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Maths — Pure Mathematics 2

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Maths — Pure Mathematics 2

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Home/Edexcel/IAL/Maths — Pure Mathematics 2/Underground Mathematics

Underground Mathematics

Exam code:YMA01

Cambridge's Underground Mathematics Combining Functions, Geometry of Equations, Circles, Introducing Calculus, Product Rule & Integration by Parts, Chain Rule & Integration by Substitution, Trigonometry: Triangles to Functions, Trigonometry: Compound Angles, Exponentials & Logarithms, Sequences, Counting & Binomials, Thinking about Algebra, Divisibility & Induction, Calculus Of Powers, Calculus Meets Functions, Calculus Trig Log and Polynomials & Rational Functions collections for Edexcel IAL Maths — 302 resources mapped across 19 specification points, each with suggestions and worked solutions.

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19 specification points

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

1.1 · Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof stated below:

57 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
If an AP has $S_{10}=3S_5$ , what's the ratio of $u_{10}$ to $u_5$ ?Review question
Common termsPackage of problems · Developing
Change one thingRich example · Introducing
If AP terms $a_4,a_8, a_{16}$ are in GP, are $a_3, a_6, a_{12}$ in GP too?Review question
How are the $p$ th, $q$ th and $r$ th terms of an AP related?Review question
If $u_n = a + bn + c2^n$ , what's the sum of the first $n$ terms?Review question
Can we sum the first $2n$ terms of $1,1,2,\frac{1}{2},4,\frac{1}{4},8,\frac{1}{8},..$ ?Review question
Can we use the trapezium rule to estimate this integral?Review question
If a GP has $S_n=(3^n - 2^n)/2^{n-5}$ , what's its common ratio?Review question
If $f(n) > k$ for all $n \geq 1$ , what can we say about $k$ ?Review question
If $27$ , $x$ , $y$ are in GP, with sum $21$ , what are $x$ and $y$ ?Review question
When does the sum of this series equal $60$ ?Review question
If $S_n = 6 - 2^{n+1}/3^{n-1}$ , can we show that we have a GP?Review question
What is the sum of the multiples of 5 or 8 but not both?Review question
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$ ?Review question
Should I hire a TV or borrow to buy it?Review question
What can we tell about the signs of this arithmetic sequence?Review question
Given $S_n = S_{3n}$ , can we express $a$ in terms of $n$ ?Review question
Can we sum $1^2-2^2+3^2-4^2+\dotsb+ (2n-1)^2-(2n)^2$ ?Review question
What's the next number to occur in both sequences?Review question
Can we find the sum of the integers from $2k$ to $4k$ inclusive?Review question
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression?Review question
Can we sum from $1000$ to $2000$ excluding multiples of 5?Review question
When does the sum of this series first exceed $2999/4000$ ?Review question
How are the sums of $n$ , $2n$ and $3n$ terms connected here?Review question
What are the first term and common ratio given two facts about this geometric progression?Review question
Can we compose these polynomial functions?Review question
Can we compose two simple functions?Review question

1.2 · Proof by exhaustion: • Proof by exhaustion. This involves trying all the options. Suppose x and y are odd integers less than 7. Prove that their sum is divisible by 2.

28 resources

If an AP has $S_{10}=3S_5$ , what's the ratio of $u_{10}$ to $u_5$ ? recommendedReview question
Common termsPackage of problems · Developing
Change one thingRich example · Introducing
If AP terms $a_4,a_8, a_{16}$ are in GP, are $a_3, a_6, a_{12}$ in GP too?Review question
How are the $p$ th, $q$ th and $r$ th terms of an AP related?Review question
If $u_n = a + bn + c2^n$ , what's the sum of the first $n$ terms?Review question
Can we sum the first $2n$ terms of $1,1,2,\frac{1}{2},4,\frac{1}{4},8,\frac{1}{8},..$ ?Review question
Can we use the trapezium rule to estimate this integral?Review question
If a GP has $S_n=(3^n - 2^n)/2^{n-5}$ , what's its common ratio?Review question
If $f(n) > k$ for all $n \geq 1$ , what can we say about $k$ ?Review question
If $27$ , $x$ , $y$ are in GP, with sum $21$ , what are $x$ and $y$ ?Review question
When does the sum of this series equal $60$ ?Review question
If $S_n = 6 - 2^{n+1}/3^{n-1}$ , can we show that we have a GP?Review question
What is the sum of the multiples of 5 or 8 but not both?Review question
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$ ?Review question
Should I hire a TV or borrow to buy it?Review question
What can we tell about the signs of this arithmetic sequence?Review question
Given $S_n = S_{3n}$ , can we express $a$ in terms of $n$ ?Review question
Can we sum $1^2-2^2+3^2-4^2+\dotsb+ (2n-1)^2-(2n)^2$ ?Review question
What's the next number to occur in both sequences?Review question
Can we find the sum of the integers from $2k$ to $4k$ inclusive?Review question
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression?Review question
Can we sum from $1000$ to $2000$ excluding multiples of 5?Review question
When does the sum of this series first exceed $2999/4000$ ?Review question
How are the sums of $n$ , $2n$ and $3n$ terms connected here?Review question
What are the first term and common ratio given two facts about this geometric progression?Review question
Can we compose these polynomial functions?Review question
Can we compose two simple functions?Review question

1.3 · Disproof by counter example. Algebra and functions: • Disproof by counter example – show that the statement “n² − n + 1 is a prime number for all values of n” is untrue.

28 resources

If an AP has $S_{10}=3S_5$ , what's the ratio of $u_{10}$ to $u_5$ ? recommendedReview question
Common termsPackage of problems · Developing
Change one thingRich example · Introducing
If AP terms $a_4,a_8, a_{16}$ are in GP, are $a_3, a_6, a_{12}$ in GP too?Review question
How are the $p$ th, $q$ th and $r$ th terms of an AP related?Review question
If $u_n = a + bn + c2^n$ , what's the sum of the first $n$ terms?Review question
Can we sum the first $2n$ terms of $1,1,2,\frac{1}{2},4,\frac{1}{4},8,\frac{1}{8},..$ ?Review question
Can we use the trapezium rule to estimate this integral?Review question
If a GP has $S_n=(3^n - 2^n)/2^{n-5}$ , what's its common ratio?Review question
If $f(n) > k$ for all $n \geq 1$ , what can we say about $k$ ?Review question
If $27$ , $x$ , $y$ are in GP, with sum $21$ , what are $x$ and $y$ ?Review question
When does the sum of this series equal $60$ ?Review question
If $S_n = 6 - 2^{n+1}/3^{n-1}$ , can we show that we have a GP?Review question
What is the sum of the multiples of 5 or 8 but not both?Review question
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$ ?Review question
Should I hire a TV or borrow to buy it?Review question
What can we tell about the signs of this arithmetic sequence?Review question
Given $S_n = S_{3n}$ , can we express $a$ in terms of $n$ ?Review question
Can we sum $1^2-2^2+3^2-4^2+\dotsb+ (2n-1)^2-(2n)^2$ ?Review question
What's the next number to occur in both sequences?Review question
Can we find the sum of the integers from $2k$ to $4k$ inclusive?Review question
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression?Review question
Can we sum from $1000$ to $2000$ excluding multiples of 5?Review question
When does the sum of this series first exceed $2999/4000$ ?Review question
How are the sums of $n$ , $2n$ and $3n$ terms connected here?Review question
What are the first term and common ratio given two facts about this geometric progression?Review question
Can we compose these polynomial functions?Review question
Can we compose two simple functions?Review question

2.1 · Simple algebraic division; use of the Factor Theorem and the Remainder Theorem. Coordinate geometry in the (x, y) plane: • Only division by (ax + b) or (ax – b) will be required, e.g. students should know that if f(x) = 0 when x = b⁄a, then (ax – b) is a factor of f(x). Students may be required to factorise cubic expressions such as x³ + 3x² − 4 and 6x³ + 11x² − x − 6. Students should be familiar with the terms ‘quotient’ and ‘remainder’ and be able to determine the remainder when the polynomial f(x) is divided by (ax + b).

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

3.1 · Coordinate geometry of the circle using the equation of a circle in the form (x − a)² + (y − b)² = r² and including use of the following circle properties: • (i) the angle in a semicircle is a right angle; (ii) the perpendicular from the centre to a chord bisects the chord; (iii) the perpendicularity of radius and tangent. • Students should be able to find the radius and the coordinates of the centre of the circle, given the equation of the circle and vice versa.

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

4.1 · Sequences, including those given by a formula for the nth term and those generated by a simple relation of the form xₙ₊₁ = f(xₙ).

30 resources

Sort it out recommendedBuilding blocks · Introducing
Connect three?Food for thought · Developing
Common termsPackage of problems · Developing
If an AP has $S_{10}=3S_5$ , what's the ratio of $u_{10}$ to $u_5$ ?Review question
Change one thingRich example · Introducing
If AP terms $a_4,a_8, a_{16}$ are in GP, are $a_3, a_6, a_{12}$ in GP too?Review question
How are the $p$ th, $q$ th and $r$ th terms of an AP related?Review question
If $u_n = a + bn + c2^n$ , what's the sum of the first $n$ terms?Review question
Can we sum the first $2n$ terms of $1,1,2,\frac{1}{2},4,\frac{1}{4},8,\frac{1}{8},..$ ?Review question
Can we use the trapezium rule to estimate this integral?Review question
If a GP has $S_n=(3^n - 2^n)/2^{n-5}$ , what's its common ratio?Review question
If $f(n) > k$ for all $n \geq 1$ , what can we say about $k$ ?Review question
If $27$ , $x$ , $y$ are in GP, with sum $21$ , what are $x$ and $y$ ?Review question
When does the sum of this series equal $60$ ?Review question
If $S_n = 6 - 2^{n+1}/3^{n-1}$ , can we show that we have a GP?Review question
What is the sum of the multiples of 5 or 8 but not both?Review question
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$ ?Review question
Should I hire a TV or borrow to buy it?Review question
What can we tell about the signs of this arithmetic sequence?Review question
Given $S_n = S_{3n}$ , can we express $a$ in terms of $n$ ?Review question
Can we sum $1^2-2^2+3^2-4^2+\dotsb+ (2n-1)^2-(2n)^2$ ?Review question
What's the next number to occur in both sequences?Review question
Can we find the sum of the integers from $2k$ to $4k$ inclusive?Review question
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression?Review question
Can we sum from $1000$ to $2000$ excluding multiples of 5?Review question
When does the sum of this series first exceed $2999/4000$ ?Review question
How are the sums of $n$ , $2n$ and $3n$ terms connected here?Review question
What are the first term and common ratio given two facts about this geometric progression?Review question
Can we compose these polynomial functions?Review question
Can we compose two simple functions?Review question

4.2 · Understand and work with arithmetic sequences and series, including the formula for the nth term and the sum of a finite arithmetic series; the sum of the first n natural numbers.: • The proof of the sum formula should be known. Understanding of Σ notation will expected.

28 resources

If an AP has $S_{10}=3S_5$ , what's the ratio of $u_{10}$ to $u_5$ ? recommendedReview question
Common termsPackage of problems · Developing
Change one thingRich example · Introducing
If AP terms $a_4,a_8, a_{16}$ are in GP, are $a_3, a_6, a_{12}$ in GP too?Review question
How are the $p$ th, $q$ th and $r$ th terms of an AP related?Review question
If $u_n = a + bn + c2^n$ , what's the sum of the first $n$ terms?Review question
Can we sum the first $2n$ terms of $1,1,2,\frac{1}{2},4,\frac{1}{4},8,\frac{1}{8},..$ ?Review question
Can we use the trapezium rule to estimate this integral?Review question
If a GP has $S_n=(3^n - 2^n)/2^{n-5}$ , what's its common ratio?Review question
If $f(n) > k$ for all $n \geq 1$ , what can we say about $k$ ?Review question
If $27$ , $x$ , $y$ are in GP, with sum $21$ , what are $x$ and $y$ ?Review question
When does the sum of this series equal $60$ ?Review question
If $S_n = 6 - 2^{n+1}/3^{n-1}$ , can we show that we have a GP?Review question
What is the sum of the multiples of 5 or 8 but not both?Review question
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$ ?Review question
Should I hire a TV or borrow to buy it?Review question
What can we tell about the signs of this arithmetic sequence?Review question
Given $S_n = S_{3n}$ , can we express $a$ in terms of $n$ ?Review question
Can we sum $1^2-2^2+3^2-4^2+\dotsb+ (2n-1)^2-(2n)^2$ ?Review question
What's the next number to occur in both sequences?Review question
Can we find the sum of the integers from $2k$ to $4k$ inclusive?Review question
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression?Review question
Can we sum from $1000$ to $2000$ excluding multiples of 5?Review question
When does the sum of this series first exceed $2999/4000$ ?Review question
How are the sums of $n$ , $2n$ and $3n$ terms connected here?Review question
What are the first term and common ratio given two facts about this geometric progression?Review question
Can we compose these polynomial functions?Review question
Can we compose two simple functions?Review question

4.3 · Increasing sequences, decreasing sequences and periodic sequences.

30 resources

Sort it out recommendedBuilding blocks · Introducing
Connect three?Food for thought · Developing
Common termsPackage of problems · Developing
If an AP has $S_{10}=3S_5$ , what's the ratio of $u_{10}$ to $u_5$ ?Review question
Change one thingRich example · Introducing
If AP terms $a_4,a_8, a_{16}$ are in GP, are $a_3, a_6, a_{12}$ in GP too?Review question
How are the $p$ th, $q$ th and $r$ th terms of an AP related?Review question
If $u_n = a + bn + c2^n$ , what's the sum of the first $n$ terms?Review question
Can we sum the first $2n$ terms of $1,1,2,\frac{1}{2},4,\frac{1}{4},8,\frac{1}{8},..$ ?Review question
Can we use the trapezium rule to estimate this integral?Review question
If a GP has $S_n=(3^n - 2^n)/2^{n-5}$ , what's its common ratio?Review question
If $f(n) > k$ for all $n \geq 1$ , what can we say about $k$ ?Review question
If $27$ , $x$ , $y$ are in GP, with sum $21$ , what are $x$ and $y$ ?Review question
When does the sum of this series equal $60$ ?Review question
If $S_n = 6 - 2^{n+1}/3^{n-1}$ , can we show that we have a GP?Review question
What is the sum of the multiples of 5 or 8 but not both?Review question
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$ ?Review question
Should I hire a TV or borrow to buy it?Review question
What can we tell about the signs of this arithmetic sequence?Review question
Given $S_n = S_{3n}$ , can we express $a$ in terms of $n$ ?Review question
Can we sum $1^2-2^2+3^2-4^2+\dotsb+ (2n-1)^2-(2n)^2$ ?Review question
What's the next number to occur in both sequences?Review question
Can we find the sum of the integers from $2k$ to $4k$ inclusive?Review question
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression?Review question
Can we sum from $1000$ to $2000$ excluding multiples of 5?Review question
When does the sum of this series first exceed $2999/4000$ ?Review question
How are the sums of $n$ , $2n$ and $3n$ terms connected here?Review question
What are the first term and common ratio given two facts about this geometric progression?Review question
Can we compose these polynomial functions?Review question
Can we compose two simple functions?Review question

4.4 · Understand and work with For example, given the sum of a series students should be geometric sequences and series, able to use logs to find the value of n . including the formulae for the n th The proof of the sum formula for a finite series should be term and the sum of a finite known. geometric series; the sum to infinity of a convergent geometric series, The sum to infinity may be expressed as S ∞ . including the use of | r| < 1.

14 resources

Square spirals recommendedFood for thought · Introducing
Can we find the sum of the integers from $2k$ to $4k$ inclusive?Review question
What can we say if $a, x_1, x_2, x_3, x_4, x_5, b$ are in arithmetic progression?Review question
Can we sum from $1000$ to $2000$ excluding multiples of 5?Review question
When does the sum of this series first exceed $2999/4000$ ?Review question
How are the sums of $n$ , $2n$ and $3n$ terms connected here?Review question
What are the first term and common ratio given two facts about this geometric progression?Review question
Bouncing to nothingRich example · Developing
Can you find... series editionMany ways problem · Developing
A puzzling pentagonGo and think about it... · Developing
Given the sequence $(x_n)$ , what is $\sum_{k=0}^\infty 1/x_k$ ?Review question
How do we add the odd-numbered terms of a geometric series?Review question
When does $1 + 2/3 + (2/3)^2 + \dotsb$ first exceed $0.9999S_\infty$ ?Review question
Can we find the area between $y=e^{-x}$ and $y=e^{-x}\sin x$ ?Review question

4.5 · Binomial expansion of (a + bx)ⁿ for positive integer n. onentials and logarithms: • n n The notations n!, and Cᵣ may be used. r

13 resources

Can we find the coefficient of $x^2$ in $(5 + 4x)(1 - x/2)^{10}$ ? recommendedReview question
Which power of $x$ has the greatest coefficient?Review question
Can we find the minimum value of $x^2+2/x$ for $x$ positive?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

5.1 · y = aˣ and its graph.: • a > 0, a ≠ 1

3 resources

An irrational inequality recommendedGo and think about it... · Developing
Can we sketch the inverse of this composite function?Review question
To the limitScaffolded task · Introducing

5.2 · Laws of logarithms. To include log a ( xy ) ≡ log a x + log a y , log a  x  ≡ log a x − log a y ,   y   log a ( x k ) ≡ k log a x , log a  1  ≡ − log a x ,   x   log a a = 1 where a, x, y > 0, a ≠ 1.

19 resources

1950s calculators recommendedBuilding blocks · Developing
Changing basesPackage of problems · Developing
Factorial fragmentsFluency exercise · Developing
Logarithm latticeFluency exercise · Developing
Logarithm lineupFluency exercise · Developing
Proving the laws of logarithmsScaffolded task · Developing
Which is the smallest of these surd, log and trig expressions?Review question
How are the $p$ th, $q$ th and $r$ th terms of an AP related?Review question
If $yz=a^2$ , can we prove that $1/(a+y) + 1/(a+z) = 1/a$ ?Review question
Which of these log and trig expressions is the largest?Review question
Which is the largest of these four logarithms?Review question
Can we find these log expressions in terms of $\log_9 x$ ?Review question
Can we use the laws of logs to find $x$ and $y$ ?Review question
Can we find the ranges of $f$ and $g$ , and the function $f \circ g$ ?Review question
Can we find bounds for $\log_2 3$ ?Review question
Which of these four log expressions is the smallest?Review question
Can we evaluate these log expressions without using a calculator?Review question
See the powerBuilding blocks · Introducing
Summing to oneBuilding blocks · Developing

5.3 · The solution of equations of the form aˣ = b.: • Students may use the change of base formula.

15 resources

Powerful fractions recommendedFood for thought · Developing
How many roots are there to $x=8^{\log_2x} -9^{\log_3x}-4^{\log_2x}+\log_{0.5}{0.25}$ ?Review question
Can we solve these three simultaneous log equations?Review question
Can we solve these simultaneous exponential equations?Review question
How large is $2^{100}$ , and what is its starting digit?Review question
When does this inequality have a finite number of integer solutions?Review question
Given this log equation, when can $a$ be largest?Review question
Can we prove these bounds for this fraction involving surds?Review question
When does $9^x - 3^{x+1} = k$ have one or more real solutions?Review question
How many real roots does $8^x+4=4^x+2^{x+2}$ have?Review question
Can we solve $\log_3 x + 2 \log_x 3 = -3$ ?Review question
Can we solve the equation $\log_2 x + \log_3 x = 1 + \log_4 x$ ?Review question
Can we find $a$ and $b$ as powers of $10$ ?Review question
Can we use $\log_{10}$ to find the first digit of $3^{100}$ ?Review question
To log or not to log?Problem requiring decisions · Developing · teacher support included

6.1 · Knowledge and use of sinθ tan θ = , and sin² cosθ⁄θ+cos² θ = 1.

38 resources

Can we prove this trig identity to solve this equation? recommendedReview question
Equation or identity? (I)Package of problems · Developing
How many solutions does $\sin^2x+3\sin x\cos x+2\cos^2 x=0$ have?Review question
How many solutions do these simultaneous trig equations have?Review question
When does this equation in $x, y$ and $\theta$ represent a circle?Review question
How many solutions does this equation involving $\sin^8x$ have?Review question
Can we show this polynomial has roots $1$ , $\cos\theta$ and $\sin\theta$ ?Review question
How many solutions does $2 \cos^2x + 5\sin x = 4$ have?Review question
As $\theta$ varies, what's the largest this ratio of areas can be?Review question
When is $\sin^8 x+\cos^6 x$ equal to $1$ ?Review question
How many solutions does $7\sin x +2\cos ^2 x =5$ have?Review question
Slices of $\pi$ Many ways problem · Developing · teacher support included
When does this trig function have its maximum value?Review question
How many solutions does this composite function equation have?Review question
Equation or identity? (II)Package of problems · Developing · teacher support included
Inspecting identitiesMany ways problem · Developing
Lost but lovely: the haversineBigger picture · Developing
Proving half-angle formulaeScaffolded task · Developing
Can we solve the equation $3\cos x + 1 = 2\sin x$ ?Review question · teacher support included
If $bc=a^2+1$ , what is $\arctan[(a+b)^{-1}]+\arctan[(a+c)^{-1}]$ ?Review question · teacher support included
Can we solve these trig equations for $x, y$ and $z$ ?Review question
Can we find the value of $\sin 18^\circ$ as a surd?Review question · teacher support included
Can we solve $\tan\theta\tan(\theta+\pi/3)=2$ ?Review question · teacher support included
Can we write $\cos \theta - 2 \cos 3 \theta + \cos 5\theta$ using $\sin$ ?Review question · teacher support included
Can we write $\sin\theta$ and $\cos\theta$ in terms of $\tan(\theta/2)$ ?Review question
Can we simplify $\cos 3\theta \sin 2\theta - \cos 4\theta \sin \theta$ ?Review question · teacher support included
Using angles in a pentagon...Review question
An exact value for $\sin 15^{\circ}$ ...Review question · teacher support included
Can we sketch the curve $y = \cos x - \sin x +2$ ?Review question
Can we solve these simultaneous trigonometric equations?Review question
Can we prove $\sin a \sin b \leq \sin^2\frac{1}{2}(a+b)$ ?Review question · teacher support included
Can we find the height of a tower using its reflection?Review question · teacher support included
Can we solve $\cos\theta - \sin (2\theta) + \cos (3\theta) - \sin (4\theta) = 0$ ?Review question · teacher support included
For which $0^\circ < \theta < 500^\circ$ is $3\cos \theta + 4\sin \theta > 2$ ?Review question · teacher support included
If $\sin \alpha = 3/5, \cos \beta = 12/13$ , what's $\cos(\alpha + \beta)$ ?Review question · teacher support included
Why is $\cos \theta + \cos (\theta + 2\pi/3) + \cos(\theta + 4\pi/3)=0$ ?Review question · teacher support included
$t$ for $\tan$ Building blocks · Developing · teacher support included
Transformation... or not?Investigation · Developing

6.2 · Solution of simple trigonometric Students should be able to solve equations such as equations in a given interval.  π  3 sin  x +  = 4 for 0 < x < 2 π ,  2  cos ( x + 30 ° ) = 1 for −180 ° < x < 180 ° , 2 tan 2 x = 1 for 90 ° < x < 270 ° , 6 cos 2 x + sin x − 5 = 0 for 0  x < 360 ° , sin 2  x + π  = 1 for – π  x < π .  6  2  

35 resources

How many solutions does $\sin^2x+3\sin x\cos x+2\cos^2 x=0$ have? recommendedReview question
General solutionsFood for thought · Introducing
When is $\sin x$ or $\sin 2x$ larger than a half?Review question
Can we prove this trig identity to solve this equation?Review question
If $\sin X$ and $\cos Y$ are both negative, what can we say about $X + Y$ ?Review question
Which points in the $x$ - $y$ plane satisfy $\sin y=\sin x$ ?Review question
How many solutions does this equation involving $\sin^8x$ have?Review question
How many solutions does $\cos (\sin x)=\frac{1}{2}$ have?Review question
When is $\sin\theta > \sin 3\theta$ ?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
Equation or identity? (II)Package of problems · Developing · teacher support included
Inspecting identitiesMany ways problem · Developing
Lost but lovely: the haversineBigger picture · Developing
Proving half-angle formulaeScaffolded task · Developing
Can we solve the equation $3\cos x + 1 = 2\sin x$ ?Review question · teacher support included
If $bc=a^2+1$ , what is $\arctan[(a+b)^{-1}]+\arctan[(a+c)^{-1}]$ ?Review question · teacher support included
Can we solve these trig equations for $x, y$ and $z$ ?Review question
Can we find the value of $\sin 18^\circ$ as a surd?Review question · teacher support included
Can we solve $\tan\theta\tan(\theta+\pi/3)=2$ ?Review question · teacher support included
Can we write $\cos \theta - 2 \cos 3 \theta + \cos 5\theta$ using $\sin$ ?Review question · teacher support included
Can we write $\sin\theta$ and $\cos\theta$ in terms of $\tan(\theta/2)$ ?Review question
Can we simplify $\cos 3\theta \sin 2\theta - \cos 4\theta \sin \theta$ ?Review question · teacher support included
Using angles in a pentagon...Review question
An exact value for $\sin 15^{\circ}$ ...Review question · teacher support included
Can we sketch the curve $y = \cos x - \sin x +2$ ?Review question
Can we solve these simultaneous trigonometric equations?Review question
Can we prove $\sin a \sin b \leq \sin^2\frac{1}{2}(a+b)$ ?Review question · teacher support included
Can we find the height of a tower using its reflection?Review question · teacher support included
Can we solve $\cos\theta - \sin (2\theta) + \cos (3\theta) - \sin (4\theta) = 0$ ?Review question · teacher support included
For which $0^\circ < \theta < 500^\circ$ is $3\cos \theta + 4\sin \theta > 2$ ?Review question · teacher support included
If $\sin \alpha = 3/5, \cos \beta = 12/13$ , what's $\cos(\alpha + \beta)$ ?Review question · teacher support included
Why is $\cos \theta + \cos (\theta + 2\pi/3) + \cos(\theta + 4\pi/3)=0$ ?Review question · teacher support included
$t$ for $\tan$ Building blocks · Developing · teacher support included
Transformation... or not?Investigation · Developing

7.1 · Applications of differentiation to maxima and minima and stationary points, increasing and decreasing functions. To include applications to curve sketching. Maxima and minima problems may be set in the context of a practical problem.

38 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

8.1 · Evaluation of definite integrals.

10 resources

Additional integralsPackage of problems · Developing
Integral chasingFluency exercise · Developing
What does it mean if a function $f(x)=f(2\alpha-x)$ ?Review question
If we know $f(x)$ , can we find $\int_{-1}^1 f(x^2-1) \:dx$ ?Review question
Can we find the area under a parabola? recommendedReview question
When does this definite integral have a minimum?Review question
If $f(x)$ always satisfies this equation, what is $\int_{-1}^1 f(x)\: dx$ ?Review question
Slippery areasBuilding blocks · Introducing
Stretch the functionFood for thought · Developing
What else do you know?Food for thought · Developing

8.2 · Interpretation of the definite integral as the area under a curve.: • Students will be expected to be able to evaluate the area of a region bounded by a curve and given straight lines. For example, find the finite area bounded by the curve y = 6x − x² and the line y = 2x. ∫xdy will not be required. Students will be expected to be able to evaluate the area of a region bounded by two curves.

12 resources

Meaningful areasInvestigation · Developing
What volume is generated when $y = ax - x^2$ is rotated about the x-axis?Review question
Can we find the centre of mass of a solid hemisphere?Review question
Can we sketch the graph of $y^2 = x^3$ ?Review question
What's the area enclosed by $y=x(x-3)^2$ and a tangent?Review question
Can we find the area inside a straight line, a parabola and the x-axis?Review question
If we take a slice from this sphere, where is its centre of gravity?Review question
When does a cubic curve have two turning points?Review question
What's the area between $f_k(x) = x(x-k)(x-2)$ and the $x$ -axis?Review question
When is the area of this region bigger than the area of this one?Review question
Can we find the area enclosed by this quadratic and the $x$ -axis?Review question
Can we find the area between a parabola and a line? recommendedReview question

8.3 · Approximation of area under a For example, curve using the trapezium rule. use the trapezium rule to approximate 1 (2 x + 1) d x using four strips. ∫ 0 Use of increasing number of trapezia to improve accuracy and an estimate of the error may be required.

7 resources

Approximating areasProblem requiring decisions · Developing
When does the trapezium rule give the exact answer here?Review question
Can we use the trapezium rule to estimate this integral?Review question
Can the trapezium rule help us find the area under $y = x\ln x$ ? recommendedReview question
Can we approximate the area under this sine curve?Review question
Can we apply the trapezium rule to $f(x) = x/2 - \lfloor x/2 \rfloor$ ?Review question
Underneath the archesInvestigation · Developing