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Pure Maths A Level #1
Terms in this set (80)
Length of a line
How do you know if two lines are perpendicular?
When you multiply the gradients together they should equal -1. The general rule is perpendicular = -1/m. Or you could use Pythagoras to show there is a right angle
How do you know if two lines are parallel?
The gradients should be the same
What should you always remember when finding the square roots?
The answer could be positive or negative
What is the quadratic formula?
x = (-b ± √b^2 - 4ac)/2a (will get 2 answers)
Describe the proof for the quadratic formula
1) Divide by a on both sides 2) Complete the square for ax2/a + bx2/a 3) Expand the second bracket, (b/2a)2 4) Multiply by c/a by 4 and add the fractions 5) Factorise into brackets 6) Add the second bracket to the right side 7) Square root both sides 8) Subtract the b/2a on both sides 9) Simplify the answer
What's the alternative formula you can use for completing the sqauare?
ax2+bx+c = a(x + b/2a)2 + (c-b2/2a) You then simplify the second bracket and solve the equation
Horizontal translation, +a move left, -a move right, use vector (-a,0)
Vertical translation, +a move up, -a move down, use vector (0/a)
Horizontal stretch by scale factor 1/a
Vertical stretch by scale factor a
Reflection of the graph in the x axis
The graph is reflected in the y-axis
Equation for real roots
b2-4ac>0 Quadratic graph passes through x axis
Equation for equal roots
b2-4ac=0 Quadratic graph touches the x axis
Equation for no real roots
b2-4ac<0 Quadratic graph doesn't touch x axis
What is a linear model?
Shows the relationship between two variables, x and y. The graph has a straight line and the variables are related by y = mx + c
What are the qualities of direct proportion?
The two quantities increase at the same rate and the straight line passes through the origin (0,0). Therefore y is directly proportional to x so y = k x where k is the real constant
How do you decide whether a linear model is appropriate?
- Points should be on the line or close to the straight line, the further away the points then the less appropriate it is
- A mathematical model is an attempt to represent a real life situation, however assumptions are needed to sketch it
What does k represent?
K represents the constant and the gradient, so on a extension graph then k represents the increase in extension in cm when the mass is increased by 1 gram.
What is the equation of a circle?
(x-a)^2 + (y-b)^2 = r^2 center (a,b) radius r
What is a perpendicular bisector, tangent and a chord?
- The perpendicular bisector is the shortest distance between the center and a chord is a right angle and is also passes through the center.
- The tangent is a straight line that intersects the circle at one point.
- A chord is a line segment that joins two points on the circumference.
What are the rules for a triangle in a circle?
A triangle consists of three vertices and you can draw a circle around those three vertices. The circle is called the circumcircle and the center is called the circumcenter, this is also the point where all the perpendicular bisectors intersect.
What are the rules for a right-angled triangle in a circle?
For a right angled triangle, the hypotenuse of the triangle is a diameter of the circumcircle. If angle PRQ is 90o then R lies on the circle with the diameter PQ. The angle in the semi circle is always a right angle.
How do you find the center of a circle given any three points on the circumference?
Find the equations of the perpendicular bisectors of two different chords. Find the coordinates of the point of intersection of the perpendicular bisectors.
What is the graph for real roots?
The regions outside the graph are shaded, above the x-axis if the graph is positive, below if it's negative. -2 > x & 2 < x
What is the graph for no real roots?
The regions inside the graph are shaded, below the x-axis if the graph is positive, above if it's negative. -2 < x < 2
Formula to rationalise 1/ square root (a)
Multiply the numerator and denominator by square root (a)
Formula to rationalise 1/a + square root (b)
Multiply the numerator and denominator by a - square root (a)
Formula to rationalise 1/a - square root (b)
Multiply the numerator and denominator by b + square root (a)
Coordinates of the turning point
Completing the square formula: a(x+p)^2 + q, turning points = (-p,q)
Colour code for linear inequality number lines
Black circles: less/greater than or equal to. White circles: less/greater than
Inequality for when the graph of y = f(x) is below the graph of g(x)
f(x) < g(x)
Inequality for when the graph y = f(x) is above the curve y = g(x)
f(x) > g(x)
Lines to represent inequalities on graphs
Dotted: greater/less. Solid line: greater/less than or equal to
Equation of a cubic graph
y = (ax)^3 + (bx)^2 + cx + d
Equation of a quartic graph
y = (ax)^4 + (bx)^3 + (cx)^2 + dx + e (looks like a w if it is positive and an m if it is negative)
What is a polynomial?
A polynomial is a finite expression with positive whole number indices and you can use long division to divide them
What is the factor theorem?
If f(x) is a polynomial then if f(p) = 0, then (x-p) is a factor of f(x) or if (x+p) is a factor if f(x), then f(-p) = 0
What is a theorem and what is a conjecture?
A theorem is a statement that has been proven and a conjecture has not been proven
How do you prove by deduction?
You start from know facts and definitions, then use logical steps to reach a conclusion. You use numbers for a demonstration and then use algebra to prince that statement and you then write a statement of proof
How do you prove by exhaustion?
You break the statement into smaller cases and break each case separately, but this is better for statements with small cases.
How do you disprove a mathematical statement?
You can use counter-examples which is an example that does not work for the statement and you only need to give one example.
A pattern for finding the coefficients of the terms of a binomial expansion and it is formed by adding the adjacent pairs of numbers to find the numbers on the next row. The first row expands brackets/numbers to the power of 0.
What is factorial notion?
3! = 3 x 2 x 1 or 2! = 2 x 1
What is the formula using factorial notion to find entries on Pascal's triangle?
n C r = (n/r) = n!/r!(n-r)!, the rth entry would be n-1 C r-1 = (n-1 / r-1)
What is the binomial expansion?
(a+b)^n = a^n + (n C 1)(a)^n-1(b) + (n C 2)(a)^n-2(b)^2 ..... (n C r)(a)^n-r(b)^n
a² = b² + c² - 2bcCosA
a/sinA = b/sinB = c/sinC
Sine rule (angle)
sinA/a = sinB/b = sinC/c
Two outcomes of sine rule for a missing angle
sin(X) = sin (180 - X)
Area of a triangle
Cosine Rule for Angles
Trigonometric ratio relating sin and cos
sin^2(X) + cos^2(X) = 1
1 - sin^2(X) = cos^2(X)
1 - cos^2(X) = sin^2(X)
Trigonometric rations relating sin, cos and tan
tan(X) = sin(X) / cos (X)
What is the principle value?
The angle you get when you use the inverse trigonometric functions on your calculator
What is a vector?
A quantity that has both magnitude and direction
What is the resultant vector?
The sum of two or more vectors
What are the two forms of writing vectors?
Column vector and component vector (using i for east and west and j for north and south). Using pi + qj is also a two dimensional vector
How do you find the magnitude of the vector?
[a] = square root (x^2 + y^2)
What is a unit vector?
A vector with the magnitude of 1, a unit vector in the direction of a is a/[a]
How do you calculate vector AB?
AB = OB -OA where O is the origin
How do you calculate speed and distance from vectors?
Speed is the magnitude of the velocity vector and distance between A and B is the magnitude of AB
What is the formula for the derivative to differentiate from first principles?
Equation of the tangent to the curve
y - f(a) = f'(a)(x-a)
Equation to the normal of the curve
y - f(a) = -1/f'(a) (x-a)
How do you know if a function is increasing on a interval?
[a,b] f' > 0 for a < x < b
How do you know if a function is decreasing on a interval?
[a,b] f' < 0 for a < x < b
Types of stationary points: Local minimum
f'(x-h) - Positive
f'(0) - 0
f'(x+h) - Negative
f''(a) > 0 or = 0
Types of stationary points: Local maximum
f'(x-h) - Negative
f'(0) - 0
f'(x+h) - Positive
f''(a) < 0 or = 0
Types of stationary points: Points of inflection
f'(x-h) - Negative / Positive
f'(0) - 0 / 0
f'(x+h) - Positive / Negative
f''(a) = 0
Sketching gradient functions
Min/max = cuts x-axis
Inflection = touches x axis
Positive gradient = above
Negative gradient = below
Vertical asymptote = same
Horizontal asymptote = horizontal asymptote at x-axis
How do you use differentiation in modelling?
f'(x) represents the rate of change, so you can use it to find the rate in volume for example
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