limits go where?

Highest value on the top

Which limit value gets taken away from which one?

Top minus bottom

Stages of definite integrals?

Squiggly line, square bracket, two normal brackets

When integrating to find area, what area does it find? If it's under the x-axis what must you do?

The area between the curve and the x-axis, if it is under the x-axis it will be negative so you need to flip it

Why is e special?

The derivative of e^x = e^x

What is a^x = n as a log?

log(a)n = x

log x + log y =

log xy

log x - log y =

log x/y

log x^k =

k * log x

ln x =

log(e) x

e^(ln x) =

ln e^x = x

How can you use logs to understand non-linear data?

Mess around with the equation till you have a linear equation with logs, e.g. (log y = n * log x + log a) is the same as y = mx + c

If y = ax^n then the graph of log y against log x will be

A straight line with gradient n and vertical intercept log a

If y = an^x then the graph of log y against x will be

A straight line with gradient log n and vertical intercept log a