Show that the equation \[ x^4 - 3x + 1 = 0 \] has only two real roots and evaluate the smaller of the two correct to three decimal places.
(i) \(a,b,c\) and \(d\) are distinct complex numbers. By an appeal to the Argand diagram or otherwise, show that, if any two of the numbers \[ \frac{a-b}{c-d}, \quad \frac{b-c}{a-d}, \quad \frac{c-a}{b-d} \] are pure imaginaries, then so is the third. (ii) What complex numbers correspond, in the Argand diagram, to the centroid and the orthocentre of the triangle whose vertices are represented by the numbers \(0, 1+3i\) and \(5i\)?
Find the maxima, minima and points of inflexion of the curve \(y = \sqrt{x} \cos \log \sqrt{x}\), where \(x > 0\).
Evaluate the definite integrals \[ \text{(i)} \quad \int_0^{\pi/4} \tan^8 x \, dx, \quad \text{(ii)} \quad \int_{\sqrt{2}}^{2\sqrt{2}} \frac{x^2-1}{x^4+1} \, dx. \]
Sketch the curve \[ x = \cos t, \quad y = \sin 2t \] and find the area enclosed by one of the loops. Write down the equation of the normal to the curve at a general point \(t\). Hence, or otherwise, show that the centre of curvature of the curve at \(t=0\) is the point \((-3,0)\) and find the position of the centre of curvature at \(t=\pi/4\).
The sides \(a,b,c\) of a triangle are measured with a small percentage error \(\epsilon\) and the area is calculated from the measured values of the sides, using the formula \[ \sqrt{s(s-a)(s-b)(s-c)}, \] where \(s = \frac{1}{2}(a+b+c)\). Show that, if the triangle is acute-angled, the percentage error in the area cannot exceed \(2\epsilon\), approximately; but that, if the angle \(A\) of the triangle is obtuse, the percentage error in the area may be as large as \(2\epsilon \cot B \cot C\).
If \(y = e^{\frac{1}{2}x^2+bx^2}\) and \(c_n = \left(\frac{d^n y}{dx^n}\right)_{x=0}\), show that \(c_{n+1} = c_n + nc_{n-1}\). Prove also that \(c_n\) is an integer and \((n+1)(c_n-1)\) is always divisible by 3.
If \(\xi=x^2-y^2\), \(\eta=2xy\) and \(f(x,y)=g(\xi,\eta)\), show that \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 4(x^2+y^2)\left(\frac{\partial^2 g}{\partial \xi^2} + \frac{\partial^2 g}{\partial \eta^2}\right). \]
Find the limits as \(n\) tends to infinity of
Show that \[ \frac{1}{x}e^{-\frac{1}{2}x^2} = \int_x^\infty e^{-\frac{1}{2}y^2} \left(1+\frac{1}{y^2}\right) \, dy \] and that \[ \int_x^\infty e^{-\frac{1}{2}y^2} \, dy \] lies between \(\displaystyle\frac{1}{x}e^{-\frac{1}{2}x^2}\) and \(\displaystyle\frac{1}{x}e^{-\frac{1}{2}x^2} - \frac{1}{x^3}e^{-\frac{1}{2}x^2}\).