Course

Cambridge A Level Mathematics: Pure 1 (9709 Paper 1)

Name: Cambridge A Level Mathematics: Pure 1 (9709 Paper 1)
Price: 2499 BDT

Paper 1 of 9709, the paper every AS and A Level candidate sits: quadratics, functions, coordinate geometry, circular measure, trigonometry, series, differentiation and integration, taught step by step with models you can move.

100 lessons18h 3mEnglish

What you'll learn

Solve quadratic equations and inequalities, complete the square, and use the discriminant to count roots
Find the range, composite and inverse of a function, and describe translations, reflections and stretches of a graph
Work with straight lines and circles on coordinate axes, including tangents and points of intersection
Use radians to find arc lengths, sector areas and segment areas
Sketch the graphs of sin, cos and tan, use exact values and identities, and find every solution of an equation in an interval
Expand (a + b)ⁿ and solve problems with arithmetic and geometric progressions, including the sum to infinity
Differentiate powers of x and composite functions, and use the derivative for tangents, normals, stationary points and rates of change
Integrate, evaluate definite integrals, and find areas and volumes of revolution
Answer Paper 1 questions with full working, the command words and the rounding the mark scheme expects

Requirements

Cambridge IGCSE or O Level Mathematics, or equivalent
Comfortable expanding brackets, factorising, rearranging formulas and reading a graph
A scientific calculator

Course content

100 lessons · 18h 3m

1.1 · Quadratics

Solve quadratic equations three ways, complete the square to find the vertex, solve inequalities, count roots with the discriminant, and decide whether a line meets, touches or misses a curve.

0/13

Solving quadratic equations by factorisation

Put the equation equal to zero, split it into two brackets, and set each bracket to zero.

Preview12m 2.

Completing the square

Halve the number in front of x, square it, and write ax² + bx + c as a(x + p)² + q.

Preview12m

Using the completed square form to sketch a graph

Read the vertex and the line of symmetry straight from a(x + p)² + q, then sketch the curve.

12m

The quadratic formula

Complete the square on ax² + bx + c = 0 once, and the formula falls out. Use it when factorising fails.

12m

Solving simultaneous equations (one linear, one quadratic)

Make one letter the subject of the linear equation, substitute, and solve the quadratic you get.

12m

Solving more complex quadratic equations

Spot a quadratic hiding in x⁴, √x or tan x, replace it with u, solve, then go back to x.

12m

Maximum and minimum values of a quadratic function

A square is never negative, so the completed square gives the greatest or least value. Fence the plant stall.

12m

Solving quadratic inequalities

Find where the curve crosses the x-axis, sketch it, and read off where it is above or below.

12m

The number of roots of a quadratic equation

The discriminant b² − 4ac says two roots, one repeated root or no real roots before you solve.

12m

10.

Intersection of a line and a quadratic curve

Substitute the line into the curve, then the discriminant decides: meets twice, touches, or misses.

12m

11.

Quadratics: timed practice

A timed drill of mixed quadratics questions, then practice problems from easy to exam level, each with a worked solution.

10m

12.

Quadratics: exam practice

Paper 1 style questions with marks, full worked solutions and the mark scheme for every step.

15m

13.

Quadratics: summary

The key points, formulas and common mistakes of the chapter on one screen, with flashcards.

1.2 · Functions

Domain and range, one-one functions, composite and inverse functions, the mirror line y = x, and moving graphs by translations, reflections and stretches.

0/14

Definition of a function

A function gives exactly one output for each input. The inputs are the domain; the outputs are the range.

Preview12m 2.

Finding the range of a function

Sketch the graph over its domain and read the range off the y-axis.

Preview12m

One-one functions

Each output comes from only one input. A horizontal line meets the graph at most once.

12m

Composite functions

fg(x) means do g first, then f. Put the whole of g(x) into f.

12m

When a composite function can be formed

gf exists only when the range of f fits inside the domain of g.

12m

Inverse functions

Only a one-one function has an inverse. Swap x and y, then make y the subject.

12m

The graph of a function and its inverse

The graph of the inverse is the reflection of the graph in the line y = x.

12m

Transformations: translations

y = f(x) + a moves the graph up; y = f(x + a) moves it to the left.

12m

Transformations: reflections

y = −f(x) reflects the graph in the x-axis; y = f(−x) reflects it in the y-axis.

12m

10.

Transformations: stretches

y = af(x) stretches the graph up and down; y = f(ax) squeezes it sideways by 1/a.

12m

11.

Combined transformations

Do two transformations one at a time, and see why the order can change the final graph.

12m

12.

Functions: timed practice

A timed drill of mixed functions questions, then practice problems from easy to exam level, each with a worked solution.

10m

13.

Functions: exam practice

Paper 1 style questions with marks, full worked solutions and the mark scheme for every step.

15m

14.

Functions: summary

The key points, formulas and common mistakes of the chapter on one screen, with flashcards.

1.3 · Coordinate geometry

Lengths, midpoints and gradients, the three forms of a straight line, the equation of a circle in both forms, tangents and chords, and where lines meet circles.

0/13

Length of a line segment

Draw the right-angled triangle under the segment and use Pythagoras on the coordinates.

Preview12m 2.

Midpoint of a line segment

The midpoint is the mean of the two x-coordinates and the mean of the two y-coordinates.

Preview12m

Parallel and perpendicular lines

Parallel lines have equal gradients. Perpendicular gradients multiply to −1.

12m

Equations of straight lines

Find a line from two points or a point and a gradient, and write it as y = mx + c or ax + by + c = 0.

12m

Points of intersection of two lines

Where two lines cross, both equations are true, so solve them together.

12m

The equation of a circle

(x − a)² + (y − b)² = r² is Pythagoras for every point on a circle with centre (a, b) and radius r.

12m

The expanded form of the equation of a circle

Complete the square twice to find the centre and radius from x² + y² + 2gx + 2fy + c = 0.

12m

Tangents to a circle

A tangent is perpendicular to the radius at the point where it touches.

12m

Chords, symmetry and the angle in a semicircle

The perpendicular from the centre cuts a chord in half, and the angle in a semicircle is 90°.

12m

10.

Problems involving intersections of lines and circles

Substitute the line into the circle: two roots cut it, one root touches it, none misses it.

12m

11.

Coordinate geometry: timed practice

A timed drill of mixed coordinate geometry questions, then practice problems from easy to exam level, each with a worked solution.

10m

12.

Coordinate geometry: exam practice

Paper 1 style questions with marks, full worked solutions and the mark scheme for every step.

15m

13.

Coordinate geometry: summary

The key points, formulas and common mistakes of the chapter on one screen, with flashcards.

1.4 · Circular measure

Measure angles in radians, find arc lengths with s = rθ and sector areas with A = ½r²θ, and use triangles to find chords and segments.

0/10

Radians

Lay the radius along the edge of the circle. The angle it makes at the centre is one radian.

Preview12m 2.

Converting between degrees and radians

π radians is 180°. Multiply or divide by that to change from one to the other.

Preview12m

Length of an arc

An arc is a fraction of the whole edge, so its length is s = rθ, with θ in radians.

12m

Area of a sector

A sector is a fraction of the whole circle, so its area is A = ½r²θ, with θ in radians.

12m

Lengths, angles and areas of triangles in circle problems

Two radii and a chord make a triangle. Find the chord and the area with trigonometry.

12m

Area of a segment

A segment is a sector with its triangle taken away.

12m

Problems involving arcs and sectors

Perimeters and shaded areas made from arcs, sectors, triangles and segments.

12m

Circular measure: timed practice

A timed drill of mixed circular measure questions, then practice problems from easy to exam level, each with a worked solution.

10m

Circular measure: exam practice

Paper 1 style questions with marks, full worked solutions and the mark scheme for every step.

15m

10.

Circular measure: summary

The key points, formulas and common mistakes of the chapter on one screen, with flashcards.

1.5 · Trigonometry

Exact values, angles of any size, the graphs of sin, cos and tan and their transformations, principal values, the two identities, and every solution of an equation in an interval.

0/15

Exact values of sin, cos and tan of 30°, 45° and 60°

Cut a square and an equilateral triangle in half, and the exact values appear as surds.

Preview12m 2.

The general definition of an angle

Angles turn from the positive x-axis: anticlockwise is positive, and they can go past 360°.

Preview12m

Trigonometric ratios of general angles

Find the related acute angle, then the quadrant decides the sign.

12m

Graphs of sine and cosine

The height of a seat on the big wheel draws a wave that repeats every 360°, or 2π.

12m

The graph of tangent

tan repeats every 180° and has asymptotes where cos is zero.

12m

Transformations of trigonometric graphs

Sketch graphs such as y = 3 sin x, y = 1 − cos 2x and y = tan(x + π/4).

12m

Inverse trigonometric functions

sin⁻¹, cos⁻¹ and tan⁻¹ give one answer only, the principal value, from a restricted domain.

12m

Solving trigonometric equations

Get one answer from the calculator, then use the graph to find every other answer in the interval.

12m

Trigonometric equations with multiple angles

For sin 2x = k, widen the interval first, solve for 2x, then halve every answer.

12m

10.

Trigonometric identities

tan θ ≡ sin θ / cos θ and sin²θ + cos²θ ≡ 1 come from the unit circle. Use them to simplify.

12m

11.

Proving trigonometric identities

Start from the harder side and use the two identities until it matches the other side.

12m

12.

Further trigonometric equations

Use an identity to make one function, solve the quadratic, and reject values outside −1 to 1.

12m

13.

Trigonometry: timed practice

A timed drill of mixed trigonometry questions, then practice problems from easy to exam level, each with a worked solution.

10m

14.

Trigonometry: exam practice

Paper 1 style questions with marks, full worked solutions and the mark scheme for every step.

15m

15.

Trigonometry: summary

The key points, formulas and common mistakes of the chapter on one screen, with flashcards.

1.6 · Series

Expand (a + b)ⁿ with Pascal’s triangle and ⁿCᵣ, find one term, then the nth term and sum of arithmetic and geometric progressions, and the sum to infinity.

0/13

Binomial expansion of (a + b)ⁿ using Pascal’s triangle

Row n of Pascal’s triangle gives the numbers in front of each term of (a + b)ⁿ.

Preview12m 2.

Binomial coefficients: n! and ⁿCᵣ

ⁿCᵣ counts the ways to choose r things from n, and it gives the same numbers as Pascal’s triangle.

Preview12m

The binomial expansion using ⁿCᵣ

Write (a + b)ⁿ with ⁿCᵣ so large n needs no triangle. Powers of a fall as powers of b rise.

12m

Finding a particular term or coefficient

Choose r so the power is right, then work out that one term. Includes products of two brackets.

12m

Arithmetic progressions

Add the same number each time. The nth term is a + (n − 1)d.

12m

The sum of an arithmetic progression

Write the sum forwards and backwards and add: each pair gives the same total.

12m

Geometric progressions

Multiply by the same number each time. The nth term is arⁿ⁻¹.

12m

The sum of a geometric progression

Multiply the sum by r and subtract: almost every term cancels.

12m

The sum to infinity of a geometric progression

When −1 < r < 1 the terms shrink and the total settles at a / (1 − r).

12m

10.

Further arithmetic and geometric series

Three numbers in AP or GP, and questions that use two progressions at once.

12m

11.

Series: timed practice

A timed drill of mixed series questions, then practice problems from easy to exam level, each with a worked solution.

10m

12.

Series: exam practice

Paper 1 style questions with marks, full worked solutions and the mark scheme for every step.

15m

13.

Series: summary

The key points, formulas and common mistakes of the chapter on one screen, with flashcards.

1.7 · Differentiation

The gradient of a curve as a limit of chords, the derivative of xⁿ for any rational n, the chain rule, tangents and normals, and second derivatives.

0/11

The gradient of a curve

Take chords closer and closer to one point; their gradients settle on the gradient of the curve there.

Preview12m 2.

The derivative of xⁿ

Bring the power down and take one off the power. Read dy/dx and f′(x) as the gradient function.

Preview12m

Differentiating constant multiples, sums and differences

Differentiate a polynomial one term at a time; a number on its own differentiates to zero.

12m

Differentiating negative and fractional powers

Rewrite 1/x² and √x as powers of x first, then the same rule works for any rational n.

12m

The chain rule

Differentiate the outside, keep the inside, then multiply by the derivative of the inside.

12m

Equations of tangents

dy/dx at the point gives the gradient; y − y₁ = m(x − x₁) gives the tangent.

12m

Equations of normals

The normal is perpendicular to the tangent, so its gradient is −1 divided by dy/dx.

12m

Second derivatives

Differentiate twice. d²y/dx² tells you how the gradient itself is changing.

12m

Differentiation: timed practice

A timed drill of mixed differentiation questions, then practice problems from easy to exam level, each with a worked solution.

10m

10.

Differentiation: exam practice

Paper 1 style questions with marks, full worked solutions and the mark scheme for every step.

15m

11.

Differentiation: summary

The key points, formulas and common mistakes of the chapter on one screen, with flashcards.

1.7 · Further differentiation

Increasing and decreasing functions, stationary points and the second derivative test, sketching curves, practical maximum and minimum problems, and connected rates of change.

0/3

Increasing and decreasing functions

Where dy/dx is positive the curve goes up; where it is negative the curve goes down.

Preview12m 2.

Stationary points

At a stationary point the gradient is zero. Solve dy/dx = 0 to find it.

Preview12m

Determining the nature of a stationary point

A negative second derivative means a maximum; a positive one means a minimum.

12m

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.