CAIE Further Paper 1 (Further Paper 1) 2024 November

The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k\) (\(k \neq 0\)), followed by a shear, \(x\)-axis fixed, with \((0, 1)\) mapped to \((k, 1)\).

1. Show that \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]

The unit square \(S\) in the \(x\)-\(y\) plane is transformed by \(\mathbf{M}\) onto the parallelogram \(P\).

1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]

Prove by mathematical induction that, for all positive integers \(n\), $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\tan^{-1}x\right) = P_n(x)\left(1+x^2\right)^{-n},$$ where \(P_n(x)\) is a polynomial of degree \(n-1\). [6]

The quartic equation \(x^4 + 2x^3 - 1 = 0\) has roots \(\alpha, \beta, \gamma, \delta\).

1. Find a quartic equation whose roots are \(\alpha^4, \beta^4, \gamma^4, \delta^4\) and state the value of \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\). [5]
2. Find the value of \(\alpha^5 + \beta^5 + \gamma^5 + \delta^5\). [3]
3. Find the value of \(\alpha^8 + \beta^8 + \gamma^8 + \delta^8\). [2]

1. Use the method of differences to find \(\sum_{r=1}^{n} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. [4]

It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).

1. Find the value of \(k\). [2]
2. Hence find \(\sum_{r=1}^{n-1} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\). [2]

1. Show that the curve with Cartesian equation $$\left(x^2+y^2\right)^2 = 6xy$$ has polar equation \(r^2 = 3\sin 2\theta\). [2]

The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
2. Find the area of the region enclosed by \(C\). [2]
3. Find the maximum distance of a point on \(C\) from the initial line. [6]

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

1. Find the equations of the asymptotes of \(C\). [2]
2. Find the coordinates of any stationary points on \(C\). [4]
3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]

The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu(\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) respectively. The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).

1. Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\). [4]

The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [4]

The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find a vector equation for \(PQ\). [7]