Paper 1
SL
- Time: 45 minutes (30 marks)
- 30 multiple – choice questions (core)
- No marks deducted from incorrect answers
- NO CALCULATOR ALLOWED
- Data booklet provided
- 20% weight
Topic 1 : Algebra
- Topic 1.1 :
- Topic 1.2 :
- Topic 1.3 :
Topic 2 : Functions and equations
- Topic 2.1
- Topic 2.2
- The graph of a function; its equation \(y = f\left( x \right)\) .
- Function graphing skills.
- Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range.
- Use of technology to graph a variety of functions, including ones not specifically mentioned.
- The graph of \(y = {f^{ – 1}}\left( x \right)\) as the reflection in the line \(y = x\) of the graph of \(y = f\left( x \right)\).
- Topic 2.3
- Transformations of graphs.
- Translations: \(y = f\left( x \right) + b\) ; \(y = f(x – a)\) .
- Reflections (in both axes): \(y = – f\left( x \right)\) ; \(y = f( – x)\) .
- Vertical stretch with scale factor \(p\): \(y = pf\left( x \right)\) .
- Stretch in the x-direction with scale factor \(\frac{1}{q}\): \(y = f(qx)\) .
- Composite transformations.
- Topic 2.4
- The quadratic function \(x \mapsto a{x^2} + bx + c\) : its graph, \(y\)-intercept \((0, c)\). Axis of symmetry.
- The form \(x \mapsto a\left( {x – p} \right)\left( {x – q} \right)\) , \(x\)-intercepts \((p, 0)\) and \((q,0)\) .
- The form \(x \mapsto a{\left( {x – h} \right)^2} + k\) , vertex \((h, k)\) .
- Topic 2.5
- Topic 2.6
- Exponential functions and their graphs: \(x \mapsto {a^x}\) , \(a > 0\) , \(x \mapsto {{\text{e}}^x}\) .
- Logarithmic functions and their graphs: \(x \mapsto {\log _a}x\) , \(x > 0\) , \(x \mapsto \ln x\), \(x > 0\)
- Relationships between these functions: \({a^x} = {{\text{e}}^{x\ln a}}\) ; \({\log _a}{a^x} = x\) ; \({a^{{{\log }_a}x}} = x\) , \(x > 0\) .
- Topic 2.7
- Solving equations, both graphically and analytically.
- Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
- Solving \(a{x^2} + bx + c = 0\) , \(a \ne 0\) .
- The quadratic formula.
- The discriminant \(\Delta = {b^2} – 4ac\) and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
- Solving exponential equations.
- Topic 2.8
- Applications of graphing skills and solving equations that relate to real-life situations.
Topic 3 – Core: Circular functions and trigonometry
- Topic 3.1
- Topic 3.2
- Definition of \(\cos \theta \) and \(\sin \theta \) in terms of the unit circle.
- Definition of \(\tan \theta \) as \(\frac{{\sin \theta }}{{\cos \theta }}\) .
- Exact values of trigonometric ratios of \(0\), \(\frac{\pi }{6}\), \(\frac{\pi }{4}\), \(\frac{\pi }{3}\), \(\frac{\pi }{2}\) and their multiples.
- Topic 3.3
- Topic 3.4
- Topic 3.5
- Topic 3.6
- Solution of triangles.
- The cosine rule.
- The sine rule, including the ambiguous case.
- Area of a triangle, \(\frac{1}{2}ab\sin C\) .
- Applications.
Topic 4 – Core: Vectors
- Topic 4.1
- Vectors as displacements in the plane and in three dimensions.
- Components of a vector; column representation; \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
- Algebraic and geometric approaches to the sum and difference of two vectors; the zero vector, the vector \( – v\).
- Algebraic and geometric approaches to multiplication by a scalar, \(kv\) ; parallel vectors.
- Algebraic and geometric approaches to magnitude of a vector, \(\left| v \right|\) .
- Algebraic and geometric approaches to unit vectors; base vectors; \(i\), \(j\) and \(k\).
- Algebraic and geometric approaches to position vectors \(\overrightarrow {OA} = a\) .
- Algebraic and geometric approaches to \(\overrightarrow {AB} = \overrightarrow {OB} – \overrightarrow {OA} = b – a\) .
- Topic 4.2
- Topic 4.3
- Topic 4.4
- Distinguishing between coincident and parallel lines.
- Finding the point of intersection of two lines.
- Determining whether two lines intersect.
Topic 5 – Core: Statistics and probability
- Topic 5.1
- Concepts of population, sample, random sample, discrete and continuous data.
- Presentation of data: frequency distributions (tables); frequency histograms with equal class intervals
- Box-and-whisker plots; outliers.
- Grouped data: use of mid-interval values for calculations; interval width; upper and lower interval boundaries; modal class.
- Topic 5.2
- Topic 5.3
- Topic 5.4
- Linear correlation of bivariate data.
- Pearson’s product–moment correlation coefficient \(r\).
- Scatter diagrams; lines of best fit.
- Equation of the regression line of \(y\) on \(x\).
- Use of the equation for prediction purposes.
- Mathematical and contextual interpretation.
- Topic 5.5
- Concepts of trial, outcome, equally likely outcomes, sample space ( \(U\) ) and event.
- The probability of an event \(A\) is \(P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( U \right)}}\) .
- The complementary events \(A\) and \({A’}\) (not \(A\)).
- Use of Venn diagrams, tree diagrams and tables of outcomes.
- Topic 5.6
- Combined events, \(P\left( {A \cup B} \right)\) .
- Mutually exclusive events: \(P(A \cap B) = 0\) .
- Conditional probability; the definition \(P\left( {\left. A \right|B} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\) .
- Independent events; the definition \(P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)\) .
- Probabilities with and without replacement.
- Topic 5.7
- Topic 5.8
- Binomial distribution.
- Mean and variance of the binomial distribution.
- Topic 5.9
- Normal distributions and curves.
- Standardization of normal variables (\(z\)-values, \(z\)-scores).
- Properties of the normal distribution.
Topic 6 – Core: Calculus
- Topic 6.1
- Informal ideas of limit and convergence.
- Limit notation.
- Definition of derivative from first principles as \(f’\left( x \right) = {\mathop {\lim }\limits_{h \to 0 } } \left( {\frac{{f\left( {x + h} \right) – f\left( x \right)}}{h}} \right)\) .
- Derivative interpreted as gradient function and as rate of change.
- Tangents and normals, and their equations.
- Topic 6.2
- Derivative of \({x^n}\left( {n \in \mathbb{Q}} \right)\) , \(\sin x\) , \(\cos x\) , \(\tan x\) , \({{\text{e}}^x}\) and \(\ln x\) .
- Differentiation of a sum and a real multiple of these functions.
- The chain rule for composite functions.
- The product and quotient rules.
- The second derivative.
- Extension to higher derivatives.
- Topic 6.3
- Topic 6.4
- Indefinite integration as anti-differentiation.
- Indefinite integral of \({x^n}\left( {n \in \mathbb{Q}} \right)\) , \(\sin x\) , \(\cos x\) , \(\frac{1}{x}\) and \({{\text{e}}^x}\) .
- The composites of any of these with the linear function \(ax + b\) .
- Integration by inspection, or substitution of the form \(\mathop \int \nolimits f\left( {g\left( x \right)} \right)g’\left( x \right){\text{d}}x\) .
- Topic 6.5
- Topic 6.6