**Question**

**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 a^{y} = 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*

*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**

**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**

**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**

**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**

**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(3*k* + *x)* + ln(2*k* − *x)*, expressing *x* in terms of *k*. [5]

**Answer/Explanation**

Ans:

Use 2ln *x = *ln *x ^{2}*

Use law for addition or subtraction of logarithms

Obtain

*x*= (3 +

^{2}*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

5^{x+3} = 7^{x-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 = Ae^{p(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 *A *

Obtain *A* = 3.2

Alternative:

Obtain an equation either e^{1.6} = Ae^{p} or e^{2.92} = Ae^{4p}

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**

**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(3*k* + *x)* + ln(2*k* − *x)*, expressing *x* in terms of *k*. [5]

**Answer/Explanation**

Ans:

Use 2ln *x = *ln *x ^{2}*

Use law for addition or subtraction of logarithms

Obtain

*x*= (3 +

^{2}*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

5^{x+3} = 7^{x-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 = Ae^{p(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 *A *

Obtain *A* = 3.2

Alternative:

Obtain an equation either e^{1.6} = Ae^{p} or e^{2.92} = Ae^{4p}

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