Sketch graphs to show root existence

A question is this type if and only if it asks to sketch suitable graphs to demonstrate that an equation has exactly one (or a specific number of) real root(s).

29 questions · Standard +0.1

1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams
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CAIE P2 2022 November Q4
5 marks Standard +0.3
4
  1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\mathrm { e } ^ { - \frac { 1 } { 2 } x } = x ^ { 5 }$$ has exactly one real root.
  2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } }\) to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2003 June Q5
8 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has exactly one root.
  2. Verify by calculation that the root lies between 1.0 and 1.4 .
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ to determine the root correct to 2 decimal places, showing the result of each iteration.
CAIE P2 2007 June Q5
8 marks Standard +0.3
5
  1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - x$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 1.0 and 1.2.
  3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x } \right)$$
  4. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2011 June Q7
9 marks Standard +0.3
7
  1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 14 - x ^ { 2 }$$ has exactly two real roots.
  2. Show by calculation that the positive root lies between 1.2 and 1.3.
  3. Show that this root also satisfies the equation $$x = \frac { 1 } { 2 } \ln \left( 14 - x ^ { 2 } \right) .$$
  4. Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places.
  5. Express \(4 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  6. Solve the equation \(4 \sin \theta - 6 \cos \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  7. Find the greatest and least possible values of \(( 4 \sin \theta - 6 \cos \theta ) ^ { 2 } + 8\) as \(\theta\) varies.
CAIE P2 2014 June Q4
8 marks Moderate -0.3
4
  1. By sketching a suitable pair of graphs, show that the equation $$3 \ln x = 15 - x ^ { 3 }$$ has exactly one real root.
  2. Show by calculation that the root lies between 2.0 and 2.5.
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2017 June Q3
5 marks Moderate -0.3
3
  1. By sketching a suitable pair of graphs, show that the equation $$x ^ { 3 } = 11 - 2 x$$ has exactly one real root.
  2. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 11 - 2 x _ { n } \right)$$ to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P3 2011 June Q6
7 marks Standard +0.3
6
  1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 1 + x ^ { 2 }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.5 and 0.8.
  3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2014 June Q8
9 marks Standard +0.3
8
  1. By sketching each of the graphs \(y = \operatorname { cosec } x\) and \(y = x ( \pi - x )\) for \(0 < x < \pi\), show that the equation $$\operatorname { cosec } x = x ( \pi - x )$$ has exactly two real roots in the interval \(0 < x < \pi\).
  2. Show that the equation \(\operatorname { cosec } x = x ( \pi - x )\) can be written in the form \(x = \frac { 1 + x ^ { 2 } \sin x } { \pi \sin x }\).
  3. The two real roots of the equation \(\operatorname { cosec } x = x ( \pi - x )\) in the interval \(0 < x < \pi\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
    1. Use the iterative formula $$x _ { n + 1 } = \frac { 1 + x _ { n } ^ { 2 } \sin x _ { n } } { \pi \sin x _ { n } }$$ to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
    2. Deduce the value of \(\beta\) correct to 2 decimal places.
CAIE P3 2017 March Q3
7 marks Moderate -0.3
3
  1. By sketching suitable graphs, show that the equation \(\mathrm { e } ^ { - \frac { 1 } { 2 } x } = 4 - x ^ { 2 }\) has one positive root and one negative root.
  2. Verify by calculation that the negative root lies between - 1 and - 1.5 .
  3. Use the iterative formula \(x _ { n + 1 } = - \sqrt { } \left( 4 - e ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2010 November Q4
7 marks Standard +0.3
4
  1. By sketching suitable graphs, show that the equation $$4 x ^ { 2 } - 1 = \cot x$$ has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.6 and 1 .
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \sqrt { } \left( 1 + \cot x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2014 November Q9
10 marks Standard +0.3
9
  1. Sketch the curve \(y = \ln ( x + 1 )\) and hence, by sketching a second curve, show that the equation $$x ^ { 3 } + \ln ( x + 1 ) = 40$$ has exactly one real root. State the equation of the second curve.
  2. Verify by calculation that the root lies between 3 and 4 .
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 40 - \ln \left( x _ { n } + 1 \right) \right)$$ with a suitable starting value, to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  4. Deduce the root of the equation $$\left( \mathrm { e } ^ { y } - 1 \right) ^ { 3 } + y = 40$$ giving the answer correct to 2 decimal places.
CAIE P3 2019 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
  2. Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
  3. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2011 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation $$\frac { 1 } { x } = \sin x$$ where \(x\) is in radians, has only one root for \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between \(x = 1.1\) and \(x = 1.2\).
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { \sin x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2015 November Q4
7 marks Moderate -0.3
4
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
  2. Verify by calculation that \(4.5 < \alpha < 5.0\).
  3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 Specimen Q4
7 marks Moderate -0.3
4
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
  2. Verify by calculation that \(4.5 < \alpha < 5.0\).
  3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2021 June Q6
7 marks Standard +0.3
6
  1. By sketching a suitable pair of graphs, show that the equation \(\cot \frac { 1 } { 2 } x = 1 + \mathrm { e } ^ { - x }\) has exactly one root in the interval \(0 < x \leqslant \pi\).
  2. Verify by calculation that this root lies between 1 and 1.5.
  3. Use the iterative formula \(x _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 1 } { 1 + \mathrm { e } ^ { - x _ { n } } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2022 June Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(\ln x = 3 x - x ^ { 2 }\) has one real root.
  2. Verify by calculation that the root lies between 2 and 2.8.
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2020 March Q3
7 marks Standard +0.3
3
  1. By sketching a suitable pair of graphs, show that the equation \(\sec x = 2 - \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.8 and 1 .
  3. Use the iterative formula \(x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { 4 - x _ { n } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2022 March Q7
7 marks Standard +0.3
7
  1. By sketching a suitable pair of graphs, show that the equation \(4 - x ^ { 2 } = \sec \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \pi\).
  2. Verify by calculation that this root lies between 1 and 2 .
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt { 4 - \sec \frac { 1 } { 2 } x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2020 November Q5
5 marks Standard +0.8
5
  1. By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } x = 1 + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) has exactly two roots in the interval \(0 < x < \pi\).
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \pi - \sin ^ { - 1 } \left( \frac { 1 } { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } + 1 } \right)$$ with initial value \(x _ { 1 } = 2\), converges to one of these roots.
    Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
OCR C3 2007 January Q3
7 marks Standard +0.3
3
  1. It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x ^ { 5 } \quad \text { and } \quad y = a - b x$$ on the same diagram, show that the equation $$x ^ { 5 } + b x - a = 0$$ has exactly one real root.
  2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { 53 - 2 x _ { n } }\), with a suitable starting value, to find the real root of the equation \(x ^ { 5 } + 2 x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places.
OCR C3 2014 June Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_524_720_246_676} The diagram shows the curve \(y = x ^ { 4 } - 8 x\).
  1. By sketching a second curve on the copy of the diagram, show that the equation $$x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0$$ has two real roots. State the equation of the second curve.
  2. The larger root of the equation \(x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0\) is denoted by \(\alpha\).
    1. Show by calculation that \(2.1 < \alpha < 2.2\).
    2. Use an iterative process based on the equation $$x = \sqrt [ 4 ] { 9 + 8 x - x ^ { 2 } } ,$$ with a suitable starting value, to find \(\alpha\) correct to 3 decimal places. Give the result of each step of the iterative process.
CAIE P2 2015 June Q5
12 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$|3x| = 16 - x^4$$ has two real roots. [3]
  2. Use the iterative formula \(x_{n+1} = \sqrt[4]{16 - 3x_n}\) to find one of the real roots correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
  3. Hence find the coordinates of each of the points of intersection of the graphs \(y = |3x|\) and \(y = 16 - x^4\), giving your answers correct to 3 decimal places. [2]
CAIE P2 2003 November Q5
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, for \(x < 0\), show that exactly one root of the equation \(x^2 = 2^x\) is negative. [2]
  2. Verify by calculation that this root lies between \(-1.0\) and \(-0.5\). [2]
  3. Use the iterative formula $$x_{n+1} = -\sqrt{(2^{x_n})}$$ to determine this root correct to 2 significant figures, showing the result of each iteration. [3]
CAIE P3 2006 June Q6
8 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$2\cot x = 1 + e^x,$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac{1}{2}\pi\). [2]
  2. Verify by calculation that this root lies between 0.5 and 1.0. [2]
  3. Show that this root also satisfies the equation $$x = \tan^{-1}\left(\frac{2}{1 + e^x}\right).$$ [1]
  4. Use the iterative formula $$x_{n+1} = \tan^{-1}\left(\frac{2}{1 + e^{x_n}}\right),$$ with initial value \(x_1 = 0.7\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]