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Maths — Pure Mathematics 1

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Underground Mathematics
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Maths — Pure Mathematics 1

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SyllabAI Hub — revision for Edexcel IGCSE & IAL and Cambridge International qualifications.

Home/Edexcel/IAL/Maths — Pure Mathematics 1/Underground Mathematics

Underground Mathematics

Exam code:YMA01

Cambridge's Underground Mathematics Quadratics, Combining Functions, Geometry of Equations, Circles, Thinking about Geometry, Introducing Calculus, Product Rule & Integration by Parts, Chain Rule & Integration by Substitution, Trigonometry: Triangles to Functions, Trigonometry: Compound Angles, Exponentials & Logarithms, Counting & Binomials, Thinking about Algebra, Calculus Of Powers, Calculus Meets Functions, Calculus Trig Log, Power Series and Polynomials & Rational Functions collections for Edexcel IAL Maths — 301 resources mapped across 22 specification points, each with suggestions and worked solutions.

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22 specification points

© University of Cambridge (Underground Mathematics); © UCLES / Oxford MAT / NRICH for sourced exam questions. Source: undergroundmathematics.org.

1.1 · Laws of indices for all rational a m × a n = a m + n , a m ÷ a n = a m − n , ( a m ) n = a mn exponents. m n m The equivalence of a n and a should be known.

51 resources

If we increase $x$ by a small amount, what happens to $y$ ? recommendedReview question
Chain mappingBuilding blocks · Introducing
Designing a polynomialInvestigation · Developing
Gradient spottingBuilding blocks · Introducing
I can see $u$ !Fluency exercise · Developing
Producing a ruleBuilding blocks · Introducing
Given $f'(x)$ and $g'(x)$ , what is $f''(2x)$ ?Review question
Can we find the repeated root for $4x^4+x^2+3x+1=0$ ?Review question
Slippery slopes... another derivativeBuilding blocks · Introducing
Where do two tangents cross, and where do two normals cross?Review question
Can we find the area of the triangle created by a common tangent?Review question
What is the gradient of this curve at its $x$ -intercepts?Review question
Can we find the triangle areas given by a tangent?Review question
Can we find the locus of a midpoint created by a parabola?Review question
Can we show these two cubic curves touch?Review question
Can you find... curvy cubics editionMany ways problem · Developing
Curvy cubicsInvestigation · Developing
Floppy hairFluency exercise · Developing
Where does $f(x) = x^2 - 2px + 3$ have its minimum?Review question
Can we maximise the volume of this metal tin?Review question
When does $3x^4-16x^3+18x^2=k$ have exactly two real roots?Review question
When is the area of a face of this pyramid a minimum?Review question
When is the cost of producing this chemical a minimum?Review question
When is $f(x) = 2x^3 − 9x^2 + 12x + 3$ smallest over this range?Review question
Can we sketch the graph of this piecewise function?Review question
What's the most economical speed for this ship's journey?Review question
What's the area enclosed by $y=x(x-3)^2$ and a tangent?Review question
Given the line $PRQ$ , when is $OP + OQ$ a minimum?Review question
How small can this triangle be?Review question
When is the volume of this cylinder inside a cone a maximum?Review question
Can we sketch $y = x^2(2-x)$ , and from this, $y^2 = x^2(2-x)$ ?Review question
If $y=f(x)$ has a turning point when $x=\frac{1}{4}$ , can we find $\lambda$ ?Review question
When does $y=kx^3-(k+1)x^2+(2-k)x-k$ have a minimum at $x =1$ ?Review question
What's the least surface area of this rocket-shaped solid?Review question
When is the volume of this cuboid a maximum?Review question
What can we say about the combined area of these two shapes?Review question
When does this function of two variables have a minimum?Review question
When is the volume of this pyramid a maximum?Review question
Can we find the minimum value of $x^2+2/x$ for $x$ positive?Review question
Can we find the coefficient of $x^2$ in $(5 + 4x)(1 - x/2)^{10}$ ?Review question
Which power of $x$ has the greatest coefficient?Review question
What is the coefficient of $x^3y^5$ in the expansion of $(1+xy+y^2)^n$ ?Review question
Can we prove these four binomial coefficient identities?Review question
Can we estimate a difference of eighth powers?Review question
Can we find an inequality from the expansion of $[ 1+1/\sqrt{n}]^n$ ?Review question
Can we find an approximation to $(1.0099)^6$ ?Review question
Can we use a binomial expansion to evaluate $\left(19\tfrac{3}{4}\right)^6$ ?Review question
Can we show $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-2}{r-1}+\binom{n-3}{r-1}+\dotsb+\binom{r-1}{r-1}$ ?Review question
Can we find the coefficients of $x^{−12}$ and $x^2$ in this expansion?Review question
Can we factorise $(x+y)^6 - (x-y)^6$ completely?Review question
If $x - 1/x = u$ , what's $x^3 - 1/x^3$ ?Review question

1.2 · Use and manipulation of surds. Students should be able to rationalise denominators.

37 resources

Factorisable quadraticsPackage of problems · Developing
Inequalities for some occasionsFluency exercise · Developing
Irrational rootsPackage of problems · Developing
Pick a card...Fluency exercise · Developing
Quadratic solving sorter recommendedFluency exercise · Developing
When is $12x^2+7x-10$ negative?Review question
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$ ?Review question
Can we find both roots if one is double the other?Review question
Can we solve $12/(x-3) < x+1$ ?Review question
Where does $y = px^2 + 8x + p - 6$ cross the $x$ -axis?Review question
When are these quadratic inequalities true together?Review question
Which quadratic?Building blocks · Developing
GeoGebra constructions... quadratic editionPackage of problems · Developing
Proving the quadratic formulaScaffolded task · Developing
Quadratic gridsRich example · Introducing
How do we increase the roots of a quadratic equation by $1$ ?Review question
Can we prove this inequality involving two square roots?Review question
If we know two values satisfying a quadratic, can we find the quadratic?Review question
Can we fit a quadratic to this given data?Review question
Into how many regions do these parabolas divide the plane?Review question
Trig tablesFluency exercise · Developing · teacher support included
Using angles in a pentagon...Review question
Can we approximate the area under this sine curve?Review question
Which of these area and perimeter statements is true?Review question
Trig countdownFluency exercise · Developing
Trig tables - teacher supportResource in action · Developing
Can we compose two simple functions?Review question
Can we draw the graph of $\left| x + [x] \right|$ ?Review question
Can we find the ranges of $f$ and $g$ , and the function $f \circ g$ ?Review question
Can we sketch the graph of $g \colon x \to x - [x]$ ?Review question
Can we work with the domain, codomain and range of a bijection?Review question
Compose!Package of problems · Developing
Composing gets me nowhereBuilding blocks · Developing
Domain and range dominoesFluency exercise · Developing
Forming functions — defining domainsBuilding blocks · Introducing
InvertiblesBuilding blocks · Developing
Making inverse functionsScaffolded task · Introducing

1.3 · Quadratic functions and their graphs.

37 resources

Name that graph recommendedPackage of problems · Developing
Quadratic symmetryPackage of problems · Developing
For what values of $k$ is $x^2 + 6kx +144$ always positive?Review question
Given the minimum point, what's this parabola's equation?Review question
If $7 - px - x^2 = 16 - (q + x)^2$ , what are $p$ and $q$ ?Review question
Which parabola?Package of problems · Developing
Which quadratic?Building blocks · Developing
Paired parabolasPackage of problems · Developing
DiscriminatingPackage of problems · Developing · teacher support included
Discriminating - teacher supportResource in action · Developing
When are the roots for $x^2-bx+c=0$ real and positive?Review question
When does $x^2-4x-1=2k(x-5)$ have equal roots?Review question
When does $x^2+(3k-7)x+(2k+6)=0$ have real roots?Review question
When does $(p+1)x^2+4px+9=0$ have a repeated root?Review question
Can we show $(b^2 - 2ac)x^2 + 4(a + c)x = 8$ always has real roots?Review question
When do these simultaneous equations have no real solutions?Review question
When does $x^2 + ax + a = 1$ have distinct real roots?Review question
Where does $y = px^2 + 8x + p - 6$ cross the $x$ -axis?Review question
When are the coefficients of a quadratic equal to its roots?Review question
Factorisable quadraticsPackage of problems · Developing
Inequalities for some occasionsFluency exercise · Developing
Irrational rootsPackage of problems · Developing
Pick a card...Fluency exercise · Developing
Quadratic solving sorterFluency exercise · Developing
When is $12x^2+7x-10$ negative?Review question
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$ ?Review question
Can we find both roots if one is double the other?Review question
Can we solve $12/(x-3) < x+1$ ?Review question
When are these quadratic inequalities true together?Review question
GeoGebra constructions... quadratic editionPackage of problems · Developing
Proving the quadratic formulaScaffolded task · Developing
Quadratic gridsRich example · Introducing
How do we increase the roots of a quadratic equation by $1$ ?Review question
Can we prove this inequality involving two square roots?Review question
If we know two values satisfying a quadratic, can we find the quadratic?Review question
Can we fit a quadratic to this given data?Review question
Into how many regions do these parabolas divide the plane?Review question

1.4 · The discriminant of a quadratic function.: • Need to know and to use b² − 4ac > 0, b² − 4ac = 0 and b² − 4ac < 0

12 resources

Discriminating recommendedPackage of problems · Developing · teacher support included
Discriminating - teacher supportResource in action · Developing
When are the roots for $x^2-bx+c=0$ real and positive?Review question
When does $x^2-4x-1=2k(x-5)$ have equal roots?Review question
For what values of $k$ is $x^2 + 6kx +144$ always positive?Review question
When does $x^2+(3k-7)x+(2k+6)=0$ have real roots?Review question
When does $(p+1)x^2+4px+9=0$ have a repeated root?Review question
Can we show $(b^2 - 2ac)x^2 + 4(a + c)x = 8$ always has real roots?Review question
When do these simultaneous equations have no real solutions?Review question
When does $x^2 + ax + a = 1$ have distinct real roots?Review question
Where does $y = px^2 + 8x + p - 6$ cross the $x$ -axis?Review question
When are the coefficients of a quadratic equal to its roots?Review question

1.5 · Completing the square. Solution of Solution of quadratic equations by factorisation, use of the quadratic equations. formula, use of a calculator and completing the square. b 2  b 2  ax 2 + bx + c = a  x + +  c −  2 a  4 a     

38 resources

Name that graph recommendedPackage of problems · Developing
Quadratic symmetryPackage of problems · Developing
For what values of $k$ is $x^2 + 6kx +144$ always positive?Review question
Given the minimum point, what's this parabola's equation?Review question
If $7 - px - x^2 = 16 - (q + x)^2$ , what are $p$ and $q$ ?Review question
Which parabola?Package of problems · Developing
Which quadratic?Building blocks · Developing
Paired parabolasPackage of problems · Developing
Factorisable quadraticsPackage of problems · Developing
Inequalities for some occasionsFluency exercise · Developing
Irrational rootsPackage of problems · Developing
Pick a card...Fluency exercise · Developing
Quadratic solving sorterFluency exercise · Developing
When is $12x^2+7x-10$ negative?Review question
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$ ?Review question
Can we find both roots if one is double the other?Review question
Can we solve $12/(x-3) < x+1$ ?Review question
Where does $y = px^2 + 8x + p - 6$ cross the $x$ -axis?Review question
When are these quadratic inequalities true together?Review question
GeoGebra constructions... quadratic editionPackage of problems · Developing
Proving the quadratic formulaScaffolded task · Developing
Quadratic gridsRich example · Introducing
How do we increase the roots of a quadratic equation by $1$ ?Review question
Can we prove this inequality involving two square roots?Review question
If we know two values satisfying a quadratic, can we find the quadratic?Review question
Can we fit a quadratic to this given data?Review question
Into how many regions do these parabolas divide the plane?Review question
Can we solve the simultaneous equations $x + y + \sqrt{xy} = 39$ and $x^2 + y^2 + xy = 741$ ?Review question
How do we solve $x + 3\sqrt{x} − 1/2 = 0$ ?Review question
If $2y = a^x + a^{-x}$ , can we find $a^x$ ?Review question
Can we solve $\sqrt{3-3x} - \sqrt{2-x} = 1$ ?Review question
When does $x^4=(x-c)^2$ have four real roots?Review question
Powerful quadraticsFood for thought · Developing
How do we solve an equation containing three square roots?Review question
Can we solve this inequality with three square roots?Review question
Can we solve these simultaneous equations that involve reciprocals?Review question
When can this equation involving algebraic fractions hold?Review question
What does the curve $x^4-y^2=y+1$ look like?Review question

1.6 · Solve simultaneous equations; analytical solution by substitution.

10 resources

When does $y=mx-5$ intersect $y=x^2-1$ twice?Review question
Can we solve these simultaneous equations, one linear, one quadratic?Review question
When is this line a tangent to this parabola? recommendedReview question
Can we find the midpoint of a chord of this parabola?Review question
When does $y=kx$ intersect the parabola $y=(x-1)^2$ ?Review question
When do these simultaneous equations have no real solutions?Review question
Can we find the tangents to a circle from a point?Review question
Can we solve these simultaneous equations, both of degree two?Review question
Given two simultaneous equations of degree two, can we solve them?Review question
Into how many regions do these parabolas divide the plane?Review question

1.7 · Interpret linear and quadratic For example, inequalities graphically. ax + b > cx + d , px 2 + qx + r  0, px 2 + qx + r < ax + b. Interpreting the third inequality as the range of x for which the curve y = px 2 + qx + r is below the line with equation y = ax + b. Including inequalities with brackets and fractions. These would be reducible to linear or quadratic inequalities, e.g. a < b becomes ax < bx 2 , x ≠ 0.

20 resources

Factorisable quadraticsPackage of problems · Developing
Inequalities for some occasionsFluency exercise · Developing
Irrational rootsPackage of problems · Developing
Pick a card...Fluency exercise · Developing
Quadratic solving sorter recommendedFluency exercise · Developing
When is $12x^2+7x-10$ negative?Review question
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$ ?Review question
Can we find both roots if one is double the other?Review question
Can we solve $12/(x-3) < x+1$ ?Review question
Where does $y = px^2 + 8x + p - 6$ cross the $x$ -axis?Review question
When are these quadratic inequalities true together?Review question
Which quadratic?Building blocks · Developing
GeoGebra constructions... quadratic editionPackage of problems · Developing
Proving the quadratic formulaScaffolded task · Developing
Quadratic gridsRich example · Introducing
How do we increase the roots of a quadratic equation by $1$ ?Review question
Can we prove this inequality involving two square roots?Review question
If we know two values satisfying a quadratic, can we find the quadratic?Review question
Can we fit a quadratic to this given data?Review question
Into how many regions do these parabolas divide the plane?Review question

1.8 · Represent linear and quadratic inequalities graphically.: • Represent linear and quadratic inequalities such as y > x + r and y > ax² + bx + c graphically. Shading and use of dotted and solid line convention is required.

20 resources

Factorisable quadraticsPackage of problems · Developing
Inequalities for some occasionsFluency exercise · Developing
Irrational rootsPackage of problems · Developing
Pick a card...Fluency exercise · Developing
Quadratic solving sorter recommendedFluency exercise · Developing
When is $12x^2+7x-10$ negative?Review question
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$ ?Review question
Can we find both roots if one is double the other?Review question
Can we solve $12/(x-3) < x+1$ ?Review question
Where does $y = px^2 + 8x + p - 6$ cross the $x$ -axis?Review question
When are these quadratic inequalities true together?Review question
Which quadratic?Building blocks · Developing
GeoGebra constructions... quadratic editionPackage of problems · Developing
Proving the quadratic formulaScaffolded task · Developing
Quadratic gridsRich example · Introducing
How do we increase the roots of a quadratic equation by $1$ ?Review question
Can we prove this inequality involving two square roots?Review question
If we know two values satisfying a quadratic, can we find the quadratic?Review question
Can we fit a quadratic to this given data?Review question
Into how many regions do these parabolas divide the plane?Review question

1.9 · Solutions of linear and quadratic inequalities.: • For example, solving ax + b > cx + d, px² + qx + r 0, px² + qx + r < ax + b.

20 resources

Factorisable quadraticsPackage of problems · Developing
Inequalities for some occasionsFluency exercise · Developing
Irrational rootsPackage of problems · Developing
Pick a card...Fluency exercise · Developing
Quadratic solving sorter recommendedFluency exercise · Developing
When is $12x^2+7x-10$ negative?Review question
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$ ?Review question
Can we find both roots if one is double the other?Review question
Can we solve $12/(x-3) < x+1$ ?Review question
Where does $y = px^2 + 8x + p - 6$ cross the $x$ -axis?Review question
When are these quadratic inequalities true together?Review question
Which quadratic?Building blocks · Developing
GeoGebra constructions... quadratic editionPackage of problems · Developing
Proving the quadratic formulaScaffolded task · Developing
Quadratic gridsRich example · Introducing
How do we increase the roots of a quadratic equation by $1$ ?Review question
Can we prove this inequality involving two square roots?Review question
If we know two values satisfying a quadratic, can we find the quadratic?Review question
Can we fit a quadratic to this given data?Review question
Into how many regions do these parabolas divide the plane?Review question

1.10 · Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation.: • Students should be able to use brackets. Factorisation of polynomials of degree n, n 3, e.g. x³ + 4x² + 3x.The notation f(x) may be used.

24 resources

Function builder IIBuilding blocks · Developing
What is the remainder when $p_n (x)$ is divided by $p_{n-1} (x)$ ?Review question
Can we sketch the graph of $y = (4x-1)/(x+5)$ ?Review question
Can we factorise $6x^3+5x^2-17x-6$ completely?Review question
What's left? recommendedPackage of problems · Developing
Can you find... cubic editionMany ways problem · Developing
How not to solve a cubic...Food for thought · Developing
Can we prove that $26^n$ always ends in the same two digits?Review question
If this cubic has two equal roots, what can $p$ be?Review question
Given three remainders, what is the function?Review question
Can we sketch the reciprocal of a polynomial?Review question
Can we solve $\sqrt{4x+13}-\sqrt{x+1}=\sqrt{12-x}?$ Review question
If dividing by $x+a$ or $x-2a$ gives the same remainder, what's $a$ ?Review question
Can we prove the Arithmetic and Geometric Mean Inequality?Review question
Can we sketch the graph $y=x^3-x^2-x+1$ ?Review question
If we divide by $x-1$ , the remainder is?Review question
When is $n^2 x^{2n+3} - 25n x^{n+1} + 150x^7$ divisible by $x^2-1$ ?Review question
When is $x^n + 1$ divisible by $x + 1$ ?Review question
When is $x^2 +1$ a factor of $(3+x^4)^n-(x^2+3)^n(x^2-1)^n$ ?Review question
Can we factorise $a^3(b-c)+b^3(c-a)+c^3(a-b)$ ?Review question
If two remainders are related can we find $a$ ?Review question
What if the remainder when we divide by $x-k$ is $k$ ?Review question
What kind of roots does $x^3 - 30x^2 + 108x - 104 = 0$ have?Review question
Can we find the coefficients and factorise this quartic?Review question

1.11 · Graphs of functions; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations.: • Functions to include simple cubic functions and the reciprocal functions y = k⁄x and y = k⁄x² with x ≠ 0. Knowledge of the term asymptote is expected. Also, trigonometric graphs.

47 resources

When is the volume of this cuboid a maximum? recommendedReview question
Can you find... curvy cubics editionMany ways problem · Developing
Curvy cubicsInvestigation · Developing
Floppy hairFluency exercise · Developing
Where does $f(x) = x^2 - 2px + 3$ have its minimum?Review question
Can we maximise the volume of this metal tin?Review question
When does $3x^4-16x^3+18x^2=k$ have exactly two real roots?Review question
When is the area of a face of this pyramid a minimum?Review question
When is the cost of producing this chemical a minimum?Review question
When is $f(x) = 2x^3 − 9x^2 + 12x + 3$ smallest over this range?Review question
Can we sketch the graph of this piecewise function?Review question
What's the most economical speed for this ship's journey?Review question
What's the area enclosed by $y=x(x-3)^2$ and a tangent?Review question
Given the line $PRQ$ , when is $OP + OQ$ a minimum?Review question
How small can this triangle be?Review question
When is the volume of this cylinder inside a cone a maximum?Review question
Can we sketch $y = x^2(2-x)$ , and from this, $y^2 = x^2(2-x)$ ?Review question
If $y=f(x)$ has a turning point when $x=\frac{1}{4}$ , can we find $\lambda$ ?Review question
When does $y=kx^3-(k+1)x^2+(2-k)x-k$ have a minimum at $x =1$ ?Review question
What's the least surface area of this rocket-shaped solid?Review question
What can we say about the combined area of these two shapes?Review question
When does this function of two variables have a minimum?Review question
When is the volume of this pyramid a maximum?Review question
Is the gradient function increasing or decreasing on this curve?Review question
Talking about curvesInvestigation · Introducing
Two-way calculusMany ways problem · Developing
Average turning pointsGo and think about it... · Developing
Can we find a curve from its gradient function?Review question
When does this cubic equation have three distinct real roots?Review question
What's the minimum area of this variable triangle?Review question
When does a cubic curve have two turning points?Review question
When does $3x^4 - 16x^3 +18x^2 + k = 0$ have four real solutions?Review question
When is this parabola's turning point nearest to the origin?Review question
How many stationary points does $y = 2x^3 - 6x^2 + 5x - 7$ have?Review question
When does this cubic equation have distinct real positive solutions?Review question
Can we find the minimum value of $x^2+2/x$ for $x$ positive?Review question
When does this cone have a maximum volume?Review question
When will the arclength $QR$ take a maximum value?Review question
Can we identify the right sketch of a transformed function?Review question
Can we sketch and describe these composite functions?Review question
How do these transformations change the graph of $f(x)$ ?Review question
It's a matter of perspectiveBigger picture · Developing
Name that graph againFood for thought · Developing
Order! Order!Investigation · Introducing
TransformersMany ways problem · Developing
Translating or not?Package of problems · Developing
Which transformations give us the graph of $y=2^{x^2-4x+3}$ ?Review question

1.12 · Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax). rdinate geometry in the (x, y) plane: • Students should be able to apply one of these transformations to any of the above functions (quadratics, cubics, reciprocals, sine, cosine, and tangent) and sketch the resulting graphs. Given the graph of any function y = f(x), students should be able to sketch the graph resulting from one of these transformations.

9 resources

Can we identify the right sketch of a transformed function?Review question
Can we sketch and describe these composite functions?Review question
How do these transformations change the graph of $f(x)$ ?Review question
It's a matter of perspectiveBigger picture · Developing
Name that graph againFood for thought · Developing
Order! Order! recommendedInvestigation · Introducing
TransformersMany ways problem · Developing
Translating or not?Package of problems · Developing
Which transformations give us the graph of $y=2^{x^2-4x+3}$ ?Review question

2.1 · Equation of a straight line, including the forms y − y₁ = m(x − x₁) and ax + by + c = 0.: • To include: (i) the equation of a line through two given points (ii) the equation of a line parallel (or perpendicular) to a given line through a given point. For example, the line perpendicular to the line 3x + 4y = 18 through the point (2, 3) has equation y − 3 = 4⁄3(x − 2).

61 resources

Cartesian coordinatesBigger picture · Developing
Straight linesBuilding blocks · Introducing
Lots of lines!Building blocks · Introducing
Where is the foot of the perpendicular from a point to a line?Review question
Given a right angle and the equations of two sides, can we find the third?Review question
Can we calculate an approximation to the root of this function?Review question
Can we find a rectangle from two vertices and a diagonal?Review question
Can we reflect this point in this line?Review question
Can we find the coordinates of this tilted rectangle?Review question
A point $Y$ moves to define a point $P$ — can we find the locus of $P\,$ ?Review question
A line cuts the axes to make a triangle; can we find its area?Review question
A line through a variable point creates a triangle; can we find its area?Review question
Can we show this parallelogram is a rectangle?Review question
Can we find the area of this quadrilateral?Review question
Can we find the point midway between an intersection and a line?Review question
Simultaneous squaresInvestigation · Developing
Straight line pairs recommendedInvestigation · Developing
Between the linesFood for thought · Developing
Near missFood for thought · Developing
Negatively triangularMany ways problem · Developing
What's the locus of the intersection of these variable lines?Review question
Can we find an integer solution to three simultaneous inequalities?Review question
Which four inequalities define this region?Review question
If we know the sum and product of two numbers, what are they?Review question
Can we show that this point moves on a circle?Review question
No solutions, or infinitely many?Review question
Can we plot this graph and use it to solve two equations?Review question
Can we illustrate this pair of inequalities on a graph?Review question
One distance is double another; can we find a locus?Review question
Can we find the three inequalities that define this region?Review question
What is the largest $x$ to satisfy these three inequalities?Review question
Spot the differenceBuilding blocks · Developing
The circle of Apollonius... coordinate editionMany ways problem · Developing
The equation of a straight lineBigger picture · Developing
Conic sections in real lifeBigger picture · Developing
Finding circlesMany ways problem · Developing
Olympic ringsProblem requiring decisions · Developing
Pairs of circlesPackage of problems · Developing
Can we find the equations of two circles that touch another?Review question
Can we show that these four points lie on a circle?Review question
Can we find the equation of the inscribed circle of the triangle $ABC$ ?Review question
Can we find the tangents to this circle from the origin?Review question
Can we show that these two circles touch?Review question
Can we show these two circles lie entirely outside each other?Review question
Where do this pair of tangents to a circle meet?Review question
Can we show this quartic equation has exactly two real roots?Review question
When does the origin lie inside this circle?Review question
Can we show that (circle area)/(triangle area) $\geq \pi$ ?Review question
What can we say if two circles cut at right angles?Review question
Which point on this circle is closest to the origin?Review question
Can we find the two circles that satisfy these three conditions?Review question
What can we deduce if the loci traced out by two points touch?Review question
Can we find the areas enclosed between a circle and its tangent?Review question
If $AQ:QP=\lambda:1$ , can we show $Q$ lies on a circle?Review question
Given three clues, can we find the equation of this circle?Review question
Can we find the area bounded by these two graphs?Review question
Which point on the first circle is closest to the second?Review question
Given two lines crossing a circle, can we show these lengths are equal?Review question
What's the locus of $(p, q)$ if the length of this tangent equals $p$ ?Review question
Teddy bearPackage of problems · Developing
Teddy bear - teacher supportResource in action · Developing · teacher support included

2.2 · Conditions for two straight lines to be parallel or perpendicular to each other.

60 resources

Between the linesFood for thought · Developing
Straight linesBuilding blocks · Introducing
Lots of lines!Building blocks · Introducing
Near missFood for thought · Developing
Negatively triangularMany ways problem · Developing
Where is the foot of the perpendicular from a point to a line?Review question
What's the locus of the intersection of these variable lines?Review question
Can we find an integer solution to three simultaneous inequalities?Review question
Given a right angle and the equations of two sides, can we find the third?Review question
Which four inequalities define this region?Review question
If we know the sum and product of two numbers, what are they?Review question
Can we calculate an approximation to the root of this function?Review question
Can we find a rectangle from two vertices and a diagonal?Review question
Can we reflect this point in this line?Review question
Can we show that this point moves on a circle?Review question
Can we find the coordinates of this tilted rectangle?Review question
No solutions, or infinitely many?Review question
A point $Y$ moves to define a point $P$ — can we find the locus of $P\,$ ?Review question
A line cuts the axes to make a triangle; can we find its area?Review question
A line through a variable point creates a triangle; can we find its area?Review question
Can we plot this graph and use it to solve two equations?Review question
Can we illustrate this pair of inequalities on a graph?Review question
One distance is double another; can we find a locus?Review question
Can we find the three inequalities that define this region?Review question
Can we show this parallelogram is a rectangle?Review question
Can we find the area of this quadrilateral?Review question
Can we find the point midway between an intersection and a line?Review question
What is the largest $x$ to satisfy these three inequalities?Review question
Simultaneous squaresInvestigation · Developing
Spot the differenceBuilding blocks · Developing
Straight line pairsInvestigation · Developing
The circle of Apollonius... coordinate editionMany ways problem · Developing
The equation of a straight lineBigger picture · Developing
Conic sections in real lifeBigger picture · Developing
Finding circlesMany ways problem · Developing
Olympic ringsProblem requiring decisions · Developing
Pairs of circlesPackage of problems · Developing
Can we find the equations of two circles that touch another?Review question
Can we show that these four points lie on a circle?Review question
Can we find the equation of the inscribed circle of the triangle $ABC$ ?Review question
Can we find the tangents to this circle from the origin?Review question
Can we show that these two circles touch?Review question
Can we show these two circles lie entirely outside each other?Review question
Where do this pair of tangents to a circle meet?Review question
Can we show this quartic equation has exactly two real roots?Review question
When does the origin lie inside this circle?Review question
Can we show that (circle area)/(triangle area) $\geq \pi$ ?Review question
What can we say if two circles cut at right angles?Review question
Which point on this circle is closest to the origin?Review question
Can we find the two circles that satisfy these three conditions?Review question
What can we deduce if the loci traced out by two points touch?Review question
Can we find the areas enclosed between a circle and its tangent?Review question
If $AQ:QP=\lambda:1$ , can we show $Q$ lies on a circle?Review question
Given three clues, can we find the equation of this circle?Review question
Can we find the area bounded by these two graphs?Review question
Which point on the first circle is closest to the second?Review question
Given two lines crossing a circle, can we show these lengths are equal?Review question
What's the locus of $(p, q)$ if the length of this tangent equals $p$ ?Review question
Teddy bear recommendedPackage of problems · Developing
Teddy bear - teacher supportResource in action · Developing · teacher support included

3.1 · The sine and cosine rules, and the area of a triangle in the form 1⁄2 ab sin C.: • Including the ambiguous case of the sine rule.

29 resources

Going round in circles recommendedBuilding blocks · Introducing
Big trigFood for thought · Developing
Calculating latitudeBigger picture · Developing
Can you find... trigonometry editionMany ways problem · Developing · teacher support included
Can you find...trigonometry edition - teacher supportResource in action · Developing
Catching wavesBigger picture · Developing
From stars to wavesBigger picture · Introducing
Have a sineFood for thought · Developing
How high am I?Food for thought · Developing
Muddled trigBuilding blocks · Developing
When is $\sin x$ or $\sin 2x$ larger than a half?Review question
How are these three angles connected?Review question
If $\sin X$ and $\cos Y$ are both negative, what can we say about $X + Y$ ?Review question
Which of these log and trig expressions is the largest?Review question
Which points in the $x$ - $y$ plane satisfy $\sin y=\sin x$ ?Review question
What's the result when we reflect $y=\sin x$ twice?Review question
Can we approximate the area under this sine curve?Review question
Are any of these four trig expressions equal to each other?Review question
How many solutions does $\cos (\sin x)=\frac{1}{2}$ have?Review question
When is $\sin\theta > \sin 3\theta$ ?Review question
Slices of $\pi$ Many ways problem · Developing · teacher support included
Slices of $\pi$ - teacher supportResource in action · Developing
Trig tablesFluency exercise · Developing · teacher support included
How many solutions does $\sin^2x+3\sin x\cos x+2\cos^2 x=0$ have?Review question
General solutionsFood for thought · Introducing
Can we prove this trig identity to solve this equation?Review question
How many solutions does this equation involving $\sin^8x$ have?Review question
How many solutions does $2 \cos^2x + 5\sin x = 4$ have?Review question
How many solutions does $7\sin x +2\cos ^2 x =5$ have?Review question

3.2 · Radian measure, including use for arc length and area of sector.: • Use of the formulae s = rθ and A = 1 r²θ. 2

17 resources

Can we find a formula for an island's territorial waters? recommendedReview question
Problem areasFood for thought · Developing
When will the arclength $QR$ take a maximum value?Review question
BeltFood for thought · Developing
What's the area of this square if it sits in this sector?Review question
What's the shared area for these two circles?Review question
How much of this log is below the surface?Review question
What's the area inside both the circle $C$ and the triangle $T$ ?Review question
Can we find the ratio of the areas of these two sectors?Review question
What area do these intersecting circles share?Review question
When does a circle touch a parabola twice?Review question
Can we find the area between three touching circles?Review question
When does this area within a circle have a maximum value?Review question
If we know the difference in their areas, how big are these sectors?Review question
Sector spiralsFluency exercise · Introducing
Can we find the areas enclosed between a circle and its tangent?Review question
Can we find the area bounded by these two graphs?Review question

3.3 · Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.: • Knowledge of graphs of curves with equations such as y = 3 sin x, y = sin x+π⁄6, y = sin 2x is expected.

23 resources

Going round in circles recommendedBuilding blocks · Introducing
Big trigFood for thought · Developing
Calculating latitudeBigger picture · Developing
Can you find... trigonometry editionMany ways problem · Developing · teacher support included
Can you find...trigonometry edition - teacher supportResource in action · Developing
Catching wavesBigger picture · Developing
From stars to wavesBigger picture · Introducing
Have a sineFood for thought · Developing
How high am I?Food for thought · Developing
Muddled trigBuilding blocks · Developing
When is $\sin x$ or $\sin 2x$ larger than a half?Review question
How are these three angles connected?Review question
If $\sin X$ and $\cos Y$ are both negative, what can we say about $X + Y$ ?Review question
Which of these log and trig expressions is the largest?Review question
Which points in the $x$ - $y$ plane satisfy $\sin y=\sin x$ ?Review question
What's the result when we reflect $y=\sin x$ twice?Review question
Can we approximate the area under this sine curve?Review question
Are any of these four trig expressions equal to each other?Review question
How many solutions does $\cos (\sin x)=\frac{1}{2}$ have?Review question
When is $\sin\theta > \sin 3\theta$ ?Review question
Slices of $\pi$ Many ways problem · Developing · teacher support included
Slices of $\pi$ - teacher supportResource in action · Developing
Trig tablesFluency exercise · Developing · teacher support included

4.1 · The derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change; second order derivatives.: • For example, knowledge that dy⁄dx is the rate of change of y with respect to x. Knowledge of the chain rule is not required. The notation f ′(x) and f ′′(x) may be used.

5 resources

Binomials are the answer! recommendedInvestigation · Introducing
Estimating gradientsProblem requiring decisions · Introducing
Gradients of gradientsBuilding blocks · Introducing
Mapping a derivativeBuilding blocks · Introducing
Zooming inScaffolded task · Introducing

4.2 · Differentiation of xⁿ, and related sums, differences and constant multiples.: • The ability to differentiate expressions such as (2x + 5)(x − 1) and (x² + 5x − 3)⁄3x is expected.

39 resources

If we increase $x$ by a small amount, what happens to $y$ ? recommendedReview question
Chain mappingBuilding blocks · Introducing
Designing a polynomialInvestigation · Developing
Gradient spottingBuilding blocks · Introducing
I can see $u$ !Fluency exercise · Developing
Producing a ruleBuilding blocks · Introducing
Given $f'(x)$ and $g'(x)$ , what is $f''(2x)$ ?Review question
Can we find the repeated root for $4x^4+x^2+3x+1=0$ ?Review question
Slippery slopes... another derivativeBuilding blocks · Introducing
Where do two tangents cross, and where do two normals cross?Review question
Can we find the area of the triangle created by a common tangent?Review question
What is the gradient of this curve at its $x$ -intercepts?Review question
Can we find the triangle areas given by a tangent?Review question
Can we find the locus of a midpoint created by a parabola?Review question
Can we show these two cubic curves touch?Review question
Can you find... curvy cubics editionMany ways problem · Developing
Curvy cubicsInvestigation · Developing
Floppy hairFluency exercise · Developing
Where does $f(x) = x^2 - 2px + 3$ have its minimum?Review question
Can we maximise the volume of this metal tin?Review question
When does $3x^4-16x^3+18x^2=k$ have exactly two real roots?Review question
When is the area of a face of this pyramid a minimum?Review question
When is the cost of producing this chemical a minimum?Review question
When is $f(x) = 2x^3 − 9x^2 + 12x + 3$ smallest over this range?Review question
Can we sketch the graph of this piecewise function?Review question
What's the most economical speed for this ship's journey?Review question
What's the area enclosed by $y=x(x-3)^2$ and a tangent?Review question
Given the line $PRQ$ , when is $OP + OQ$ a minimum?Review question
How small can this triangle be?Review question
When is the volume of this cylinder inside a cone a maximum?Review question
Can we sketch $y = x^2(2-x)$ , and from this, $y^2 = x^2(2-x)$ ?Review question
If $y=f(x)$ has a turning point when $x=\frac{1}{4}$ , can we find $\lambda$ ?Review question
When does $y=kx^3-(k+1)x^2+(2-k)x-k$ have a minimum at $x =1$ ?Review question
What's the least surface area of this rocket-shaped solid?Review question
When is the volume of this cuboid a maximum?Review question
What can we say about the combined area of these two shapes?Review question
When does this function of two variables have a minimum?Review question
When is the volume of this pyramid a maximum?Review question
Can we find the minimum value of $x^2+2/x$ for $x$ positive?Review question

4.3 · Applications of differentiation to gradients, tangents and normals. Use of differentiation to find equations of tangents and normals at specific points on a curve.

29 resources

What (approximately) is the square root of $26$ ? recommendedReview question
Where do two tangents cross, and where do two normals cross?Review question
Can we find the area of the triangle created by a common tangent?Review question
What is the gradient of this curve at its $x$ -intercepts?Review question
Can we find the triangle areas given by a tangent?Review question
Can we find the locus of a midpoint created by a parabola?Review question
Can we show these two cubic curves touch?Review question
If we increase $x$ by a small amount, what happens to $y$ ?Review question
A tangent is ...Building blocks · Introducing
Gradient matchBuilding blocks · Developing
Why are gradients important in the real world?Bigger picture · Developing
Can we find a percentage increase using the gradient?Review question
When does the normal to a parabola pass through this point?Review question
Can we find the area inside a parabola, a tangent, and the $x$ -axis?Review question
Can we track this constantly growing patch of fluid?Review question
Given three clues, can we find the equation of the curve?Review question
How many common tangents can two parabolas have?Review question
A cubic has one real root—can we find an approximation to it?Review question
What can we say if $y = m(x-a)$ is tangent to $y = x^3 - x$ ?Review question
If we know the object's speed, can we find the speed of the image?Review question
Can we find the area enclosed by this quadratic and the $x$ -axis?Review question
Can we find this area by integrating with respect to $y$ ?Review question
Slippery slopesFood for thought · Developing
Speed vs velocityBuilding blocks · Developing
Tangent or normalPackage of problems · Developing
Walk-sortingBuilding blocks · Developing
Is the gradient function increasing or decreasing on this curve?Review question
Talking about curvesInvestigation · Introducing
Two-way calculusMany ways problem · Developing

5.1 · Indefinite integration as the reverse of differentiation.: • Students should know that a constant of integration is required.

6 resources

What happens if we integrate, then differentiate? recommendedReview question
Can we find $\int 1/(\sqrt{x+2} - \sqrt{x+1}) dx$ ?Review question
Can we find a curve from its gradient function?Review question
If $dy/dx$ is inversely proportional to $x^2$ , can we find $y$ ?Review question
Given three clues, can we find the equation of the curve?Review question
Thinking constantlyBuilding blocks · Developing

5.2 · Integration of xⁿ and related sums, differences and constant multiples.: • (Excluding n = −1 and related sums, differences and multiples). For example, the ability to integrate expressions such as 1⁄2x² − 3x⁻¹ and (x + 2)²⁄x is expected. Given f ′(x) and a point on the curve, students should be able to find an equation of the curve in the form y = f(x).

2 resources

What happens if we integrate, then differentiate? recommendedReview question
Can we find $\int 1/(\sqrt{x+2} - \sqrt{x+1}) dx$ ?Review question