### Question

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

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.

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.

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.

[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.

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.

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).

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

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

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]

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
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]

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]

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 $$\ln (3-2x))-2\ln x=\ln 5$$.

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.

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

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.

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

.

### 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]

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
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]

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]

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 $$\ln (3-2x))-2\ln x=\ln 5$$.

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.

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

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.

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