Home / CIE A level -Pure Mathematics 2 : Topic : 2.6 Numerical solution of equations- root of an equation : Exam Style Questions Paper 2

Question

A curve has equation \(y=x^3e^{0.2x}\) where \(x\geq0\). At the point P on the curve, the gradient of the curve is 15.
(a) Show that the x-coordinates of P satisfies the equation \(x=\sqrt{75e^{0.2x}}{15+x}\).
(b) Use the equation in part (a) to show by calculation that the x-coordinate of P lies between 1.7 and 1.8
(c) Use an iterative formula, based on the equation in part (a), to find the x-coordinate of P correct to 4 significant figures. Gives the result of each iteration to 6 significant figures.

Answer/Explanation

Ans:

(a) Differentiate using the product rule
(b) Obtain \(3x^2e^{0.2x}+0.2x^3e^{0.2x}\)
Equate first derivative to 15 and rearrange to x = …..
Confirm \(x=\sqrt{\frac{75e^{-0.2x}}{15+x}}\)
(b) Consider sign of \(x-\sqrt{\frac{75e^{-0.2x}}{15+x}}\) or equivalent for 1.7 and 1.8
Obtain -0.08…. and 0.03….. or equivalent for 1.7 and 1.8
Obtain -0.08… and 0.03… or equivalents and justify conclusion
(c) Use iterative process correctly at least once
Obtain final answer 1.771
Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [1.7705, 1.7715]

Question

(a) Given that 2 In(x+1) + In x = In(x+9), show that \(x=\sqrt{\frac{9}{x+2}}\).
(b) It is given that the equation \(x=\sqrt{\frac{9}{x+2}}\) has single root
Show by calculation that this root lies between 1.5 and 2.0
(c) Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

Answer/Explanation

Ans:

  1. Use the power law correctly
    Use correct process to obtain equation with no logarithms
    Confirm \(x=\sqrt{\frac{9}{x+2}}\)
  2. Consider sign of \(x-\sqrt{\frac{9}{x+2}}\)
    Obtain -0.1…. and 0.5 or equivalents and justify conclusion
  3. Use iteration process correctly at least once
    Obtain final answer 1.58
    Show sufficient iterations to 5 sf to justify answer or show a sign change in interval [1.575, 1.585]

Question


The diagram shows part of the curve with equation \(y=\frac{5x}{4x^3+1}\). The shaded region is bounded by the curve and the lines x=1, x=3 and y=0.

(a) Find \(\frac{dy}{dx}\) and hence find the the x-coordinate of the maximum point.
(b) Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
(c) State, with a reason, whether your answer to part (b) is an over-estimate or under-estimate of the exact area of the shaded region.

Answer/Explanation

Ans:

  1.  Differentiate using quotient rule (or product rule)
    Obtain \(\frac{5(4x^3+1)-60x^3}{(4x^3+1)^2}\)
    Equate first derivative to zero and attempt solution
    Obtain \(x=\frac{1}{2}\)
  2. Use y values \(\frac{5}{5}, \frac{10}{33}, \frac{15}{109}\) or decimal equavalents
    Use correct formula, or equivalent, with h=1
    Obtain \(\frac{1}{2}(1+\frac{20}{30}+\frac{15}{109})\) or equivalent and hence 0.87
  3. State over-estimate with reference to top of each trapezium above curve

Question

  The sequence of values given by the iterative formula \(x_{n+1}=\frac{6+8x_{n}}{8+x^{2}_{n}}\) with initial value x1 = 2 converges
     to \(\alpha \)

      (a) Use the iterative formula to find the value of ! correct to 4 significant figures. Give the result of
             each iteration to 6 significant figures.                                                                                                                                                                             [3]

      (b) State an equation satisfied by \(\alpha \)  and hence determine the exact value of \(\alpha \)                                                                         [2] 

Answer/Explanation

Ans

 (a) Use iteration correctly at least once 
          Obtain final answer 1.817 A1 Answer required to exactly 4 significant figures
          Show sufficient iterations to 6 significant figures to justify answer
          or show sign change in interval [1.8165, 1.8175]

 (b) State equation \(x=\frac{6+8X}{8+x^{2}}\ or \ equivalent \ using \ \alpha \)

          Obtain \(\sqrt[3]{6}\) or exact equivalent 

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