10273 problems found
A uniform heavy chain of length 10 feet is given two complete turns and a half turn round a smooth circular cylinder of diameter 1 foot whose axis is horizontal. The chain is to be assumed to lie in a vertical plane perpendicular to the axis of the cylinder and its free ends to hang symmetrically one on each side of the cylinder. Investigate whether the chain remains in contact with the lowest generator of the cylinder.
A mass \(m\) is connected by an inelastic string to the end \(B\) of a uniform rod \(AB\) of mass \(M\). The system is at rest on a smooth horizontal table with the mass \(m\) at \(B\). The mass is projected along the table with velocity \(v\) perpendicular to \(AB\); find its velocity immediately after the string becomes taut. Prove that the loss of kinetic energy of the system due to the tightening of the string is \(\frac{1}{2}Mmv^2/(M+4m)\).
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 \[ \left(\frac{\partial x}{\partial y}\right)_z = \frac{\frac{\partial x}{\partial u}\frac{\partial z}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial z}{\partial u}}{\frac{\partial y}{\partial u}\frac{\partial z}{\partial v} - \frac{\partial y}{\partial v}\frac{\partial z}{\partial u}}. \]