Question

Given that
              ln (2x + 1) − ln (x − 3) = 2,
     find x in terms of e.                                                                                  [4]

Answer/Explanation

Ans

Use correct logarithm property to simplify left-hand side 
    Use correct process to obtain equation without logarithms

    Obtain \(\frac{2x+1}{x-3}=e^{2}\)

    Obtain \(x=\frac{3e^{2}+1}{e^{2-2}}\)

Question

The variables x and y satisfy the equation ay = kx, where a and k are constants. The graph of y against ln x is a straight line passing through the points (1.03, 6.36) and (2.58, 9.00), as shown in the diagram.

Find the values of a and k, giving each value correct to 2 significant figures.

Answer/Explanation

Ans:

State or imply equation is y ln a = ln k + ln x

Equate gradient of line to \(\frac{1}{ln a}\)

Obtain \(\frac{1}{ln a}\)  =  \(\frac{2.64}{1.55}\) or equivalent and hence a =1.8

Substitute appropriate values to find ln k

Obtain ln 2.7… k = and hence k =15

Question

A curve has equation y = 7 + 4 ln (2x + 5)

Find the equation of the tangent to the curve at the point (-2, 7), giving your answer in the form y = mx + c. 

Answer/Explanation

Ans:

Differentiate to obtain form \(\frac{k}{2x+5}\)

Obtain correct \(\frac{8}{2x+5}\)

Substitute x = − 2 to obtain gradient 8

Attempt equation of tangent through ((-2, 7) with numerical gradient

Obtain y= 8x + 23

Question

The polynomial p(x )is defined by \(p(x)=ax^{3}+ax^{2}-15x-18\), where a is a constant. It is given that(x − 2 )is a factor of p(x).

(i) Find the value of a.

(ii) Using this value of a, factorise p(x )completely.

(iii) Hence solve the equation \(p(e\sqrt{y})=0,\) giving the answer correct to 2 significant figures.

Answer/Explanation

[2]

4(i) Substitute x = 2 , equate to zero and attempt solution

Obtain a = 4

4(ii) Divide by x − 2 at least as far as the x term 

Obtain\( 4x^{2}+12x+9\)

Conclude\( (x-2 (2x+3)^{2}\)

4(iii) Attempt correct process to solve \(e^{\sqrt{y}} \)= k where k >0

Obtain 0.48 and no others

Question

(i) Solve the inequality 2x − 7 < 2x − 9.

(ii) Hence find the largest integer n satisfying the inequality 2 ln n − 7 < 2 ln n − 9.

Answer/Explanation

1(i) State or imply non-modular inequality \((2x-7)^{2}<(2x-9)^{2}\)or corresponding  equation or linear equation (with signs of 2x different)Obtain critical value 4 State x<4 only

1(ii) Attempt to find n from ln n = their critical value from part (i) 

Obtain or imply  n< \(e^{4}\) and hence 54

Question

The variables x and y satisfy the equation  \(y=Ae^{px+p}\) where A and p are constants. The graph of

ln y against x is a straight line passing through the points (1, 2.835 )and

(6, 6.585), as shown in the
diagram. Find the values of A and p.

Answer/Explanation

State or imply equation is ln ln y= A+ px+ p

Equate gradient of line to p

Obtain p = 0.75

Substitute appropriate values to find ln A

Obtain ln 1.335… A = and hence A= 3.8

Question

Solve the equation 2 ln
(2x )− ln(x + 3 )= ln(3x + 5).

Answer/Explanation

Use 2In(2x)=In\( (2x)^{2}\) Use addition or subtraction property of logarithms

Obtain \(4x^{2}=(x+3)(3x+5) \) or equivalent without logarithms

Solve 3-term quadratic equation

Conclude with x = 15 only

Question

 It is given that k is a positive constant. Solve the equation 2 ln x = ln(3k + x) + ln(2kx), expressing x in terms of k. [5]

Answer/Explanation

Ans:

 Use 2ln x = ln x2
    Use law for addition or subtraction of logarithms 
    Obtain x2 = (3 + x) (2-x) or equivalent with no logarithms 
    Solve 3-term quadratic equation 
    Obtain \(x\frac{3}{2}\) and no other solutions 

Question

 Use logarithms to solve the equation

5x+3 = 7x-1,

   giving the answer correct to 3 significant figures. [4]

Answer/Explanation

Ans:

Introduce logarithms and use power law twice 
    Obtain (x + 3)log 5 = (x −1)log 7 or equivalent 
    Solve linear equation for x 
    Obtain 20.1 

Question

 

The variables x and y satisfy the equation

y = Aep(x−1),

where A and p are constants. The graph of ln y against x is a straight line passing through the points (2, 1.60) and (5, 2.92), as shown in the diagram. Find the values of A and p correct to 2 significant figures. [5]

Answer/Explanation

Ans:

2 State or imply that ln y = ln A + p(x-1)
    Equate gradient to p or obtain two equations for ln A and p
    Obtain p = 0.44 
    Substitute values correctly, to find value of ln
    Obtain A = 3.2 

    Alternative:
    Obtain an equation either e1.6 = Aep or e2.92 = Ae4p
    Obtain both equations correctly 
    Solve to obtain p = 0.44 
    Substitute value correctly to find A 
    Obtain A = 3.2 

Question

Solve the equation \(\ln (3-2x))-2\ln x=\ln 5\).

Answer/Explanation

Use \(2\ln x=\ln (x^{2})\)
Use law for addition or subtraction of logarithms

Obtain correct quadratic equation in x A1
Make reasonable solution attempt at a 3-term quadratic 
(dependent on previous M marks)
State \(x=\frac{3}{5}\)  and no other solutions

Question

Use logarithms to solve the equation \(5^{x}=3^{2x-1}\), giving your answer correct to 3 significant figures.

Answer/Explanation

Use law for the logarithm of a product, a quotient or a power 
Obtain  \(x\log 5=(2x-1)\log 3\) or equivalent 
Solve for x
Obtain answer x = 1.87

Question

The variables x and y satisfy the equation \((y=A(b^{x})\) ,where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 2.14) and (5, 4.49), as shown in the diagram.
Find the values of A and b, correct to 1 decimal place.

Answer/Explanation

State or imply that \(\ln y=\ln A+x\ln b\)

Equate intercept on y-axis to \(\ln A\)

Obtain  \(\ln A=2.14\) and hence A=8.5

Attempt gradient of line or equivalent (or use of correct substitution)

Obtain 0.47= or equivalent and hence b=1.6

Question

(a) Find  \(\int \frac{1+Cos^{4}2x}{cos^{2}2x}\)

(b) Without using a calculator, find the exact value of dx, giving your answer in the form ln , where a and b are integers

Answer/Explanation

.

Question

 It is given that k is a positive constant. Solve the equation 2 ln x = ln(3k + x) + ln(2kx), expressing x in terms of k. [5]

Answer/Explanation

Ans:

 Use 2ln x = ln x2
    Use law for addition or subtraction of logarithms 
    Obtain x2 = (3 + x) (2-x) or equivalent with no logarithms 
    Solve 3-term quadratic equation 
    Obtain \(x\frac{3}{2}\) and no other solutions 

Question

 Use logarithms to solve the equation

5x+3 = 7x-1,

   giving the answer correct to 3 significant figures. [4]

Answer/Explanation

Ans:

Introduce logarithms and use power law twice 
    Obtain (x + 3)log 5 = (x −1)log 7 or equivalent 
    Solve linear equation for x 
    Obtain 20.1 

Question

 

The variables x and y satisfy the equation

y = Aep(x−1),

where A and p are constants. The graph of ln y against x is a straight line passing through the points (2, 1.60) and (5, 2.92), as shown in the diagram. Find the values of A and p correct to 2 significant figures. [5]

Answer/Explanation

Ans:

2 State or imply that ln y = ln A + p(x-1)
    Equate gradient to p or obtain two equations for ln A and p
    Obtain p = 0.44 
    Substitute values correctly, to find value of ln
    Obtain A = 3.2 

    Alternative:
    Obtain an equation either e1.6 = Aep or e2.92 = Ae4p
    Obtain both equations correctly 
    Solve to obtain p = 0.44 
    Substitute value correctly to find A 
    Obtain A = 3.2 

Question

Solve the equation \(\ln (3-2x))-2\ln x=\ln 5\).

Answer/Explanation

Use \(2\ln x=\ln (x^{2})\)
Use law for addition or subtraction of logarithms

Obtain correct quadratic equation in x A1
Make reasonable solution attempt at a 3-term quadratic 
(dependent on previous M marks)
State \(x=\frac{3}{5}\)  and no other solutions

Question

Use logarithms to solve the equation \(5^{x}=3^{2x-1}\), giving your answer correct to 3 significant figures.

Answer/Explanation

Use law for the logarithm of a product, a quotient or a power 
Obtain  \(x\log 5=(2x-1)\log 3\) or equivalent 
Solve for x
Obtain answer x = 1.87

Question

The variables x and y satisfy the equation \((y=A(b^{x})\) ,where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 2.14) and (5, 4.49), as shown in the diagram.
Find the values of A and b, correct to 1 decimal place.

Answer/Explanation

State or imply that \(\ln y=\ln A+x\ln b\)

Equate intercept on y-axis to \(\ln A\)

Obtain  \(\ln A=2.14\) and hence A=8.5

Attempt gradient of line or equivalent (or use of correct substitution)

Obtain 0.47= or equivalent and hence b=1.6