A level maths pure 1

Expanding brackets

Expand and simplify (2x+3)(x-4)(3x-2)

Simplifying algebraic fractions

Simplify fully x²+5x/x²+7x+10

Factorising quadratics

Factorise 2x²-9x+10

Index laws

Simplify fully (32x^5)^-2/5

Harder index laws

Simplify fully x²(4x^-1/2)²

Surds

Expand and simplify (5+2√2)(3-√8)

Rationalising the denominator

Rationalise 2+√2/5-3√2

Solving quadratics

Solve 2x²-x-1=0

The quadratic formula

Solve 3x²-11x-13=0 Give your solutions correct to 3 significant figures.

Completing the square

By completing the square, find the turning point for the graph with equation y=2x²+16x+26

Solving equations by completing the square

Write 2x²-8x-16 in the form p(x+q)²+r where p, q and r are integers.

Hence solve the equation 2x²-8x-16=0, giving your answer in the form a±b√3 where a and b are integers.

Negative quadratics

Sketch the graph of y=5-3x-x², showing the coordinates of the turning point and any points of intersection with the coordinate axes.

The discriminant explained

Does the equation y=x²+4x+2 have two distinct real roots, two equal real roots or no real roots?

Solving problems with the discriminant

4x-5-x²=q-(x+p)² where q and p are integers.

a)Find the value of p and the value of q

b)Calculate the discriminant of 4x-5-x²

c)Sketch the curve with equation y=4x-5-x², showing clearly the coordinates of any points of intersection with the axes.

Modelling with quadratics

An arena has 25,000 seats. The arena known from past events that they will only sell 10,000 tickets if the each ticket costs £30. They also expect to sell 1,000 more tickets every time the price goes down by £1. The number of tickets sold t can be modelled by the linear equation t=M-1000p where £p is the price of each ticket and M is a constant.

a)Find the value of M

The total revenue, £r, can be calculated by multiplying the number of tickets sold by the price of each ticket. This can also be written as r=p(M-1000p).

b)Rearrange r into the form A-B(p-C)², where A, B and C are constants to be found.

c)Using your answer to part b or otherwise, work out how much the arena should charge for each ticket if they want to make the maximum amount of money.

Linear simultaneous equations

Solve the simultaneous equations: 8x+3y=2 and 3x-2y=-5.5

Quadratic simultaneous equations with a circle meets a line

Solve the simultaneous equations: x²+y²=13 and x=y-5

Quadratic simultaneous equations with a curve meets a line

Solve the simultaneous equations: y+4x+1=0 and y²+5x²+2x=0

Graphical simultaneous equations

Solve the simultaneous equations: x²+y”=13 and x=y-5

Linear inequalities using set notation

Use set notation to describe the set of values for x for which: 3x-5<x+7 and 5x>x-8

Quadratic inequalities

Find the set of values of x for which both 2x²-5x-3>0 and 8x-7<5x+5

Regions

On the grid, shade the region that satififes these inequalities: y<3x, y>0 and 4x+3y<12

Sketching cubic graphs

Sketch the graph of y=x³-6x²+9x showing clearly the coordinates of the points where the curve meets the coordinate axes.

Sketching quartic graphs

Sketch the graph of y=x(3-x)(x²+2x+4), showing clearly the coordinates of the points where the curve meets the coordinate axes.

Reciprocal graphs and asymptotes

Sketch the graph of y=3/x-1, showing any points at which the curve crosses the coordinate axes and writing down the equations of the asymptotes of the curve.

Intersecting graphs problems

On the same axes, sketch the graphs with equations: y=x(x+1)(x-4) and y=2/x+1

Graph transformations explained

The diagram shows part of the curve with the equation y=f(x). The maximum point of the curve has coordinates (2,3). Write down the coordinates of the maximum point with the equation: y=f(x-2)

y=f(x)-1

y=f(2x)

y=3f(x)

y=f(-x)

y=-f(x)

Translating functions

The diagram shows part of a curve with equation y=f(x). Draw a sketch of the graph with equation y=f(x-4)

y=f(-x)

Equation of a line

Find the equation of the line that passes through the coordinates (5,7) and (3,-1)

Perpendicular lines

Find the equation of a line perpendicular to y=2-3x that passes through the point (6,5)

Area with coordinate geometry

The straight line l1 has equation 4x-y=0 and l2 with equation 2x+3y-21=0 intersect and point A. Work out the triangle of AOB where B is the point where l2 meets the x axes.

Modelling with linear graphs

A container is leaking water at a constant rate. The water remaining is recorded at certain intervals. At the start the depth was 19.1 cm and after 100 seconds the depth was 6.1 cm. Deduce an equation in the form d=at+b and interpret the meaning of the coefficients a and b.

Use the mode to find the time when the container will be empty.

Midpoints and perpendicular bisectors

The line segment AB is the diameter of circle centre C, where A is (-1, 4) and B is (5,2). The line l passes through C and is perpendicular to AB. Find tue equation of line l.

Equation of a circle

A circle C has centre (3,1) and passes through the point (-2,5). Write down an equation for circle C.

Equation of a circle to find the centre

The circle C has equation x²+y²-6x+2y-15=0. Find the coordinates of the centre of C and the radius of C.

Find the coordinates of the points where C crosses the y axes.

Intersections of linear graphs and circles

Show that the line x-y-10=0 does not intersect with the circle x²-4x+y²=21

Tangents to a circle

The point P (2,-1) lies on the cirlce centre (4,6). Find the equation of the tangent to the circle at P.

Chord properties

The points A, B and C lie on the circumference of the circle. Show that AB is the diameter of the circle.

Algebraic fractions

Simplify fully 3x²+11x-4/x²-16

The factor theorem

f(x)=x³-9x²+20x-12. Use the factor theorem to show that (x-2) is a factor of f(x).

Hence, factorise fully x³-9x²+20x-12

Methods of proof with inequalities

The equation x²+(k-3)x-4k=0 has two distinct real roots. Show that k satififes k²+10k+9>0

Hence find the set of possible values for k.

Methods of algebraic proof

Given nEN, prove that n³+2 is not divisible by 8

Binomial expansion explained

Expand and simplify (x+3)³

The binomial expansion

Find the first 4 terms, in ascending powers of x, of the binomial expansion of (3-1/3x)^5 giving each term in its simplest form.

Solving binomial problems

Find the first 3 terms, in ascending powers of x, of the binomial expansion of (1+px)^10 where p is a non-zero constant. Give each term in its simplest form.

Given that, in this expansion the coefficient of x² is 9 times the coefficient of x, find the value of p.

Hence, write down the coefficient of x²

Binomial estimation

Find the first 4 terms in ascending powers of x of the binomial expansion of (1-x/4)^10. Use your expansion to estimate the value of 0.975^10, giving your answer to 3 d.p

The cosine rule

Calculate the length of QR

The sine rule

Work out the size of angle x. Give your answer correct to 3 significant figures.

Areas of triangles

The area of the triangle is 105 cm². Work out the value of x. Give your answer to 3 significant figures.

Solving triangle problems with bearings

A, B and C are 3 villages. B is 6.4 km due east of A. C is 3.8 km from A on a bearing of 210°. Work out the bearing of B from C. Give your answer to the nearest degree.

Transforming trigonometric graphs

The graph of y=sin x° for x values from -270 to 270 is shown below. On the same axes, draw the graph of y=1-sin(x) for values from -270 to 270

Graphs of sine, cosine and tangent

Exact values of trigonometric ratios

Trigonometric identities

Sin²theta +cos²theta

Equations and identities

Harder trigonometric equations

Vectors

Representing vectors

Magnitude and direction of vectors

Position vectors

Solving geometric problems

Modelling with vectors

Differentiation explained

Differentiation from first principles

Differentiating quadratics

Harder differentiation

Gradients of tangents and normals

Increasing and decreasing functions

Second order derivatives

Stationery points

Modelling with differentiation

Integration explained

Indefinite integrals

Finding functions by integrating

Definite integrals

Evaluate

Areas under curves

Areas under the x-axis

Areas between curves and lines

Logarithms explained

Laws of logarithms

Solving simple equations using logarithms

Laws of logs (Adding)

Express log₂ 4+log₂16 as a single logarithm to base 2

Laws of logs (Subtracting)

Laws of Logs (Multiplying)

Solving harder logarithmic equations

Exponential functions

Differentiating e^x

Differentiate y=2e^3x with respect to x

Solving exponential equations using natural logarithms

Solving exponential quadratics with natural logarithms

Modelling with exponentials