Specialist Mathematics · QCAA
Full Specialist Mathematics syllabus
Drill from units to topics, subtopics and individual learning objectives. Every LO is wired up to AI-marked practice questions.
▶1 — Combinatorics, proof, vectors and matrices56 LOs
▶Combinatorics13 LOs
▶Introduction to counting techniques3 LOs
●Use the addition principle.
●Use the inclusion-exclusion principle formulas to determine the number of elements in the union of two and the union of three sets. |A ∪ B| = |A| + |B| − |A ∩ B|] |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|
●Use the multiplication principle.
▶Permutations (ordered arrangements) and combinations (unordered selections)10 LOs
●Define and use combinations.
●Define and use permutations.
●Model and solve problems that involve permutations and combinations including probability problems, with and without technology. Specialist Mathematics 2025 v1.4
●Solve problems that involve combinations with restrictions including specific objects grouped together and selection from multiple groups.
●Solve problems that involve combinations.
●Solve problems that involve permutations with restrictions including repeated objects, specific objects grouped together and selection from multiple groups.
●Solve problems that involve permutations.
●Use factorial notation.
●Use the notation (𝑛 𝑟) and 𝐶𝑟 𝑛 to represent the number of ways of selecting 𝑟 objects from 𝑛 distinct objects where order is not important. 𝐶𝑟 𝑛 = (𝑛 𝑟) = 𝑛! 𝑟!(𝑛−𝑟)!
●Use the notation 𝑃𝑛 𝑟 to represent the number of ways of selecting 𝑟 objects from 𝑛 distinct objects where order is important. 𝑃𝑟 𝑛 = 𝑛! (𝑛−𝑟)! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) × … × (𝑛 − 𝑟 + 1)
▶Introduction to proof10 LOs
▶The nature of proof7 LOs
●Define and use set notation of number systems, including integers (ℤ), positive integers (ℤ+), negative integers (ℤ−), rational numbers (ℚ), irrational numbers (ℚ′), and real numbers (ℝ).
●Use examples and counterexamples.
●Use implication, converse, equivalence, negation, contrapositive.
●Use proof by contradiction.
●Use the quantifiers ‘for all’ (∀) and ‘there exists’ (∃).
●Use the set notation symbol ‘is an element of’ (∈).
●Use the symbols for implication ( ⇒), equivalence ( ⟺), and equality ( =).
▶Rational and irrational numbers3 LOs
●Express rational numbers as terminating or eventually recurring decimals and vice versa.
●Prove irrationality by contradiction.
●Prove results involving integers, e.g. proving that the product of two consecutive odd numbers is an odd number and 5𝑛2 + 3𝑛 + 6 ∀𝑛 ∈ ℤ is an even number.
▶Vectors in the plane16 LOs
▶Representing vectors in the plane by directed line segments8 LOs
●Define and use the magnitude and direction of a vector.
●Examine examples of vectors including displacement, velocity and force.
●Represent a vector in the plane using a combination of the sum, difference and scalar multiple of other vectors. Specialist Mathematics 2025 v1.4
●Represent and use a scalar multiple of a vector.
●Understand and use vector equality.
●Understand and use vector notation: 𝐴𝐵⃗⃗⃗⃗⃗, 𝑐 ~, 𝒅 and unit vector notation 𝒏̂.
●Understand the difference between a scalar and a vector including distance and displacement, speed and velocity, and magnitude of force and force.
●Use the triangle rule to represent the resultant vector from the sum and difference of two vectors.
▶Vectors in two dimensions8 LOs
●Calculate and use a unit vector, 𝒏̂, in the plane. 𝒏̂ = 𝒏 |𝒏|
●Calculate the magnitude and direction of a vector. |𝒂| = |(𝑎1 𝑎2)| = √𝑎12 + 𝑎22 tan(𝜃) = 𝑦 𝑥, 𝑥 ≠ 0
●Convert between Cartesian form and polar form, with and without technology.
●Define and use unit vectors and the perpendicular unit vectors 𝒊̂ and 𝒋̂.
●Express a vector in Cartesian (component) form using the unit vectors 𝒊̂ and 𝒋̂.
●Understand and express a vector in the plane in polar form using the notation (𝑟, 𝜃).
●Understand and use the Cartesian form and polar form of a vector.
●Use ordered pair notation (𝑥, 𝑦) and column vector notation (𝑥 𝑦) to represent a position vector in two dimensions.
▶Algebra of vectors in two dimensions11 LOs
▶Algebra of vectors in two dimensions11 LOs
●Apply the scalar product to vectors expressed in Cartesian form.
●Define and use a vector representing a section of a line segment, including the midpoint of a line segment.
●Define and use multiplication by a scalar of a vector in Cartesian form.
●Define and use scalar and vector projections of vectors. scalar projection of 𝒂 on 𝒃: |𝒂| cos(𝜃) = 𝒂 ⋅ 𝒃̂ vector projection of 𝒂 on 𝒃: |𝒂| cos(𝜃) 𝒃̂ = (𝒂 ⋅ 𝒃̂)𝒃̂ = (𝒂⋅𝒃 𝒃⋅𝒃) 𝒃
●Define and use the scalar (dot) product. 𝒂 ⋅ 𝒃 = |𝒂||𝒃| cos(𝜃) (𝑎1 𝑎2) ⋅ (𝑏1 𝑏2) = 𝑎1 𝑏1 + 𝑎2 𝑏2
●Determine a vector between two points.
●Examine and use addition and subtraction of vectors in Cartesian form.
●Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular.
●Model and solve problems that involve displacement, force, velocity and relative velocity using the above concepts.
●Model and solve problems that involve motion of a body in equilibrium situations, including vector applications related to smooth inclined planes (excluding situations with pulleys and connected bodies). Specialist Mathematics 2025 v1.4
●Resolve vectors into 𝒊̂ and 𝒋̂ components.
▶Matrices6 LOs
▶Matrix arithmetic and algebra6 LOs
●Calculate the determinant and multiplicative inverse of 2 × 2 matrices, with and without technology. If 𝑨 = [𝑎 𝑏 𝑐 𝑑] then det(𝑨) = 𝑎𝑑 − 𝑏𝑐 𝑨−1 = [𝑎 𝑏 𝑐 𝑑]−1 = 1 det(𝑨) [ 𝑑 −𝑏 −𝑐 𝑎 ], det(𝑨) ≠ 0
●Define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication, multiplicative identity and multiplicative inverse.
●Model and solve problems that involve matrices of up to dimension 2 × 2, including the solution of systems of linear equations, with and without technology.
●Understand the matrix definition and notation.
●Use matrix algebra properties, including 𝑨 + 𝑩 = 𝑩 + 𝑨 (commutative law for addition) 𝑨 + 0 = 𝑨 (additive identity) 𝑨 + (−𝑨) = 0 (additive inverse) 𝑨𝑰 = 𝑨 = 𝑰𝑨 (multiplicative identity) 𝑨𝑨−1 = 𝑰 = 𝑨−1𝑨 (multiplicative inverse) 𝑨(𝑩 + 𝑪) = 𝑨𝑩 + 𝑨𝑪 (left distributive law) (𝑩 + 𝑪)𝑨 = 𝑩𝑨 + 𝑪𝑨 (right distributive law)
●Use matrix algebra to solve matrix equations that involve matrices of up to dimension 2 × 2, including those of the form 𝑨𝑿 = 𝑩, 𝑿𝑨 = 𝑩 and 𝑨𝑿 + 𝑩𝑿 = 𝑪, with and without technology.
▶2 — Complex numbers, further proof, trigonometry, functions and transformations40 LOs
▶Complex numbers9 LOs
▶Introduction to complex numbers5 LOs
●Define and use set notation of the number system for complex numbers (ℂ).
●Define the imaginary number 𝑖 as a root (solution) of the equation 𝑥2 = −1.
●Determine and use complex conjugates.
●Perform complex-number arithmetic: addition, subtraction, multiplication and division, with and without technology.
●Use complex numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are the real and imaginary parts (components) Re(𝑧) and Im(𝑧) of a complex number 𝑧.
▶The complex plane (the Argand plane)4 LOs
●Sketch and use complex numbers as points in the complex plane with real and imaginary parts as Cartesian coordinates.
●Understand and use addition of complex numbers as vector addition in the complex plane.
●Understand and use location of complex conjugates in the complex plane.
●Understand and use multiplication by a complex number as a linear transformation in the complex plane.
▶Complex arithmetic and algebra9 LOs
▶Complex arithmetic using polar form6 LOs
●Convert between Cartesian form and polar form.
●Express a complex number in Cartesian form 𝑧 = 𝑎 + 𝑏𝑖 and polar form. 𝑧 = 𝑟 (cos(𝜃) + 𝑖 sin(𝜃)) or 𝑧 = 𝑟 cis(𝜃)
●Sketch and use complex numbers in polar form as polar coordinates. Specialist Mathematics 2025 v1.4
●Understand and use multiplication, division of complex numbers in polar form and the geometric interpretation of these. 𝑧1 𝑧2 = 𝑟1 𝑟2 cis(𝜃1 + 𝜃2) 𝑧1 𝑧2 = 𝑟 1 𝑟 2 cis(𝜃1 − 𝜃2)
●Understand the difference between the argument, arg(𝑧), and the principal argument, Arg(𝑧) of a non-zero complex number 𝑧. arg(𝑧) = Arg(𝑧) + 2𝜋𝑛, 𝑛 ∈ ℤ
●Use the modulus |𝑧| of a complex number 𝑧 and the principal argument Arg(𝑧) of a non-zero complex number 𝑧. |𝑧| = √𝑎2 + 𝑏2 Arg(𝑧) = 𝜃, tan(𝜃) = 𝑏 𝑎, −𝜋 < 𝜃 ≤ 𝜋, 𝑎 ≠ 0
▶Subsets of the complex plane (the Argand plane)1 LO
●Identify and sketch subsets of the complex plane determined by straight lines and circles, e.g. |𝑧 − 3𝑖| < 4, 𝜋 4 ≤ Arg(𝑧) ≤ 3𝜋 4, Re(𝑧) > Im(𝑧) and |𝑧 − 1| = 2|𝑧 − 𝑖|.
▶Roots of real quadratic equations2 LOs
●Determine and use linear factors of quadratic polynomials with real coefficients that involve the complex conjugate root theorem, e.g. determine the coefficients of a real quadratic equation given one complex root.
●Determine complex conjugate solutions of real quadratic equations with real coefficients using factorisation, completing the square and the quadratic formula, with and without technology.
▶Circle and geometric proofs6 LOs
▶Circle properties and their proofs2 LOs
●Prove the circle properties – the angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc – an angle in a semicircle is a right angle – angles at the circumference of a circle subtended by the same arc are equal – the alternate segment theorem – the opposite angles of a cyclic quadrilateral are supplementary and its converse – a tangent drawn to a circle is perpendicular to the radius at the point of contact and its converse.
●Solve problems finding unknown angles and lengths and prove further results using the circle properties listed above.
▶Geometric proofs using vectors4 LOs
●Prove an angle in a semicircle is a right angle. Specialist Mathematics 2025 v1.4
●Prove midpoints of the sides of a quadrilateral join to form a parallelogram.
●Prove the diagonals of a parallelogram meet at right angles if and only if it is a rhombus.
●Prove the sum of the squares of the lengths of a parallelogram’s diagonals is equal to the sum of the squares of the lengths of the sides.
▶Trigonometry and functions9 LOs
▶Sketching graphs2 LOs
●Understand and use the relationship between the graph of 𝑦 = 𝑓(𝑥) and the graphs of 𝑦 = 1 𝑓(𝑥), 𝑦 = |𝑓(𝑥)| and 𝑦 = 𝑓(|𝑥|).
●Use and apply the notation |𝑥| for the absolute value for the real number 𝑥 and the graph of 𝑦 = |𝑥|.
▶The reciprocal trigonometric functions, secant, cosecant and cotangent (31 LO
●Define and use the reciprocal trigonometric functions to determine their simplified exact values and sketch their graphs.
▶Trigonometric identities6 LOs
●Convert sums 𝑎 cos(𝑥) + 𝑏 sin(𝑥) to 𝑅 cos(𝑥 ± 𝛼) or 𝑅 sin(𝑥 ± 𝛼) and apply these to sketch graphs.
●Model and solve problems that involve equations of the form 𝑎 cos(𝑥) + 𝑏 sin(𝑥) = 𝑐.
●Prove and apply multi-angle trigonometric identities up to angles of 4𝑥 using the identities listed above, e.g. cos(4𝑥) = 8 cos4(𝑥) − 8 cos2(𝑥) + 1 and cosec(2𝑥) − cot(2𝑥) = tan(𝑥). Specialist Mathematics 2025 v1.4
●Prove and apply the angle sum, difference and double-angle identities for sines and cosines. sin(𝐴 + 𝐵) = sin(𝐴) cos(𝐵) + cos(𝐴) sin(𝐵) sin(𝐴 − 𝐵) = sin(𝐴) cos(𝐵) − cos(𝐴) sin(𝐵) cos(𝐴 + 𝐵) = cos(𝐴) cos(𝐵) − sin(𝐴) sin(𝐵) cos(𝐴 − 𝐵) = cos(𝐴) cos(𝐵) + sin(𝐴) sin(𝐵) sin(2𝐴) = 2 sin(𝐴) cos(𝐴) cos(2𝐴) = cos2(𝐴) − sin2(𝐴) = 1 − 2 sin2(𝐴) = 2 cos2(𝐴) − 1
●Prove and apply the identities for products of sines and cosines expressed as sums and differences. sin(𝐴) sin(𝐵) = 1 2 (cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)) cos(𝐴) cos(𝐵) = 1 2 (cos(𝐴 − 𝐵) + cos(𝐴 + 𝐵)) sin(𝐴) cos(𝐵) = 1 2 (sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)) cos(𝐴) sin(𝐵) = 1 2 (sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵))
●Prove and apply the Pythagorean identities. sin2(𝐴) + cos2(𝐴) = 1 tan2(𝐴) + 1 = sec2(𝐴) cot2(𝐴) + 1 = cosec2(𝐴)
▶Matrices and transformations7 LOs
▶Transformations in the plane7 LOs
●Apply these transformations to points in the plane and polygons.
●Determine geometric results by matrix multiplications, e.g. showing that the combined effect of two reflections in lines through the origin is a rotation.
●Understand and use composition of linear transformations and the corresponding matrix products.
●Understand and use inverses of linear transformations and the relationship with the matrix inverse.
●Understand and use the relationship between the determinant and the effect of a linear transformation on area.
●Understand translations and their representation as column vectors.
●Use basic linear transformations: dilations of the form (𝑥, 𝑦) → (𝑎𝑥, 𝑏𝑦), rotations about the origin and reflection in a line that passes through the origin, and the representations of these transformations by 2 × 2 matrices. dilation of factor 𝑎 parallel to the 𝑥-axis and factor 𝑏 parallel to the 𝑦-axis: [𝑎 0 0 𝑏] rotation of angle 𝜃 anticlockwise about the origin: [cos(𝜃) − sin(𝜃) sin(𝜃) cos(𝜃)] reflection in the line 𝑦 = 𝑥 tan(𝜃): [cos(2𝜃) sin(2𝜃) sin(2𝜃) − cos(2𝜃)]
▶3 — Further complex numbers, proof, vectors and matrices47 LOs
▶Further complex numbers7 LOs
▶Complex arithmetic using polar form2 LOs
●Prove complex number identities involving modulus and argument, e.g. 𝑧 𝑧̅ = |𝑧|2, |𝑧1| |𝑧2| = |𝑧1 𝑧2| and arg(𝑧1 𝑧2) = arg(𝑧1) + arg(𝑧2).
●Use De Moivre’s theorem for integral powers. 𝑧𝑛 = 𝑟𝑛 cis (𝑛𝜃)
▶Roots of complex numbers2 LOs
●Determine and examine the 𝑛th roots of complex numbers and their location in the complex plane.
●Determine and examine the 𝑛th roots of unity and their location on the unit circle.
▶Factorisation of polynomials3 LOs
●Apply the factor theorem and the remainder theorem for polynomials.
●Solve polynomial equations over ℂ to order 4 including those with real and imaginary coefficients, e.g. solve 𝑧4 + 𝑧3 − 𝑧2 + 𝑧 − 2 = 0 and 𝑧3 − 2𝑖 𝑧2 + 𝑧 − 2𝑖 = 0.
●Understand and use the complex conjugate root theorem for polynomials with real coefficients, e.g. factorise a cubic polynomial with real coefficients given one factor.
▶Mathematical induction and trigonometric proofs6 LOs
▶Mathematical induction5 LOs
●Prove De Moivre’s theorem for powers of positive integers.
●Prove divisibility results for any positive integer 𝑛.
●Prove results for sums for any positive integer 𝑛.
●Understand the nature of inductive proof including the use of initial statement, assumption statement, inductive step and conclusion.
●Use sigma notation (Σ) to represent a sum, e.g. and
▶Trigonometric proofs using De Moivre’s theorem1 LO
●Prove multi-angle trigonometric identities up to angles of 4𝑥 by equating parts using the binomial expansion and De Moivre’s theorem, e.g. cos(3𝑥) = 4 cos3(𝑥) − 3 cos(𝑥) and sin(3𝑥) = 3 sin(𝑥) − 4 sin3(𝑥). Specialist Mathematics 2025 v1.4
▶Vectors in two and three dimensions22 LOs
▶Vectors in three dimensions7 LOs
●Calculate and use a unit vector, 𝒏̂, in three-dimensional space. 𝒏̂ = 𝒏 |𝒏|
●Calculate the magnitude of a vector |𝒂| = |( 𝑎1 𝑎2 𝑎3)| = √𝑎12 + 𝑎22 + 𝑎32
●Define and use the altitude angle 𝜑.
●Define and use unit vectors and the perpendicular unit vectors 𝒊̂, 𝒋̂ and 𝒌̂.
●Express a vector in Cartesian (component) form using the unit vectors 𝒊̂, 𝒋̂ and 𝒌̂.
●Use Cartesian coordinates for three-dimensional space, including plotting points.
●Use ordered triple notation (𝑥, 𝑦, 𝑧) and column vector notation ( 𝑥 𝑦 𝑧) to represent a position vector in three dimensions.
▶Algebra of vectors in three dimensions10 LOs
●Apply the scalar product to vectors expressed in Cartesian form.
●Determine a vector between two points.
●Examine and use addition and subtraction of vectors in Cartesian form.
●Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular.
●Model and solve problems that involve displacement, force, velocity and relative velocity using the above concepts.
●Use a vector representing a section of a line segment, including the midpoint of a line segment.
●Use multiplication by a scalar of a vector in Cartesian form.
●Use scalar and vector projections of vectors. scalar projection of 𝒂 on 𝒃: |𝒂| cos(𝜃) = 𝒂 ⋅ 𝒃̂ vector projection of 𝒂 on 𝒃: |𝒂| cos(𝜃) 𝒃̂ = (𝒂 ⋅ 𝒃̂)𝒃̂ = (𝒂⋅𝒃 𝒃⋅𝒃) 𝒃
●Use the scalar (dot) product. 𝒂 ⋅ 𝒃 = |𝒂||𝒃| cos(𝜃) ( 𝑎1 𝑎2 𝑎3) ⋅ ( 𝑏1 𝑏2 𝑏3) = 𝑎1 𝑏1 + 𝑎2 𝑏2 + 𝑎3 𝑏3
●Use vectors to prove geometric results in two dimensions (other than those listed in Unit 2 Topic 3) and in three dimensions. Specialist Mathematics 2025 v1.4
▶Vector and Cartesian equations5 LOs
●Define and use the vector (cross) product to determine a vector normal to a given plane, with and without technology. 𝒂 × 𝒃 = |𝒂| |𝒃| sin(𝜃) 𝒏̂ 𝒂 × 𝒃 = ( 𝑎1 𝑎2 𝑎3) × ( 𝑏1 𝑏2 𝑏3) = ( 𝑎2𝑏3 − 𝑎3𝑏2 𝑎3𝑏1 − 𝑎1𝑏3 𝑎1𝑏2 − 𝑎2𝑏1)
●Determine vector, parametric and Cartesian equations of straight lines and straight-line segments given the position of two points, or equivalent information, in both two and three dimensions. vector equation of line: 𝒓 = 𝒂 + 𝑡𝒅 parametric equations of line: 𝑥 = 𝑎1 + 𝑡 𝑑1 𝑦 = 𝑎2 + 𝑡 𝑑2 𝑧 = 𝑎3 + 𝑡 𝑑3 Cartesian equation of line: 𝑥−𝑎1 𝑑1 = 𝑦−𝑎2 𝑑2 = 𝑧−𝑎3 𝑑3
●Understand and use equations of spheres. equation of sphere: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 + (𝑧 − 𝑙)2 = 𝑟2
●Use vector equations of curves in two or three dimensions involving a parameter, and determine a ‘corresponding’ Cartesian equation in the two-dimensional case.
●Use vector methods in applications, including areas of shapes and determining vector and Cartesian equations of a plane and of regions in a plane. vector equation of plane: 𝒓 ⋅ 𝒏 = 𝒂 ⋅ 𝒏 Cartesian equation of plane: 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0
▶Vector calculus6 LOs
▶Vector calculus6 LOs
●Apply vector calculus to model and solve problems that involve motion in a plane, including projectile and circular motion, with and without technology. Specialist Mathematics 2025 v1.4
●Differentiate and integrate a vector function with respect to time.
●Understand and use position of vectors as a function of time.
●Understand and use the Cartesian equation of a path given as a vector equation in two dimensions, including circles, ellipses and hyperbolas. equation of circle: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2 equation of ellipse: (𝑥−ℎ)2 𝑎2 + (𝑦−𝑘)2 𝑏2 = 1 equation of hyperbola: (𝑥−ℎ)2 𝑎2 − (𝑦−𝑘)2 𝑏2 = 1 or (𝑦−𝑘)2 𝑎2 − (𝑥−ℎ)2 𝑏2 = 1
●Understand and use the position of two particles, each described as a vector function of time, and determine if their paths cross or if the particles meet.
●Use vector calculus to determine equations of motion of a particle travelling in a straight line with both constant and variable acceleration.
▶Further matrices6 LOs
▶Matrix algebra and systems of equations5 LOs
●Calculate the determinant and multiplicative inverse of square matrices of any order, with technology.
●Examine the three cases for solutions of systems of equations — a unique solution, no solution and infinitely many solutions — and the geometric interpretation of a solution of a system of equations with three variables including a unique solution no solution infinitely many solutions
●Model and solve problems that involve matrices of beyond dimension 2 × 2, including the solution of systems of linear equations, with technology.
●Use matrix algebra to solve matrix equations that involve matrices of beyond dimension 2 × 2, including those of the form 𝑨𝑿 = 𝑩, 𝑿𝑨 = 𝑩 and 𝑨𝑿 + 𝑩𝑿 = 𝑪, with technology.
●Use the determinant to determine whether a square matrix of any order is singular or non- singular.
▶Applications of matrices1 LO
●Model and solve problems that involve real-life situations using matrices, including Dominance and Leslie matrices.
▶4 — Further calculus and statistical inference34 LOs
▶Integration techniques9 LOs
▶Integration techniques9 LOs
●Establish and use the formula ∫ 1 𝑥 𝑑𝑥 = ln|𝑥| + 𝑐 for 𝑥 ≠ 0 and ∫ 𝑓′(𝑥) 𝑓(𝑥) 𝑑𝑥 = ln|𝑓(𝑥)| + 𝑐 for 𝑓(𝑥) ≠ 0.
●Establish and use the formula ∫ sec2(𝑥) 𝑑𝑥 = tan(𝑥) + 𝑐.
●Integrate by parts. ∫ 𝑢 𝑑𝑣 𝑑𝑥 𝑑𝑥 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢 𝑑𝑥 𝑑𝑥 Specialist Mathematics 2025 v1.4
●Integrate expressions of the form ±1 √𝑎2−𝑥2 and 𝑎 𝑎2+𝑥2. ∫ 1 √𝑎2−𝑥2 𝑑𝑥 = sin−1 (𝑥 𝑎) + 𝑐 ∫ −1 √𝑎2−𝑥2 𝑑𝑥 = cos−1 (𝑥 𝑎) + 𝑐 ∫ 𝑎 𝑎2+𝑥2 𝑑𝑥 = tan−1 (𝑥 𝑎) + 𝑐
●Integrate using the trigonometric identities sin2(𝑥) = 1 2 (1 − cos(2𝑥)), cos2(𝑥) = 1 2 (1 + cos(2𝑥)), 1 + tan2(𝑥) = sec2(𝑥) and cot2(𝑥) + 1 = cosec2(𝑥).
●Understand and use the inverse trigonometric functions: arcsine, arccosine and arctangent.
●Use partial fractions for integration involving two distinct linear factors in the denominator, e.g. 2𝑥−1 (𝑥+1)(𝑥−2).
●Use substitution 𝑢 = 𝑔(𝑥) to integrate expressions of the form 𝑓(𝑔(𝑥))𝑔′(𝑥).
●Use the derivative of the inverse trigonometric functions: arcsine, arccosine and arctangent. 𝑑 𝑑𝑥 sin−1 (𝑥 𝑎) = 1 √𝑎2−𝑥2 𝑑 𝑑𝑥 cos−1 (𝑥 𝑎) = −1 √𝑎2−𝑥2 𝑑 𝑑𝑥 tan−1 (𝑥 𝑎) = 𝑎 𝑎2+𝑥2
▶Applications of integral calculus5 LOs
▶Applications of integral calculus5 LOs
●Apply techniques from Unit 4 Topic 1 Sub-topic: Integration techniques to calculate areas between curves determined by functions, with and without technology.
●Determine volumes of solids of revolution about either axis, with and without technology. about the 𝑥-axis: 𝑉 = 𝜋 ∫ [𝑓(𝑥)]2𝑏 𝑎 𝑑𝑥 about the 𝑦-axis: 𝑉 = 𝜋 ∫ [𝑓(𝑦)]2𝑏 𝑎 𝑑𝑦
●Model and solve problems that involve exponential random variables and associated probabilities and quantiles, with and without technology. Specialist Mathematics 2025 v1.4
●Understand and use the probability density function, 𝑓(𝑡) = 𝜆𝑒−𝜆𝑡 for 𝑡 ≥ 0, of the exponential random variable with parameter 𝜆 > 0. mean: 1 𝜆 standard deviation: 1 𝜆
●Use Simpson’s rule to approximate an area and the value of a definite integral, with and without technology. ∫ 𝑓(𝑥) 𝑑𝑥 𝑏 𝑎 ≈ 𝑤 3 [𝑓(𝑥0) + 4[𝑓(𝑥1) + 𝑓(𝑥3)+... ] + 2[𝑓(𝑥2) + 𝑓(𝑥4)+... ] + 𝑓(𝑥𝑛)] where 𝑤 = 𝑏−𝑎 𝑛
▶Rates of change and differential equations5 LOs
▶Rates of change2 LOs
●Model and solve related rates problems as instances of the chain rule including situations that involve surface area and volume of cones, pyramids and spheres, with and without technology.
●Use implicit differentiation to determine the gradient of curves whose equations are given in implicit form.
▶Differential equations3 LOs
●Determine general and particular solutions of first-order differential equations of the form 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥), differential equations of the form 𝑑𝑦 𝑑𝑥 = 𝑔(𝑦) and differential equations of the form 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥)𝑔(𝑦) using separation of variables.
●Model and solve problems using provided differential equations, including the logistic equation, Newton’s law of cooling and radioactive decay, with and without technology.
●Understand and use slope (direction or gradient) fields of a first-order differential equation.
▶Modelling motion4 LOs
▶Modelling motion4 LOs
●Model and solve problems that involve motion in a straight line with both constant and non- constant acceleration, including simple harmonic motion, vertical motion under gravity with and without air resistance, and motion of a body in non-equilibrium situations on a smooth inclined plane (excluding situations with pulleys and connected bodies). If 𝑑2𝑥 𝑑𝑡2 = −𝜔2𝑥 then 𝑥 = 𝐴 sin(𝜔𝑡 + 𝛼) or 𝑥 = 𝐴 cos(𝜔𝑡 + 𝛽) 𝑣2 = 𝜔2(𝐴2 − 𝑥2) 𝑇 = 2𝜋 𝜔 𝑓 = 1 𝑇 Specialist Mathematics 2025 v1.4
●Understand and use momentum, constant force, non-constant force, resultant force, action and reaction.
●Understand and use motion of a body in non-equilibrium situations under concurrent forces.
●Understand and use the expressions 𝑑𝑣 𝑑𝑡, 𝑑2𝑥 𝑑𝑡2, 𝑣 𝑑𝑣 𝑑𝑥 and 𝑑 𝑑𝑥 (1 2 𝑣2) to represent the acceleration of an object moving in a straight line.
▶Statistical inference11 LOs
▶Sample means4 LOs
●Model and solve problems that involve sample means, with and without technology.
●Understand the concept of the sample mean 𝑋 as a random variable whose value varies between samples where 𝑋 is a random variable with mean 𝜇 and the standard deviation 𝜎.
●Use repeated random sampling data from a variety of distributions and a range of sample sizes to examine properties of the distribution of 𝑋 across samples of a fixed size 𝑛, including its mean 𝜇, its standard deviation 𝜎 √𝑛 (where 𝜇 and 𝜎 are the mean and standard deviation of 𝑋) and its approximate normality if 𝑛 is large.
●Use repeated random sampling data from a variety of distributions and a range of sample sizes to examine the approximate standard normality of 𝑋−𝜇 𝑠 √𝑛 for large samples (𝑛 ≥ 30), where 𝑠 is the sample standard deviation (Central limit theorem).
▶Confidence intervals for means7 LOs
●Model and solve problems that involve interval estimates for sample means, with and without technology.
●Understand and use the approximate confidence interval (𝑥̅ − 𝑧 𝑠 √𝑛, 𝑥̅ + 𝑧 𝑠 √𝑛), as an interval estimate for 𝜇, the population mean, where 𝑧 is the appropriate quantile for the standard normal distribution.
●Understand and use the approximate margin of error. 𝐸 = 𝑧 𝑠 √𝑛
●Understand and use the concept that there are variations in confidence intervals between samples and that most but not all confidence intervals contain 𝜇.
●Understand and use the relationship between margin of error, level of confidence and sample size.
●Understand the concept of an interval estimate for a parameter associated with a random variable.
●Use 𝑥̅ and 𝑠 to estimate 𝜇 and 𝜎, to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for 𝜇.
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