Question1.
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
Question2.
2(i) Given that\( tan2\Theta cot\Theta = 8\), show that \(tan^{2}\Theta =\frac{3}{4}\)
2(ii) Hence solve the equation tan 21 cot 1 = 8 for 0Å < 1 < 180Å
Answer/Explanation
.
(i) Use identity \(cot\Theta =\frac{1}{tan\Theta }\)
Attempt use of identity for \(tan2\Theta\)
Confirm given \(tan^{2}\Theta =\frac{3}{4}\)
(ii) Obtain 40.9 B1
Obtain 139.1
Question3.
(i) Solve the inequality $|2x − 5| < |x + 3 |$.
(ii) Hence find the largest integer y satisfying the inequality$| 2 ln y − 5 |<| ln y + 3 |$.
Answer/Explanation
(i) State or imply non-modulus inequality \((2x-5)^{2}<(x+3)^{2}\)or
corresponding equation or pair of linear equations Attempt solution of 3-term quadratic inequality or equation or of 2 linear equations
Obtain critical values \(\frac{2}{3}\) and 8
State answer (\frac{2}{3}\)< <x< 8
(ii) Attempt to find y from ln y = upper limit of answer to part (i)
Obtain 2980
Question4.
Find the gradient of the curve \(x^{2}siny+cos3y=4\)
at the point\( (2,\frac{1}{2}\pi ).\)
Answer/Explanation
Use product rule for derivative of\( x^{2}sin y\)
Obtain\( 2xsiny +x^{2}cosy\frac{dy}{dx}\)
Obtain\( -3sin3y\frac{dy}{dx}\) as derivative of cos 3y
Obtain \(2xsiny+x^{2}cosy\frac{dy}{dx}-3sin3y\frac{dy}{dx}\)=0
Substitute\( x=2 y=\frac{1}{2}\pi \)to find value of \(\frac{dy}{dx}\)
Obtain\( -\frac{4}{3}\)
Question5.
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)
Question6.
The polynomial p(x )is defined by\( p(x)=ax^{3}+bx^{2}-17x-a\)
where a and b are constants. It is given that(x + 2 )is a factor of p(x )and that the remainder is 28 when p
(x )is divided by (x − 2).
(i) Find the values of a and b.
(ii) Hence factorise p(x )completely.
(iii) State the number of roots of the equation \(p(2^{y})=0\),justifying your answer.
Answer/Explanation
(i) Substitute x = –2 and equate to zero
Substitute x = 2 and equate to 28
Obtain -9a+4b+34=0and 7a+4b-62=0 or equivalents
Solve a relevant pair of simultaneous equations for a or b
Obtain a = 6, b = 5
(ii) Divide by x + 2, or equivalent, at least as far as \(k^{1}x^{2}+k_{2}x\)
Obtain\( 6x^{2}-7x-3\)
Obtain (x+2)(3x+1)(3x-3)
(iii) Refer to, or clearly imply, fact that\( 2^{y}\) is positive State one
Question7.
(ii) Find the exact area of the shaded region.
Answer/Explanation
