**Question**

**(i)** Show that the equation \(log_{10}(x-4)=2-log_{10}x\) can be written as a quadratic equation in x.

**(ii)** Hence solve the equation \(log_{10}(x-4)=2-log_{10} \) x giving your answer correct to 3 significant figures.

**Answer/Explanation**

**1(i)** Use law for the logarithm of a product or quotient

\(Use log_{10}100=2 or 10^{2}=100\)

obtain\( x^{2}-4x-100=0\) , or equivalent

** 1(ii)** Solve a 3-term quadratic equation

Obtain answer 12.2 only

**Question**

Showing all necessary working, solve the equation ln(2x − 3 )= 2 ln x − ln (x − 1). Give your answer correct to 2 decimal places.

**Answer/Explanation**

Use law for the logarithm of a product ,quotient or power

Obtain a correct equation free of logarithms

Solve a 3-term quadratic obtaining at least one root

Obtain answer x=4.30 only

**Question**

**Question**

It is given that 2 ln(4x − 5 )+ ln(x + 1 )3 ln 3.(i) Show that \(16x^{3}-24x^{2}-15x-2=0\)**(ii)** By first using the factor theorem, factorise \(16x^{3}-24x^{2}-15x-2=0\) completely.

**(iii)** **Hence solve the equation 2 ln (4x − 5 )+ ln(x + 1) = 3 ln 3.**

**Answer/Explanation**

(i) Use law for the logarithm for a product or quotient or exponentiation AND for a power

Obtain\( (4x-5)^{2}(x+1)=27\)

Obtain given equation correctly \(16x^{3}-24x^{2}-15x-2=0\)

**(ii)** Obtain x = 2 is root or (x – 2) is a factor, or likewise with\( x=-\frac{1}{4}\)

Divide by (x – 2) to reach a quotient of the form\((16x^{3}+kx)\)

Obtain quotient \(16x^{3}+8x+1\)

**(iii)** State x = 2 only

**Question**

(i) Show that the equation

\(log_{10}\) (x-4)=2-\(log_{10}x\)can be written as a quadratic equation in x.

(ii) Hence solve the equation giving your answer correct to 3 significant figures.

**Answer/Explanation**

**(i)** Use law for the logarithm of a product or quotient Use \(log_{10}100=2or 10^{2}=100\)Obtain \(x^{2}-4x-100=0\),or equivalent

**(ii)** Solve a 3-term quadratic equation Obtain answer 12.2 only

*Question*

Two variable quantities x and y are believed to satisfy an equation of the form \(y=C(a^{x})\), where C and a are constants. An experiment produced four pairs of values of x and y. The table below gives the corresponding values of x and ln y.

By plotting ln y against x for these four pairs of values and drawing a suitable straight line, estimate the values of C and a. Give your answers correct to 2 significant figures.

**Answer/Explanation**

2 Plot the four points and draw straight line

State or imply that ln y=ln C+x lna

Carry out a completely correct method for finding lnC or ln a

Obtain answer C = 3.7

Obtain answer a = 1.5

**Question**

**Question**

It is given that x = ln (1 − y) − ln y, where 0 < y < 1.

(i) Show that \(y=\frac{e^{-x}}{1+e^{-x}}\).

(ii) Remove logarithms correctly and obtain \(e^{x}=\frac{1-y}{y}\)

(ii) Hence show that \(\int_{0}^{1}ydx=ln(\frac{2e}{e+1})\)

**Answer/Explanation**

Obtain the **given answer \(**y=\frac{e^{-x}}{1+e^{x}}\) following full working

3(ii) State integral k ln \((1+e^{-x}) \)where k = ± 1

State correct integral- ln \((1+e^{-x})\)

Use limits correctly D

Obtain the given answer

\(ln\left ( \frac{2e}{e+1} \right )\) following full working

**Question**

**Question**

Solve the equation \((1+2^{x})=2\), giving your answer correct to 3 decimal places.

**Answer/Explanation**

Remove logarithm and obtain \(1+2^{x}=e^{2}\)

Use correct method to solve an equation of the form \(2^{x}\) = a , where a > 0Obtain answer x = 2.676

*Question*

Solve the equation

log_{10 (}2x + 1) = 2 log_{10} (x + 1) − 1

Give your answers correct to 3 decimal places. [6]

**Answer/Explanation**

State or imply 10 log 10 1 = Use law of the logarithm of a power, product or quotient

Obtain a correct equation in any form, free of logs

Reduce to x^{2} − 18x − 9 = 0, or equivalent

Solve a 3-term quadratic

Obtain final answers x = 18.487 and x = –0.487

*Question*

Find the set of values of x for which \(2(3^{1-2x})< 5^{x}\). Give your answer in a simplified exact form. [4]

**Answer/Explanation**

Ans

1 Use law of the logarithm of a product or power

Obtain a correct linear inequality in any form, e.g. ln 2+ (1–2x) ln 3 <x ln 5

Solve for x

Obtain \(x> \frac{ln 6}{ln 45}\)

**Question**

**Question**

Solve the equation In 3 + In(2x + 5) = 2 In(2x + 5) = 2 In (x + 2). Give your answer in a simplified exact form.

**Answer/Explanation**

**Ans:**

Use law of logarithm of a power and sum and remove logarithms

Obtain a correct equation in any form, e.g. \(3(2x+5)=(x+2)^2\)

Use correct method to solve a 3-term quadratic, obtain at least one root

Obtain final answer final answer \(x=1+2\sqrt{3}\) or \(1+\sqrt{12}\) only

*Question*

Find the real root of the equation \(\frac{2e^{x}+e^{-x}}{2+e^{x}}=3\), giving your answer correct to 3 decimal places.

Your working should show clearly that the equation has only one real root. [5]

**Answer/Explanation**

Ans

2 Reduce to a 3-term quadratic u^{2} + 6u −1= 0 OE

Solve a 3-term quadratic for u

Obtain root \(\sqrt{10}-3\)

Obtain answer x = – 1.818 only

Reject \(-\sqrt{10}-3\) correctly

**Alternative method for Question 2**

Rearrange to obtain a correct iterative formula

Use the iterative process at least twice

Obtain answer x = – 1.818

Show sufficient iterations to at least 4 d.p. to justify x = – 1.818

Clear explanation of why there is only one real root

**Question**

**Question**

Solve the equation \(In(x^3-3)=Inx-In3\). Give your answer correct to 3 significant figures.

**Answer/Explanation**

**Ans:**

Use law of the logarithm of a product or power

Obtain a correct equation free of logarithms, e.g. \(3(x^3-3)=x^3\)

Obtain x = 1.65

**Question**

**Question**

Solve the equation \((1+2^{x})=2\), giving your answer correct to 3 decimal places.

Remove logarithm and obtain \(1+2^{x}=e^{2}\)

Use correct method to solve an equation of the form \(2^{x}\) = a , where a > 0

**Answer/Explanation**

Obtain answer x = 2.676

**Question**

**Question**

Solve the equation = 8, giving your answer correct to 3 decimal places.

**Answer/Explanation**

Solve for\(3^{x} \)and obtain\( 3^{x}=\frac{18}{7}\)

Use correct method for solving an equation of the form

Obtain answer x = 0.860 3 d.p. only

*Question*

*Question*

Solve the equation ln(*x*^{2} + 4) = 2 ln* x* + ln 4, giving your answer in an exact form. [3]

**Answer/Explanation**

Ans:

Use law of the logarithm of a power, quotient or product

Remove logarithms and obtain a correct equation in *x*, e.g. *x*^{2} + 4 = 4*x ^{2}*

Obtain final answer

*x*= 2/ √3 , or exact equivalent

*Question*

The variables x and y satisfy the equation x^{n}y^{2} = C, where n and C are constants. The graph of ln y against ln x is a straight line passing through the points (0.31, 1.21) and (1.06, 0.91), as shown in the diagram.

Find the value of n and find the value of C correct to 2 decimal places.

**Answer/Explanation**

Ans:

State or imply n In x + 2 In y = In C

Substitute values of ln y and ln x, or equate gradient of line to \(\pm \frac{1}{2}n\) but not ±n, and solve for n

Obtain n = 0.8[0] or 0.8[00] or \(\frac{4}{5}\)

Solve for C

Obtain C = 14.41

**Question**

**Question**

Solve the equation

\(5^{x−1} \)=\( 5^{x }− 5\),

giving your answer correct to 3 significant figures.

**Answer/Explanation**

EITHER Use laws of indices correctly and solve for 5^{x} or for \( 5^{–x}\)

or for \(5^{x–1}\)

Obtain\( 5^{x}\)

or for\( 5^{–x}\)

or for\( 5 ^{x–1}\)

in any correct form, e.g. \(5^{x}\)\( \frac{5}{1-\frac{1}{5}}\)

Use correct method for solving \(5^{x}\) = a, or \(5^{–x} \) = a, or \(5^{x–1}\) = a, where a 0 Obtain answer x = 1.14

OR Use an appropriate iterative formula, \(x_{n+1}\)=\(\frac{In(5^{x-1+5})} {In5} \)correctly, at least once

Obtain answer 1.14

Obtain answer 1.14

Show sufficient iterations to at least 3 d.p. to justify 1.14 to 2 d.p., or show there is a sign change in the interval (1.135, 1.145) Show there is no other root [For the solution x = 1.14 with no relevant working give , and a further B1 if

1.14 is shown to be the only solution.]

**Question**

The polynomial p(x) is defined by\(p(x)=x^{3}-3ax+4a\),where a is a constant.**(i)** Given that (x − 2) is a factor of p(x), find the value of a. **(ii) When a has this value,****(a) factorise p(x)completely, ****(b) find all the roots of the equation\( p(x^{2})\)=0**

**Answer/Explanation**

**(i)** Substitute x = 2 and equate to zero, or divide by x – 2 and equate constant remainder to zero, or equivalent

Obtain a = 4

**(ii) (a)** Find further (quadratic or linear) factor by division, inspection or factor theorem or equivalent

Obtain x^{2}+ 2x – 8 or x + 4

State (x – 2)^{2}

(x + 4) or equivalent

**(b)** State any two of the four (or six) roots

State all roots \((\pm \sqrt{2},\pm 2i) \)provided two are purely imaginary