1.01b b) Understand and be able to use the logical connectives . Learners should be familiar with the language associated with the logical connectives: “congruence”, “if.....then” and “if and only if” (or “iff”). Section 5: Specification Content Appendix B: Mathematical Notation The 7890 specification explicitly states that: ‘In all examinations candidates are expected to construct and present clear mathematical arguments, consisting of logical deductions and precise statements involving correct use of symbols and connecting language. In particular, terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’, and notations such as and should be understood and used accurately.’ The symbol is defined in Appendix B of the 7890 specification as ‘is identical to or is congruent to’. Furthermore, as defined in this Appendix as ‘if p then q’. The term ‘iff’ is not explicitly mentioned in the 7890 specification; however, the equivalent mathematical symbol appears and is mentioned above.
1.01c c) Be able to show disproof by counter example. Learners should understand that this means that, given a statement of the form “if P(x) is true then Q(x) is true”, finding a single x for which P(x) is true but Q(x) is false is to offer a disproof by counter example. Questions requiring proof will be set on content with which the learner is expected to be familiar e.g. through study of GCSE (9-1) or AS Level Mathematics. Learners are expected to understand and be able to use the terms “integer”, “real”, “rational” and “irrational”. Section 5: Specification Content Appendix B: Mathematical Notation C1 – Indices and Surds (b) & (c) The 7890 specification explicitly states that: ‘…, candidates are expected to understand the nature of a mathematical proof. In A2 units, questions that require proof by contradiction or disproof by counter-example may be set.’ The notation for the terms ‘integer’, ‘real’ and ‘rational’ can be found in Appendix B of the 7890 specification under Set Notation. The term ‘irrational’ is implicitly covered in C1 in dealing with Indices and Surds. 1.02 Algebra and Functions 1.02a a) Understand and be able to use the laws of indices for all rational exponents. Includes negative and zero indices. Problems may involve the application of more than one of the following laws: . C1 – Indices and Surds (a) & (b) (a) understand rational indices (positive, negative and zero), and use laws of indices in the course of algebraic applications (b) recognise the equivalence of surd and index notation (e.g. )
1.02b b) Be able to use and manipulate surds, including rationalising the denominator. Learners should understand and use the equivalence of surd and index notation. C1 – Indices and Surds (a), (b) & (c) (c) use simple properties of surds such as , including rationalising denominators of the form 1.02c c) Be able to solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation. The equations may contain brackets and/or fractions. e.g. and and C1 – Polynomials (e) (e) solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic 1.02d d) Be able to work with quadratic functions and their graphs, and the discriminant (D or) of a quadratic function, including the conditions for real and repeated roots. i.e. Use the conditions: 1. real distinct roots 2. repeated roots 3. roots are not real to determine the number and nature of the roots of a quadratic equation and relate the results to a graph of the quadratic function. C1 – Polynomials (c) C1 – Coordinate Geometry and Graphs (g) & (h) Polynomials (c) find the discriminant of a quadratic polynomial and use the discriminant, e.g. to determine the number of real roots of the equation Coordinate Geometry and Graphs (g) understand the relationship between a graph and its associated algebraic equation Coordinate Geometry and Graphs (h) sketch curves with equations of the form (iii) where a, b and c are constants
1.02w w) Understand the effect of simple transformations on the graph of including sketching associated graphs, describing transformations and finding relevant equations: ,, and , for any real a. Only single transformations will be requested. Translations may be specified by a two-dimensional column vector. C1 – Coordinate Geometry and Graphs (i) (i) understand and use the relationships between the graphs of where a is a constant, and express the transformations involved in terms of translations, reflections and stretches 1.02x x) Understand the effect of combinations of transformations on the graph of including sketching associated graphs, describing transformations and finding relevant equations. The transformations may be combinations of, , and , for any real a, and any function defined in the Stage 1 or Stage 2 content. C3 – Algebra and Functions (e) (e) use and recognise compositions of transformations of graphs, such as the relationship between the graphs of and where a and b are constants
1.03d d) Understand and be able to use the coordinate geometry of a circle including using the equation of a circle in the form . Learners should be able to draw a circle given its equation or to form the equation given its centre and radius. C1 – Coordinate Geometry and Graphs (e) & (f) (e) understand that the equation represents the circle with centre and radius r (f) use algebraic methods to solve problems involving lines and circles Note that in the reformed specification the use of the expanded form of the circle is not required. 1.03e e) Be able to complete the square to find the centre and radius of a circle. C1 – Polynomials (b) C1 – Coordinate Geometry and Graphs (f) Polynomials (b) carry out the process of completing the square for a quadratic polynomial Coordinate Geometry and Graphs (f) use algebraic methods to solve problems involving lines and circles 1.03f f) Be able to use the following circle properties in the context of problems in coordinate geometry: 1. the angle in a semicircle is a right angle, 2. the perpendicular from the centre of a circle to a chord bisects the chord, 3. the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point. Learners should also be able to investigate whether or not a line and a circle or two circles intersect. C1 – Coordinate Geometry and Graphs (f) (f) knowledge of the following circle properties is included: the angle in a semicircle is a right angle; the perpendicular from the centre to a chord bisects the chord; the perpendicularity of radius and tangent
1.06b b) Know that the gradient of is equal to and hence understand why the exponential model is suitable in many applications. See 1.07k for explicit differentiation of . C3 – Differentiation and Integration (a) (a) use the derivative of together with constant multiples, sums, and differences 1.06c c) Know and use the definition of (for ) as the inverse of (for all ), where is positive. Learners should be able to convert from index to logarithmic form and vice versa as . The values and should be known. C2 – Algebra (d) (d) understand the relationship between logarithms and indices 1.06d d) Know and use the function and its graph. C3 – Algebra and Functions (h) (h) understand the properties of the logarithmic function and its graph 1.06e e) Know and use as the inverse function of . e.g. In solving equations involving logarithms or exponentials. The values and should be known. C3 – Algebra and Functions (h) (h)…including their relationship as inverse functions
2.04h h) Be able to select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or normal model may not be appropriate. Includes understanding that a given binomial distribution with large can be approximated by a normal distribution. [Questions explicitly requiring calculations using the normal approximation to the binomial distribution are excluded.] S2 - The Normal Distribution (c) (c) recall conditions under which the normal distribution can be used as an approximation to the binomial distribution, and use this approximation, with a continuity correction, in solving problems 2.05 Statistical Hypothesis Testing
2.05a a) Understand and be able to use the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value. Hypotheses should be stated in terms of parameter values (where relevant) and the meanings of symbols should be stated. For example, “,, whereis the population proportion in favour of the resolution”. Conclusions should be stated in such a way as to reflect the fact that they are not certain. For example, "There is evidence at the 5% level to reject . It is likely that the mean mass is less than 500 g." "There is no evidence at the 2% level to reject . There is no reason to suppose that the mean journey time has changed." Some examples of incorrect conclusion are as follows: "is rejected. Waiting times have increased." "Accept . Plants in this area have the same height as plants in other areas." S2 – Sampling and Hypothesis Tests (g) & (h) (g) understand the nature of a hypothesis test, the difference between one-tail and two-tail tests, and the terms ‘null hypothesis’, ‘alternative hypothesis’, ‘significance level’, ‘rejection region (or critical region)’, ‘acceptance region’ and ‘test statistic’ (h) formulate hypotheses and carry out a hypothesis test of a population proportion in the context of a single observation from a binomial distribution, using direct evaluation of binomial probabilities 2.05b b) Be able to conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. S2 – Sampling and Hypothesis Tests (h)
2.05g g) Use and be able to interpret Pearson's correlation coefficient in hypothesis tests, using either a given critical value or a p-value and a table of critical values. When using Pearson's coefficient in an hypothesis test, the data may be assumed to come from a bivariate normal distribution. A table of critical values of Pearson's coefficient will be provided. [Calculation of correlation coefficients is excluded.] The use of Pearson’s product-moment correlation coefficient in hypothesis tests is new content in the reformed specification. 3 – Mechanics OCR Reference. Content Description (unshaded content is AS content) Legacy Unit and Reference Notes 3.01 Quantities and Units in Mechanics 3.01a a) Understand and be able to use the fundamental quantities and units in the S.I. system: length (in metres), time (in seconds), mass (in kilograms). Learners should understand that these three base quantities are mutually independent. Understanding the fundamental quantities and units in the S.I. system is assumed knowledge from GCSE (10.01a). moment
3.01b b) Understand and be able to use derived quantities and units: velocity (m/s or ms-1), acceleration (m/s2 or ms-2), force (N), weight (N). Learners should be able to add the appropriate unit to a given quantity. M1 – Kinematics of Motion in a Straight Line (a) M1 – Force as a Vector (a) M1 – Equilibrium of a Particle (a) Kinematics of Motion in a Straight Line (a) understand the concepts of …velocity and acceleration as vector quantities Force as a Vector (a) understand the vector nature of force Equilibrium of a Particle (a)…use the relationship between mass and weight 3.01c c) Understand and be able to use the unit for (N m). M2 – Equilibrium of a Rigid Body (a) (a) calculate the moment of a force about a point in two dimensional situations only 3.02 Kinematics 3.02a a) Understand and be able to use the language of kinematics: position, displacement, distance, distance travelled, velocity, speed, acceleration, equation of motion. Learners should understand the vector nature of displacement, velocity and acceleration and the scalar nature of distance travelled and speed. M1 – Kinematics of Motion in a Straight Line (a) (a) understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and acceleration as vector quantities
3.02b b) Understand, use and interpret graphs in kinematics for motion in a straight line. M1 – Kinematics of Motion in a Straight Line (b) (b) sketch and interpret (t, x) and (t, v) graphs 3.02c c) Be able to interpret displacement-time and velocity-time graphs, and in particular understand and be able to use the facts that the gradient of a displacement-time graph represents the velocity, the gradient of a velocity-time graph represents the acceleration, and the area between the graph and the time axis for a velocity-time graph represents the displacement. M1 – Kinematics of Motion in a Straight Line (b) (b) sketch and interpret (t, x) and (t, v) graphs, and in particular understand and use the facts that (i) the area under a (t, v) graph represents displacement (ii) the gradient of a (t, x) graph represents velocity (iii) the gradient of a (t, v) graph represents acceleration 3.02d d) Understand, use and derive the formulae for constant acceleration for motion in a straight line: Learners may be required to derive the constant acceleration formulae using a variety of techniques: 1. by integration, e.g. , 2. by using and interpreting appropriate graphs, e.g. velocity against time, 3. by substitution of one (given) formula into another (given) formula, e.g. substituting into to obtain . M1 – Kinematics of Motion in a Straight Line (d) (d) use appropriate formulae for motion with constant acceleration
3.03d d) Understand and be able to use Newton’s second law () in simple cases of forces given as two dimensional vectors. e.g. Find in vector form the force acting on a body of mass 2 kg when it is accelerating at ms-2. Questions set involving vectors may involve either column vector notation or , notation . Newton’s second law given as two dimensional vectors is new content in the reformed specification. 3.03f f) Understand and be able to use the weight () of a body to model the motion in a straight line under gravity. e.g. A ball falling through the air. M1 – Equilibrium of a Particle (a) (a) identify the forces acting in a given situation, and use the relationship between mass and weight 3.03g g) Understand the gravitational acceleration, g, and its value in S.I. units to varying degrees of accuracy. The value of g may be assumed to take a constant value of 9.8 ms-2 but learners should be aware that g is not a universal constant but depends on location in the universe. [The inverse square law for gravitation is not required.] An understanding that g is not a universal constant is new content in the reformed specification.
3.03k k) Be able to use the concept of equilibrium together with one dimensional motion in a straight line to solve problems that involve connected particles and smooth pulleys. e.g. A train engine pulling a train carriage(s) along a straight horizontal track or the vertical motion of two particles, connected by a light inextensible string passing over a fixed smooth peg or light pulley. M1 – Newton’s Laws of Motion (a) & (c) (a) apply Newton’s laws of motion to the linear motion of bodies of constant mass moving under the action of constant forces, for example, a car pulling a caravan (c) solve simple problems which may be modelled as the motion of two particles, connected by a light inextensible string which may pass over a fixed smooth peg or light pulley 3.03m m) Be able to use the principle that a particle is in equilibrium if and only if the sum of the resolved parts in a given direction is zero. Problems may involve the resolving of forces, including cases where it is sensible to: 1. resolve horizontally and vertically, 2. resolve parallel and perpendicular to an inclined plane, 3. resolve in directions to be chosen by the learner, or 4. use a polygon of forces. M1 – Equilibrium of a Particle (b) (b) understand and use the principle that a particle is in equilibrium if and only if the vector sum of the forces acting is zero, or equivalently if and only if the sum of the resolved parts in any given direction is zero
3.03n n) Be able to solve problems involving simple cases of equilibrium of forces on a particle in two dimensions using vectors, including connected particles and smooth pulleys. e.g. Finding the required force for a particle to remain in equilibrium when under the action of forces , ,… For Stage 1 learners, examples can be restricted to problems in which the forces acting on the body will be collinear, in two perpendicular directions or given as 2-D vectors. The case of equilibrium of forces on a particle in two dimensions using vectors is new content in the reformed specification. 3.03o o) Be able to resolve forces for more advanced problems involving connected particles and smooth pulleys. e.g. The motion of two particles, connected by a light inextensible string passing over a light pulley placed at the top of an inclined plane. M1 – Newton’s Laws of Motion (b) (b) model, in suitable circumstances, the motion of a body moving vertically or on an inclined plane, as motion with constant acceleration and understand any limitations of this model