**Question**

**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**

**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:**

- Use the power law correctly

Use correct process to obtain equation with no logarithms

Confirm \(x=\sqrt{\frac{9}{x+2}}\) - Consider sign of \(x-\sqrt{\frac{9}{x+2}}\)

Obtain -0.1…. and 0.5 or equivalents and justify conclusion - 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**

**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:**

- 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 \(x=\frac{1}{2}\) - 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 - State over-estimate with reference to top of each trapezium above curve

*Question*

*Question*

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

*Question*

*Question*

The diagram shows the curve with equation \(y=\frac{3x+2}{ln x} \). The curve has a minimum point M.

(a) Find an expression for \(\frac{dy}{dx}\) and show that the x-coordinate of M satisfies the equation \(x=\frac{3x+2}{3 lnx}\)

** (b) Use the equation in part (a) to show by calculation that the x-coordinate of M lies between**** 3 and 4. [2] **

** (c) Use an iterative formula, based on the equation in part (a), to find the x-coordinate of M correct**** to 5 significant figures. Give the result of each iteration to 7 significant figures. ** [3]

**Answer/Explanation**

Ans

5 (a) Use quotient rule (or equivalent) to find first derivative

Obtain \(\frac{dy}{dx}=\frac{3ln x-\frac{1}{x}(3x+2)}{(lnx)^{2}}\)

Equate first derivative to zero and confirm \(x=\frac{3x+2}{3lnx}\)

5 (b) Consider \(x-\frac{3x+2}{3lnx}\) or equivalent for values 3 and 4

Obtain −0.33… and 0.63… or equivalents and justify conclusion

5 (c) Use iteration process correctly at least once

Obtain final answer 3.3223

Show sufficient iterations to 7 s.f. to justify answer or show sign

change in the interval [3.32225, 3.32235]

**Question**

**Question**

The curve with equation y = xe2x + 5e−x has a minimum point M.

**(a)**Show that the x-coordinate of M satisfies the equation x = \(\frac{1}{3}\) ln 5 − \(\frac{1}{3}\) ln (1 + 2x).

**(b)Use an iterative formula, based on the equation in part (a), to find the x-coordinate of M correct to 3 significant figures. Use an initial value of 0.35 and give the result of each iteration to 5 significant figures.**

**Answer/Explanation**

**(a)**Attempt use of product rule to differentiate xe^{2x}

Obtain e^{2x} + 2xe^{2x} – 5e^{-x}

Equate first derivative to zero and multiply by e^{x} to obtain an equation involving e^{3x}

Obtain e^{3x}(1 + 2x) = 5 or equivalent

Confirm given result x = \(\frac{1}{3}\) ln5 – \(\frac{1}{3}\) ln (1 + 2x) with sufficient detail.

**(b)**Use iteration process correctly at least once

Obtain final answer 0.357

Show sufficient iterations to 5sf to justify answer or show sign change in interval [0.3565, 0.3575]

**Question**

**Question**

A curve has equation e^{2x}y – e^{y} = 100.

**(a)**Show that \(\frac{dy}{dx}=\frac{2e^{2x}y}{e^{y-e^{2x}}}.\)

**(b)**Show that the curve has no stationary points.

It is required to find the x-coordinate of P, the point on the curve at which the tangent is parallel to the y-axis.

**(c)Show that the x-coordinate of P satisfies the equation. **

** x = ln 10 \(-\frac{1}{2}In (2x-1)\)State or imply e ^{y} – e^{2x} = 0 and hence y = 2x**

**(d)**Use an iterative formula, based on the equation in part (c), to find the x-coordinate of P correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.

**Answer/Explanation**

(a)Use product rule to differentiate e^{2x} y

Obtain \(2e^{2x}y+e^{2x}\frac{dy}{dx}\)

Obtain \(2e^{2x}y+e^{2x}\frac{dy}{dx}- e^{y}\frac{dy}{dx} = 0\) and rearrange to confirm given result.

(b)Consider e^{2x}y = 0 and either state e^{2x} ≠ 0 or substitute y = 0 in equation of curve

Complete argument with e^{2x }≠ 0 or e^{2x}> 0 and substitution to show y cannot be zero.

(c)Substitute for y in equation of curve and attempt rearrangement as far as e^{2x} = …..

Use relevant logarithm properties

Confirm equation \(x = In 10 – \frac{1}{2}In (2x-1)\)

(d)Use iteration process correctly at least once

Obtain final answer 1.82

Show sufficient iterations to 5 sf to justify answer or show sign change in interval [1.815, 1.825]

**Question**

**Question**

It is given that\( \int_{0}^{a}(3x^{2}+4cos2x-sinx)dx=2\)where a is a constant.

**(i)** Show that a=\(\sqrt[3]{(3-2sin2a-cosa)}\)

**(ii) Using the equation in part (i), show by calculation that 0.5 < a < 0.75.**

**(iii) Use an iterative formula, based on the equation in part (i), to find the value of a correct to****3 significant figures. Give the result of each iteration to 5 significant figures.**

**Answer/Explanation**

**5(i)** Integrate to obtain form \(x^{3}+k_{1}sin2x+k_{2}cosx\)

Obtain correct \(x^{3}+2sin2x+cosx\)

Apply limits correctly and equate to 2 Confirm given result

**5(ii)** Consider sign of \( a-\sqrt[3]{3-2sin2a-cosa}\) or equivalent for 0.5 and 0.75 Obtain –0.26 and 0.10 or equivalents and justify conclusion

**5(iii)** Use iterative process correctly at least once Obtain final answer 0.651 Show sufficient iterations to 5sf to justify answer or show a sign change in the interval [0.6505, 0.6515]

**Question**

**Question**

The diagram shows the curve with equation y=\(\frac{8+x^{3}}{2-5x}\).The maximum point is denoted by M.

(i) Find an expression for \(\frac{dy}{dx}\) and determine the gradient of the curve at the point where the curve crosses the x-axis.

**(ii) Show that the x-coordinate of the point M satisfies the equation x =\(\sqrt{(0.6x+4x^{-1})}.\)**

(iii) Use an iterative formula, based on the equation in part (ii), to find the x-coordinate of M correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

**Answer/Explanation**

<p(i) Use quotient rule (or product rule) to differentiate .

Obtain \(\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{3x^{2}\left ( 2-5x \right )-\left ( -5 \right )\left ( 8+x^{3} \right )}{\left ( 2-5x \right )^{2}}\)

State or imply curve crosses x-axiz when x=-2.

Substitute -2 to obtain 1

(ii)Equate numerator of first derivative to zero and rearrange as far as \(kx^{3}\)=….or equivalent

Confirm given result \(x=\sqrt{0.6x+4x^{-1}}\)

(iii) Use iterative process correctly atleast once

Obtain final answer 1.81

Show sufficient iterations to 5sf to justify answer or show a sign change in the interval \(\left [ 1.805,1.815 \right ]\)

**Question**

**Question**

The diagram shows the curve with equation

\(y=x^{4}+2x^{3}+2x^{2}-12x-32\)

The curve crosses the x-axis at points with coordinates (\alpha ,0)(\beta ,0).

(i) Use the factor theorem to show that (x + 2 )is a factor of \(y=x^{4}+2x^{3}+2x^{2}-12x-32\)

**(ii)**** Show that \(\beta\) satisfies an equation of the form x=\(\sqrt[3]{(p+qx)}\)and state the values of p and q. **

**(iii) **Use an iterative formula based on the equation in part (ii) to find the value of\( \beta\)correct to4 significant figures. Give the result of each iteration to 6 significant figures.

**Answer/Explanation**

**4(i)** Substitute –2 and simplify

Obtain 16 -16 8 +24 -32 −and hence zero and conclude

**4(ii)** Attempt division by

x+2 to reach at least partial quotient\( x^{3}+kx\)or use of identity or inspection

Obtain\( x^{3}+2x-16\) Equate to zero and obtain x=\(\sqrt[3]{16-2x}\)

**4(iii)** Use iteration process correctly at least once

Obtain final answer 2.256

Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval (2.2555, 2.2565)

**Question**

**Question**

The equation of a curve is \(y=\frac{e^{2x}}{4x+1}\) and the point P on the curve has y-coordinate 10.

**(i)** Show that the x coordinate of P satisfies the equation\( x=\frac{1}{2}In(40x+10).\)

**(ii) Use the iterative formula \(x_{n+1}=\frac{1}{2}In(40x_{n}+10)with x_{1}= 2.3\) to find the x-coordinate of P correct****to 4 significant figures. Give the result of each iteration to 6 significant figures. **

**(iii)** Find the gradient of the curve at P, giving the answer correct to 3 significant figures.

**Answer/Explanation**

**5(i)** Attempt rearrangement of\( \frac{e^{2x}}{4x+1}=10 to x=….. \)involving ln

Confirm\( x=\frac{1}{2} In(40x+10)\)

**5(ii)** Use iteration process correctly at least once Obtain final answer 2.316 A1

Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [2.3155, 2.3165]

**5(iii)** Use quotient rule (or product rule) to find derivative Obtain \(\frac{2e^{2x}(4x+1)-4e^{2x}}{(4x+1)^{2}} \)or equivalent Substitute answer from part **(ii)** (or more accurate value) into attempt at first derivative Obtain 16.1

**Question**

**Question**

It is given that a is a positive constant such that

\( \int _{0}^{a}(1+2x+3e^{3x})dx=250.\)

**(i)** Show that\( a-\frac{1}{3}In (251-a-a^{2})\)

**(ii) Use an iterative formula based on the equation in part (i) to find the value of a correct to 4 significant figures. Give the result of each iteration to 6 significant figures.**

**Answer/Explanation**

Integrate to obtain form \(k_{1}x+k_{2}x^{2}+k_{3}e^{3x}\) for non-zero constants

Obtain\( x+x^{2}+e^{3x}\)

Apply both limits to obtain \(a+a^{2}+e^{3a}-1\)=250 or equivalent

Apply correct process to reach form without e involved

Confirm given\( a=\frac{1}{3}In (251-a-a^{2})\)

**(ii)** Use iterative process correctly at least once

Obtain final answer 1.835

Show sufficient iterations to 6 sf to justify answer or show

sign change in interval (1.8345, 1.8355)

*Question*

The equation of a curve is y =\( \frac{3x^{2}}{x^{2}+4}\) At the point on the curve with positive x-coordinate p, the gradient of the curve is -\(\frac{1}{2}\)

**(i)** Show that p =\( \sqrt{\left ( \frac{48p-16}{p^{2}+8} \right )}\)

**(ii)** Show by calculation that 2 < p < 3.

**(iii) Use an iterative formula based on the equation in part (i) to find the value of p correct to 4 significant figures. Give the result of each iteration to 6 significant figures**

**Answer/Explanation**

.

**(i)** Use quotient rule or equivalent

Obtain or equivalent

Equate first derivative to and remove algebraic denominators dep on

Obtain or equivalent

Confirm given result

**(ii)** Consider sign of at 2 and 3 or equivalent

Complete argument correctly with appropriate calculations

**(iii)** Carry out iteration process correctly at least once Obtain final answer 2.728 Show sufficient iterations to justify accuracy to 4 sf or show sign change

in interval (2.7275, 2.7285)

**Question**

The sequence of values given by the iterative formula

\(x_{n+1}=\sqrt{\frac{1}{2}x_{n}^{2}+4x_{n}^{-3}}\) ,

with initial value x_{1}= 1.5, converges to α.

** (i)** Use this iterative formula to find α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]

** (ii)** State an equation that is satisfied by α and hence find the exact value of α.[2]

**Answer/Explanation**

Ans:

**4 (i)** Use the iterative formula correctly at least once

Obtain final answer 1.516

Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval (1.5155,1.5165)

**(ii)** State equation \(x=\sqrt{\frac{1}{2}x^{2}+4x^{-3}}\) or equivalent

Obtain exact value \(\sqrt[5]{8}\) or \(8^{0.2}\)

**Question**

(i) By sketching a suitable pair of graphs, show that the equation

ln x = 4 − \(\frac{1}{2}x\)

has exactly one real root, α. [2]

(ii) Verify by calculation that 4.5 < α < 5.0. [2]

** (iii) Use the iterative formula x _{n+1}= 8 − 2 ln x_{n} to find α correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]**

**Answer/Explanation**

Ans:

**4 (i)** Make a recognisable sketch of y = ln x

Draw straight line with negative gradient crossing positive y-axis and justify one real root

** (ii)** Consider sign of ln x = 4 − \(\frac{1}{2}x\) at 4.5 and 5.0 or equivalent

Complete the argument correctly with appropriate calculations

** (iii)** Use the iterative formula correctly at least once

Obtain final answer 4.84

Show sufficient iterations to justify accuracy to 2 d.p. or show sign change in interval (4.835, 4.845)

*Question*

**(i)** Prove that 2 cosec 2θ tan θ ≡ sec^{2 }θ. [3]

** (ii) Hence**

** (a) solve the equation 2 cosec 2θ tan θ = 5 for 0 < θ < π, [3]**

** (b) find the exact value of \(\int_{0}^{\frac{1}{6}\pi }\) 2 cosec 4x tan 2x dx. [4]**

**Answer/Explanation**

Ans:

**6 (i)** State or imply \(cosec2\theta=\frac{1}{sin2\theta }\)

Express left-hand side in terms of sinθ and cosθ

Obtain given answer sec^{2} θ correctly

**(ii) (a)** State or imply \(cos\theta =\frac{1}{\sqrt{5}}\) or tan θ = 2 at least

Obtain 1.11 or awrt 1.11, allow 0.353π

Obtain 2.03 or awrt 2.03 , allow 0.648π and no other values between 0 and π

** (b)** State integrand as sec^{2} 2x

Integrate to obtain expression of form k tan mx

Obtain correct \(\frac{1}{2}\) tan 2x

Obtain \(\frac{1}{2}\sqrt{3}\) or exact equivalent

*Question*

** (i)** Given that \(\int_{0}^{a}\left ( 3e^{\frac{1}{2}x}+1 \right )dx=10\), show that the positive constant a satisfies the equation [5]

\(a = 2ln\left ( \frac{16-a}{6} \right )\)

** (ii) Use the iterative formula \(a_{n+1} = 2ln\left ( \frac{16-a_{n}}{6} \right )\) with a _{1 }= 2 to find the value of a correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]**

**Answer/Explanation**

Ans:

**5 (i)** Obtain integral of form \(ke^{\frac{1}{2}x}+ mx\)

Obtain correct \(6e^{\frac{1}{2}x}+ x\)

Apply limits and obtain correct \(6e^{\frac{1}{2}a}+a-6\)

Equate to 10 and introduce natural logarithm correctly DM1

Obtain given answer \(a=2ln\left ( \frac{16-a}{6} \right )\) correctly

** (ii)** Use the iterative formula correctly at least once

Obtain final answer 1.732

Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval (1.7315, 1.7325)

**Question**

The diagram shows the curve \(y=x^{4}+2x-9\). The curve cuts the positive x-axis at the point P.

(i) Verify by calculation that the x-coordinate of P lies between 1.5 and 1.6.

(ii) Show that the x-coordinate of P satisfies the equation

\(x=\sqrt[3]{(\frac{9}{x}-2)}\).

(iii) Use the iterative formula

\(x_{n+1}=\sqrt[3]{(\frac{9}{x_{n}}-2)}\).

to determine the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

**Answer/Explanation**

(i)Consider sign of \(x^{4}+2x-9 at x = 1.5\) at x=1.5 and x = 1.6

Complete the argument correctly with appropriate calculations

\((f(1.5))=-0.9375,f(1.6)=0.7536\)

(ii) Rearrange \(x^{4}+2x-9=0\) to given equation or vice versa

(iii) Use the iterative formula correctly at least once M1

Obtain final answer 1.56

Show sufficient iterations to justify its accuracy to 2 d.p.

**Question**

(i) By sketching a suitable pair of graphs, show that the equation

\(\cot x=4x-2\)

where x is in radians, has only one root for \(0\leq x\leq \frac{1}{2}\pi \).

(ii) Verify by calculation that this root lies between x = 0.7 and x = 0.9.

(iii) Show that this root also satisfies the equation

\(x=\frac{1+2\tan x}{4\tan x}\).

(iv) Use the iterative formula \(x_{n+1}=\frac{1+2\tan x_{n}}{4\tan x_{n}}\) to determine this root correct to 2 decimal places.

Give the result of each iteration to 4 decimal places.

**Answer/Explanation**

(i) Make a recognisable sketch of a relevant graph, e.g. y = cot x or y = 4x – 2

Sketch a second relevant graph and justify the given statement

(ii) Consider sign of 4x – 2 – cot x at x = 0.7 and x = 0.9, or equivalent

Complete the argument correctly with appropriate calculations

(iii) Show that given equation is equivalent to \(x=\frac{1+2\tan x}{4\tan x}\) , or vice versa

(iv) Use the iterative formula correctly at least once M1

Obtain final answer 0.76

Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change

in the interval (0.755, 0.765)

**Question**

A curve has parametric equations \(x=\frac{1}{(2t+1)^{2}},y=\sqrt{(t+2)}\).

The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.

(i) Show that \(p=(p+2)^{\frac{1}{6}}-\frac{1}{2}\).**(ii) Use an iterative process based on the equation in part (i) to find the value of p correct to 3 decimalplaces. Use a starting value of 0.7 and show the result of each iteration to 5 decimal places**

**Answer/Explanation**

(i) Obtain derivative of form\( k(2t+1)^{-3}\).

Obtain \(-4(2t+1)^{-3}\) or equivalent as derivative of x .

Obtain \(\frac{1}{2}(t+2)^{-\frac{1}{2}}\) or equivalent as derivative of y

Equate attempt at \(\frac{\mathrm{d} y}{\mathrm{d} x}\) to -1

Obtain \((2p+1)^{3}=8(p+2)^{\frac{1}{2}}\) or equivalent

Confirm given answer \( p=(p+2)^{\frac{1}{6}}-\frac{1}{2}\)

(ii) Use iteration process correctly at least once

Obtain final answer 0.678

Show sufficient iterations to 5 decimal places to justify answer or show a sign change in the interval (0.6775, 0.6785)

[0.7 → 0.68003 → 0.67857 → 0.67847 → 0.6784

**Question**

The diagram shows the curve y = cos x, for 0 ≤ x ≤π. A rectangle OABC is drawn, where B is the point on the curve with x-coordinate θ, and A and C are on the axes, as shown. The shaded region R is bounded by the curve and by the lines x = θ and y = 0.

(i) Find the area of R in terms of θ.

(ii) The area of the rectangle OABC is equal to the area of R. Show that \(\Theta =\frac{1-\sin \Theta }{\cos \Theta }\)

(iii) Use the iterative formula \(\Theta _{n+1}=\frac{1-\sin \Theta }{\cos \Theta }\), with initial value θ1= 0.5, to determine the value of θ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

**Answer/Explanation**

(i)Attempt to integrate and use limits θ and π

Obtain 1– sin θ

(ii) State that area of rectangle = θcos θ, equate area of rectangle to area of R

and rearrange to given equation

(iii) Use the iterative formula correctly at least once

Obtain final answer 0.56

Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a

sign change in the interval (0.555, 0.565)