Show that, if \(x\) is a root of the equation \(x^4-6x^2+1=p(x^3-x)\), then \(\frac{1+x}{1-x}\) is also a root, and find the other two roots in terms of \(x\). Hence show that if \(p\) is real all the four roots of the equation are real.
Prove that, if \(x>0\) and \(1>p>0\), then \[ x^p - 1 \le p(x-1). \] By means of the identity \[ \frac{x-\alpha}{x^n-\alpha^n} = \frac{1}{x^{n-1}+\alpha x^{n-2} + \dots + \alpha^{n-1}}, \] or otherwise, show that, if \(n\) is an integer greater than 1 and if \(x>0\), \[ (1+x)^{\frac{1}{n}} > 1 + \frac{x}{n+\frac{1}{2}(n-1)x}. \]
The polynomial \(f(x)\) has only simple zeros \(a_1, a_2, \dots, a_n\). Show that, if \[ \frac{1}{[f(x)]^2} = \sum_{i=1}^n \frac{A_i}{(x-a_i)^2} + \frac{B_i}{x-a_i}, \] then \[ A_i = \frac{1}{[f'(a_i)]^2}, \quad B_i = -\frac{f''(a_i)}{[f'(a_i)]^3}. \] Hence, or otherwise, express \[ \frac{1}{(x^{2n}-1)^2}, \] where \(n\) is a positive integer, as the sum of real partial fractions.
Show, by induction or otherwise, that \[ \tan(\theta_1+\theta_2+\dots+\theta_n) = \frac{\sum\tan\theta_1 - \sum\tan\theta_1\tan\theta_2\tan\theta_3+\dots}{1-\sum\tan\theta_1\tan\theta_2+\dots}. \] Three angles \(\theta_1, \theta_2, \theta_3\), none of which is zero or a multiple of \(\pi\), satisfy the relations \[ \tan(\theta_1+\theta_2+\theta_3) = \tan\theta_1+\tan\theta_2+\tan\theta_3 = -\tan\theta_1\tan\theta_2\tan\theta_3. \] Show that one of tan \(\theta_1\), tan \(\theta_2\), tan \(\theta_3\) must be equal to 1, another must be equal to -1, while the third is arbitrary.
A family of curves is given by the equation \[ \left(y + \frac{1}{x^3}\right)(3x-1) = 8\lambda, \] where \(\lambda\) is a variable parameter which takes positive values only, and \(x > \frac{1}{3}\). Show that if \(0<\lambda<1\), the curves have one real maximum and one real minimum, while if \(\lambda > 1\) the curves have no real maximum or minimum. Show also that the locus of the maxima and minima is \(yx^3=3x-2\), and that this locus touches the curve \(\lambda=1\) at the point \(x=1, y=1\).
Evaluate the integrals \[ \int_0^1 \frac{\sin^{-1}x}{(1+x)^2} \, dx, \quad \int_0^{\pi/2} \frac{dx}{2\cos^2x + 2\cos x \sin x + \sin^2 x}, \quad \int_{-\infty}^{\infty} \frac{dx}{(e^x-a+1)(1+e^{-x})}. \]
If \[ L_n(x) = e^x \frac{d^n}{dx^n} (x^n e^{-x}), \] show that
Three variables \(x, y, z\) are connected by a functional relation \(f(x, y, z)=0\), so that any variable can be treated as a function of the other two. Denoting by \((\partial x/\partial y)_z\) the partial derivative of \(x\) with respect to \(y\) when \(z\) is kept constant, show that \[ \left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y = -1. \] If \(x, y\) and \(z\) are each expressed parametrically as functions of two other variables \(u\) and \(v\), show that \[ \left(\frac{\partial x}{\partial y}\right)_z = \frac{\frac{\partial x}{\partial u}\frac{\partial z}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial z}{\partial u}}{\frac{\partial y}{\partial u}\frac{\partial z}{\partial v} - \frac{\partial y}{\partial v}\frac{\partial z}{\partial u}}. \]
The coordinates of a curve are given parametrically as \[ x = a(2\cos t + \cos 2t), \quad y=a(2\sin t - \sin 2t). \] Find the radius of curvature at an arbitrary point, and show that the parametric equations of the locus of the centre of curvature is \[ x = \frac{3a}{2}(2\cos t - \cos 2t), \quad y = \frac{3a}{2}(2\sin t + \sin 2t). \]
Trace the curve \(y^2 = \frac{x^2(3-x)}{1+x}\), and find the area of the loop.