Numerical solution of equations — AS & A Level Mathematics (9709) questions by topic
Sketching a pair of graphs to prove an equation has exactly one root, showing a trigonometric equation has one root within a stated interval, and proving that a convergent iterative sequence must converge to a root of a given cubic are typical of the 12 numerical solution of equations questions from 2023 to 2025, worth 3 to 9 marks.
Show, calculate, determine, sketch and state are the top command words, with show demanding a graphical or algebraic justification rather than a bare assertion. Because a root-counting argument depends on the sketch's shape and intersections being correctly reasoned, submit your working for instant marking, so a missing justification for why curves cross only once is caught before the whole proof is marked incomplete.
May–June 2025, Paper 32
By sketching a suitable pair of graphs, show that the equation x - 2 = 2 sin((1/2)x) has only one root in the interval 0 < x < pi. [2]
Answer this question and get it marked →May–June 2025, Paper 35
By sketching a suitable pair of graphs, show that the equation sec 2x = −2x − 1/2 has exactly one root in the interval 0 <= x <= (1/2)π. [2]
Answer this question and get it marked →May–June 2024, Paper 31
By sketching a suitable pair of graphs, show that the equation cosec(1/2 x) = e^x − 3 has exactly one root, denoted by α, in the interval 0 < x < π.
Answer this question and get it marked →May–June 2024, Paper 32
It is given that the equation e^(2x) = 5 + cos 3x has only one root. Show by calculation that this root lies in the interval 0.7 < x < 0.8. [2]
Answer this question and get it marked →October–November 2024, Paper 31
By sketching a suitable pair of graphs, show that the equation 2 + e^(−0.2x) = ln(1 + x) has only one root.
Answer this question and get it marked →October–November 2024, Paper 32
By sketching a suitable pair of graphs, show that the equation cot 2x = sec x has exactly one root in the interval 0 < x < ½π.
Answer this question and get it marked →October–November 2024, Paper 33
Let f(x) = 2x³ − 5x² + 4. Show that if a sequence of values given by the iterative formula x{n+1} = √(4 / (5 − 2xn)) converges, then it converges to a root of the equation f(x) =…
Answer this question and get it marked →May–June 2023, Paper 22
The diagram shows the graph of y = 3 − e^(−x/2). On the diagram, sketch the graph of y = 5x − 4 , and show that the equation 3 − e^(−x/2) = 5x − 4 has exactly two real roots.
Answer this question and get it marked →May–June 2023, Paper 23
The diagram shows the graph of y = 3 − e^(−½x). On the diagram, sketch the graph of y = 5x − 4 , and show that the equation 3 − e^(−½x) = 5x − 4 has exactly two real roots.
Answer this question and get it marked →May–June 2023, Paper 32
The equation cot ½x = 3x has one root in the interval 0 < x < π, denoted by α. Show by calculation that α lies between 0.5 and 1.
Answer this question and get it marked →October–November 2023, Paper 31
By sketching a suitable pair of graphs, show that the equation √x = e^x − 3 has only one root.
Answer this question and get it marked →October–November 2023, Paper 32
By sketching a suitable pair of graphs, show that the equation cot x = 2 − cos x has one root in the interval 0 < x ≤ (1/2)π.
Answer this question and get it marked →Other AS & A Level Mathematics topics
- Differentiation (95)
- Algebra, logarithmic and exponential functions (87)
- Trigonometry (57)
- Series (37)
- Integration (33)
- Kinematics (33)
- The normal distribution (32)
- Energy, work and power (31)
- The Poisson distribution (31)
- Complex numbers (28)
- Sampling and estimation (28)
- Permutations and combinations (27)