## Which is easier?

Below are the questions from two mathematics papers. Each paper is divided into a section A and a section B. If you are prepared to look through the questions (maybe even doing some of them if you are sufficiently conscientious) and give me your judgment about which section A is easier and which section B is easier, I would be very interested and grateful. I shall refer to the two A sections as section A1 and section A2, and similarly for the two B sections.

1. I am deliberately saying as little as possible about the questions or my motivation for this post. I will write more once I have enough answers.

2. The order in which I have presented the two A sections was decided by the toss of a coin. The order in which I have presented the two B sections was decided by the independent toss of a (different) coin.

3. When it comes to your vote, I am interested in your degree of certainty rather than in the difference of difficulty. In particular, if you are absolutely certain that one section is very slightly harder than another, then you should indicate that you are certain and not worry that the difference is slight (just as I am absolutely certain that $\sqrt{2}>1.4142$).

4. Please remember to take account not just of the difficulty of individual questions but also of the number of marks available (which are shown in square brackets).

5. Obviously the difficulty of a question depends on how familiar you are with the general type of question to which it belongs. This shouldn’t create problems because the subject matter in the two papers is very similar. However, if it helps, a good working assumption is that people taking the papers have seen similar questions before.

6. If by any chance you recognise any of the sections, then please do not take part in the votes.

7. I am interested in independent judgments, so for the time being I will hide the results of the polls. I have also disabled comments for the time being. When enough people have voted, I will make the results available and will allow comments.

8. Thank you very much in advance to all who take the trouble to participate.

Section A1 (36 marks)

1. You are given the matrix $M=\begin{pmatrix} 2&3\\-2&1\\ \end{pmatrix}.$

Find the inverse of M.

The transformation associated with M is applied to a figure of area 2 square units. What is the area of the transformed figure? [3]

2. (i) Show that $\frac{1}{r+1}-\frac 1{r+2}=\frac 1{(r+1)(r+2)}.$ [2]

(ii) Hence use the method of differences to find the sum of the series

$\sum_{r=1}^n \frac 1{(r+1)(r+2)}.$ [4]

3. (i) Solve the equation $\frac 1{x+2}=3x+4.$ [3]

(ii) Solve the inequality $\frac 1{x+2}\leq 3x+4.$ [4]

4. Find $\sum_{r=1}^n r^2(r+2),$ giving your answer in a factorized form. [6]

5. The roots of the cubic equation $x^3+2x^2+x-3=0$ are $\alpha, \beta$ and $\gamma.$

Find the cubic equation whose roots are $\alpha+1,$ $\beta+1$ and $\gamma+1,$ simplifying your answer as far as you can. [6]

6. Prove by induction that $\sum_{r=1}^n r2^{r-1}=1+(n-1)2^n.$ [8]

Section A2 (36 marks)

1. Find the values of A, B and C in the identity $4x^2-16x+c=A(x+B)^2+2.$ [4]

2. You are given that $M=\begin{pmatrix} 2 & -5\\ 3 & 7\\ \end{pmatrix}.$

$M\begin{pmatrix} x\\ y\\ \end{pmatrix}=\begin{pmatrix} 9\\ -1\\ \end{pmatrix}$ represents two simultaneous equations.

(i) Write down these two equations. [2]

(ii) Find $M^{-1}$ and use it to solve the equations. [4]

3. The cubic equation $2z^3-z^2+4z+k=0,$ where $k$ is real, has a root $z=1+2j.$

Write down the other complex root. Hence find the real root and the value of $k.$ [6]

4. The roots of the cubic equation $x^3-2x^2-8x+11=0$ are $\alpha, \beta$ and $\gamma.$

Find the cubic equation with roots $\alpha+1,$ $\beta+1$ and $\gamma+1.$ [6]

5. Use the result $\frac 1{5r-1}-\frac 1{5r+4}\equiv\frac 5{(5r-1)(5r+4)}$ and the method of differences to find

$\sum_{r=1}^n\frac 1{(5r-1)(5r+4)},$

6. A sequence is defined by $u_1=2$ and $u_{n +1}=\frac{u_n}{1+u_n}.$

(i) Calculate $u_3.$ [2]

(ii) Prove by induction that $u_n=\frac 2{2n-1}.$ [6]

Section B1

7. Fig. 7 shows an incomplete sketch of $y=\frac{(2x-1)(x+3)}{(x-3)(x-2)}.$

[I cannot reproduce the sketch, so let me describe it. The graph has two vertical asymptotes, and is sketched in its entirety except for the part to the right of those asymptotes. No numbers are marked on the sketch, but what you can read off is that there is also a horizontal asymptote towards $-\infty,$ that the vertical asymptotes are both to the right of the y-axis, that the curve crosses the x-axis once to the left of the y-axis and once between the y-axis and the first vertical asymptote, that it tends to infinity to the left of the first asymptote, comes up from $-\infty$ to the right of the first asymptote, and reaches a local maximum below the x-axis and goes back down to $-\infty$ at the second vertical asymptote. To summarize, all the qualitative features of two of the three sections of the graph are illustrated. To the right of the diagram are the words “Not to scale”.]

(i) Find the coordinates of the points where the curve cuts the axes. [2]

(ii) Write down the equations of the three asymptotes. [3]

(iii) Determine whether the curve approaches the horizontal asymptote from above or below for large positive values of x, justifying your answer. Copy and complete the sketch. [3]

(iv) Solve the inequality $\frac{(2x-1)(x+3)}{(x-3)(x-2)}<2.$ [4]

8. Two complex numbers, $\alpha$ and $\beta,$ are given by $\alpha=\sqrt{3}+j$ and $\beta=3j.$

(i) Find the modulus and argument of $\alpha$ and $\beta.$ [3]

(ii) Find $\alpha\beta$ and $\frac\beta\alpha,$ giving your answers in the form $a+bj,$ showing your working. [5]

(iii) Plot $\alpha, \beta, \alpha\beta$ and $\frac\beta\alpha$ on a single Argand diagram. [2]

9. The matrices $\mathbf{P}=\begin{pmatrix} 0&1\\-1&0\\ \end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix} 2&0\\ 0&1\\ \end{pmatrix}$ represent transformations P and Q respectively.

(i) Describe fully the transformations P and Q. [4]

[There follows a diagram of a triangle T in the Cartesian plane. The y axis has points 0, 1 and 2 marked on it and the x-axis has points 0, 1, 2 and 3. The vertices of the triangle are as specified below in the question.]

Fig. 9 shows triangle T with vertices A (2,0), B (1,2) and C (3,1).

Triangle T is transformed first by tranformation P, then by transformation Q.

(ii) Find the single matrix that represents this composite transformation. [2]

(iii) This composite transformation maps triangle T onto triangle T’, with vertices A’, B’ and C’. Calculate the coordinates of A’, B’ and C’. [2]

T’ is reflected in the line $y=-x$ to give a new triangle T”.

(iv) Find the matrix $\mathbf{R}$ that represents reflection in the line $y=-x.$ [2]

(v) A single transformation maps T” onto the original triangle, T. Find the matrix representing this transformation. [4]

Section B2

7. A curve has equation $y=\frac{(2x-3)(x+1)}{(x+4)(x-2)}.$

(i) Write down the values of x for which y=0. [1]

(ii) Write down the equations of the three asymptotes. [3]

(iii) Determine whether the curve approaches the horizontal asymptote from above or from below for

(A) large positive values of x,

(B) large negative values of x. [3]

(iv) Sketch the curve. [3]

(v) Solve the inequality $\frac{(2x-3)(x+1)}{(x+4)(x-2)}\leq 2.$ [4]

8. Two complex numbers are given by $\alpha=2-j$ and $\beta=-1+2j.$

(i) Find $\alpha+\beta, \alpha\beta$ and $\frac\alpha\beta$ in the form $a+bj,$ showing your working. [6]

(ii) Find the modulus of $\alpha,$ leaving your answer in surd form. Find also the argument of $\alpha.$ [2]

(iii) Sketch the locus $|z-\alpha|=2$ on an Argand diagram. [2]

(iv) On a separate Argand diagram, sketch the locus arg$(z-\beta)=\frac 14\pi.$ [2]

9. You are given the matrix $M=\begin{pmatrix} 0.8 & 0.6\\ 0.6 & -0.8\\ \end{pmatrix}.$

(i) Calculate $M^2.$ [1]

You are now given that the matrix $M$ represents a reflection in a line through the origin.

(ii) Explain how your answer to part (i) relates to this information. [1]

(iii) By investigating the invariant points of the reflection, find the equation of the mirror line. [3]

(iv) Describe fully the transformation represented by the matrix $P=\begin{pmatrix} 0.8 & -0.6\\ 0.6 & 0.8\\ \end{pmatrix}.$ [2]

(v) A composite transformation is formed by the transformation represented by $P$ followed by the transformation represented by $M.$ Find the single matrix that represents this composite transformation. [2]

(vi) The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection? [1]

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