Solve the simultaneous equations \begin{align*} x + y + \lambda z &= \mu, \\ 2x + 3y + 4z &= 0, \\ 3x + 4y + 5z &= 1 \end{align*} for general values of \(\lambda, \mu\). Examine the cases arising from particular values of \(\lambda, \mu\) for which the equations do not have (i) a unique solution, (ii) a finite solution.
By means of a graph, or otherwise, determine the values of \(\lambda\) for which the equation \[ (x-1)^2(x-a) + \lambda = 0 \] has three real roots, where \(a\) is a given constant greater than unity. \par [If any general formula is quoted, it must be proved.] \par Prove that, whatever the values of \(a, \lambda\), the roots \(\alpha, \beta, \gamma\) are connected by the relation \[ \beta\gamma + \gamma\alpha + \alpha\beta - 2(\alpha+\beta+\gamma) + 3 = 0. \]
(i) Solve the equation \[ x^4 - x^3 + x^2 - x + 1 = 0. \] (ii) Find, in terms of \(p\) and \(q\), the cubic equation such that, if \(x\) is any one of its roots, \(px+q\) is also a root.
Express \(\tan n\theta\) in terms of \(\tan\theta\), where \(n\) is a positive integer, and prove your result. \par Evaluate \(\tan\dfrac{\pi}{12}\).
Prove that \[ 1+\cos\theta+\cos 2\theta + \dots + \cos(n-1)\theta = \sin\frac{n\theta}{2} \cos\frac{(n-1)\theta}{2} \operatorname{cosec}\frac{\theta}{2}. \] Examine carefully whether the result remains true as \(\theta \to 0\), and enunciate precisely any theorem on limits to which you appeal.
Sketch the curves \[ \text{(i) } y = x^2-x^3; \quad \text{(ii) } y^2 = x^2-x^3, \] and find their radii of curvature at the origin.
Prove that, if \(f(x)\) is a function whose differential coefficient \(f'(x)\) is positive throughout a given interval, then \(f(x_2)>f(x_1)\), if \(x_2>x_1\), where \(x_1, x_2\) are any two values of \(x\) in the interval.
\par Prove that
\[ x - \frac{x^3}{6} + \frac{x^5}{120} > \sin x \]
for all positive values of \(x\), and that
\[ \left(1-\frac{x^2}{2}+\frac{x^4}{24}\right)\sin x > \left(x-\frac{x^3}{6}+\frac{x^5}{120}\right)\cos x \]
when \(0
(i) If \(u=xyz\), where \(x, y, z\) are connected by the relations \[ yz+zx+xy=a, \quad x+y+z=b \quad (a, b \text{ being constants}), \] prove that \[ \frac{du}{dx} = (x-y)(x-z). \] (ii) If \(\xi, \eta\) are functions of \(x, y\) such that \(\xi=e^x \cos y, \eta=e^x \sin y\), and \(x,y\) are functions of \(r, \theta\) such that \(x=e^r \cos\theta, y=e^r \sin\theta\), where \(r\) is a function of \(\theta\), prove that \[ \frac{d\xi}{d\eta} = \frac{\frac{dr}{d\theta}-\tan(y+\theta)}{1+\frac{dr}{d\theta}\tan(y+\theta)}. \]
If \(y=x(1-x)/(1+x^2)\),
(i) Prove that, if \(n\) is a positive integer, \[ \int_0^{\pi/2} e^{\lambda x} \cos nx dx = \frac{1}{\lambda^2+n^2}\{\lambda e^{\lambda\pi/2}-1\}, \] where \(\lambda\) has one of the values \(\pm 1, \pm n\), and classify the cases. \par (ii) Find the area bounded by the parabola \(y^2=ax\) and the circle \(x^2+y^2=2a^2\).