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Logarithms, quadratics and linear systems topic check

Practise logarithms, quadratics and linear systems in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.

Learning goals

Model and interpret exponential and logarithmic relationships
Interpret and draw graphs of functions
Produce exact logarithmic values with valid real bases and arguments
Produce complete real exponential equation results
Produce calibrated exponential model parameters, predictions and target times
Solve a system of linear equations completely, including systems with no or infinitely many solutions
Find the complete real solution set of a rational quadratic inequality
Find every real solution of a rational line–quadratic system
Derive complete parameter sets for the real roots of a quadratic
Derive complete parameter sets for a quadratic to be always positive or negative
Formulate, use and interpret a mathematical model

GCE A-Level H1 Mathematics 2027

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Choose a practice setProduce exact logarithmic values with valid real bases and arguments · 2 questionsProduce complete real exponential equation results · 2 questionsProduce calibrated exponential model parameters, predictions and target times · 2 questionsFind the complete real solution set of a rational quadratic inequality · 8 questionsFind every real solution of a rational line–quadratic system · 8 questionsDerive complete parameter sets for the real roots of a quadratic · 6 questionsDerive complete parameter sets for a quadratic to be always positive or negative · 4 questionsLinear systems · 4 questionsAll skills in this check

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Practise

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Check your progress

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Produce exact logarithmic values with valid real bases and arguments · Check your progress

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Produce complete real exponential equation results · Check your progress

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Produce calibrated exponential model parameters, predictions and target times · Check your progress

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Find the complete real solution set of a rational quadratic inequality · Check your progress

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Find every real solution of a rational line–quadratic system · Check your progress

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Derive complete parameter sets for the real roots of a quadratic · Check your progress

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Derive complete parameter sets for a quadratic to be always positive or negative · Check your progress

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Linear systems · Check your progress

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Check again

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Produce exact logarithmic values with valid real bases and arguments · Check again

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Produce complete real exponential equation results · Check again

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Produce calibrated exponential model parameters, predictions and target times · Check again

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Find the complete real solution set of a rational quadratic inequality · Check again

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Find every real solution of a rational line–quadratic system · Check again

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Derive complete parameter sets for the real roots of a quadratic · Check again

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Derive complete parameter sets for a quadratic to be always positive or negative · Check again

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Linear systems · Check again

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Retention check

Come back later to see whether your learning has lasted.

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Produce exact logarithmic values with valid real bases and arguments · Retention check

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Produce complete real exponential equation results · Retention check

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Produce calibrated exponential model parameters, predictions and target times · Retention check

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Find the complete real solution set of a rational quadratic inequality · Retention check

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Find every real solution of a rational line–quadratic system · Retention check

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Derive complete parameter sets for the real roots of a quadratic · Retention check

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Derive complete parameter sets for a quadratic to be always positive or negative · Retention check

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Linear systems · Retention check

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Practise: Produce exact logarithmic values with valid real bases and arguments

Practise questions on the skills you are learning.

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Practise: Produce complete real exponential equation results

Practise questions on the skills you are learning.

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Practise: Produce calibrated exponential model parameters, predictions and target times

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Practise: Find the complete real solution set of a rational quadratic inequality

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Practise: Find every real solution of a rational line–quadratic system

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Practise: Derive complete parameter sets for the real roots of a quadratic

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Practise: Derive complete parameter sets for a quadratic to be always positive or negative

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Practise: Linear systems

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Revision notes

Revise functions and graphs

Connect exponential and logarithmic models with graph features, numerical solutions and exact quadratic methods.

Key ideas

Function rules and natural logarithms
Connect function rules, exponential/logarithm graphs and valid log algebra to growth, decay and fitted model parameters.
Quadratic formula and models
Identify coefficients, retain both possible roots and reject values only for a stated domain reason.
Completing the square and fractional quadratics
Choose an exact quadratic method and distinguish solving from rewriting a function.
Quadratic extrema and sign
Connect completed squares and discriminants to strict sign conditions, model restrictions and feasible quadratic extrema.
Discriminants, intersections and inequalities
Connect substitution, root counts and tangency to sign intervals, number-line solutions and parameter cases that change degree.
Graph features and numerical solutions
Interpret graph features, numerical boundaries and model equations, then check precision and feasible solutions.
Linear systems from context
Turn three conditions into a system of linear equations, solve it with a graphing calculator and check for repeated or contradictory conditions.

What the syllabus asks

From the GCE A-Level H1 Mathematics 2027 syllabus

Include: concept of function as a rule or relationship where for every input there is only one output; functions e^x and ln x and their graphs; exponential growth and decay; logarithmic growth; equivalence of y = e^x and x = ln y; laws of logarithms; use of a graphing calculator or graphing software to graph a given function; characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes (horizontal and vertical). Exclude: problems involving finding domain and range; use of notation f: x ↦ x² + 5; change of base of logarithms.
Include: conditions for a quadratic equation to have two real roots, two equal roots or no real roots; conditions for ax² + bx + c to be always positive or always negative; solving simultaneous equations, one linear and one quadratic, by substitution; solving quadratic equations and inequalities in one unknown analytically; solving inequalities by graphical methods; formulating an equation or a system of linear equations from a problem situation; finding the approximate solution of an equation or a system of linear equations using a graphing calculator or graphing software.