10273 problems found
A particle moves inside a fine smooth straight tube which is made to rotate about a point O of itself with constant angular velocity in a horizontal plane. Initially the particle is at relative rest at a distance \(b\) from O and it subsequently rebounds from a closed end of the tube at a distance \(a\) from O. If \(e\) is the coefficient of restitution, show that after \(n\) rebounds the least distance of the particle from O is \(\{a^2(1-e^{2n}) + b^2e^{2n}\}^{\frac{1}{2}}\).
Find the moment of inertia of a uniform thin spherical shell of mass \(m\) and radius \(a\) about a diameter. If the shell is smoothly pivoted at a point of itself and if the pivot cannot support a load greater than \(\frac{1}{2}mg\), show that the greatest horizontal impulse which can be applied to the shell along a diameter when the shell is hanging in equilibrium is \(m(5ga)^{\frac{1}{2}}\) if the shell is not to break away from the pivot. It may be assumed that the pivot can sustain the initial impulsive reaction.
If \(a, b, c\) are the roots of the equation \[ x^3 + px^2 + qx + r = 0, \] prove that \[ (a^2-bc)(b^2-ca)(c^2-ab) = rp^3 - q^3. \] If \(p, q, r\) are real, \(a\) is real and \(b, c\) are complex, prove that \(a\) is numerically greater or less than \(|b|\) according as \(rp^3-q^3\) is positive or negative.
In the Argand diagram a triangle ABC is inscribed in the circle \(|z|=1\), the vertices A, B, C corresponding to the complex numbers \(a, b, c\) respectively. Prove that the orthocentre H is given by \(z=a+b+c\). Verify that the circle \[ |2z - a - b - c| = 1 \] passes through the nine points from which it takes its name.
AC and BD are two skew lines in space. A plane meets AB at P, BC at Q, CD at R and DA at S. Prove that PQ and RS meet on AC, and that QR and PS meet on BD. P' is the harmonic conjugate of P with respect to A and B; Q' is the harmonic conjugate of Q with respect to B and C; R' is the harmonic conjugate of R with respect to C and D; and S' is the harmonic conjugate of S with respect to D and A. Prove that P', Q', R', S' lie in a plane.
A variable conic touches a fixed line \(l\) at the fixed point C and also passes through two fixed points A, B. P is the point of contact of the other tangent to the conic from a fixed point D lying on \(l\). Prove that, in general, the locus of P is a conic. Investigate any exceptional case. State the theorem obtained by taking \(l\) to be the line at infinity.
The equations of conics S, S' are \[ ax^2+2hxy+by^2=1, \quad a'x^2+2h'xy+b'y^2=1. \] Prove that the asymptotes of S are conjugate diameters of S' if and only if \[ ab' + a'b - 2hh' = 0. \] If this condition is not satisfied, show that the envelope of a chord of S whose extremities lie on conjugate diameters of S' is a conic which is similar and similarly situated to S'. What is the envelope when the condition is satisfied?
Obtain in its simplest form the derivative of \[ f(x) = \tfrac{1}{2}x + \sin x + \tfrac{1}{2}\sin 2x + \dots + \tfrac{1}{n}\sin nx + \frac{\cos(n+\tfrac{1}{2})x-k}{(2n+1)\sin\tfrac{1}{2}x}. \] Prove that, if \(k > 1\), the function \(f(x)\) attains its greatest value in the interval \(0 < x < 2\pi\) for the value \(x=\pi\). Prove that, if \(k < -1\), \(f(x)\) takes its least value in \(0 < x < 2\pi\) for \(x=\pi\). Deduce that the infinite series \[ \sin x + \tfrac{1}{2}\sin 2x + \tfrac{1}{3}\sin 3x + \dots \] is convergent for \(0 < x < 2\pi\), and find its sum.
A circle of radius \(a/n\) rolls without slipping on the inside of a fixed circle of radius \(a\), where \(n\) is a positive integer. A point \(P\) on the circumference of the moving circle traces out a curve \(S\). Prove that \(S\) can be represented by the equations \[ x = \frac{a}{n}\{(n-1)\cos\theta + \cos(n-1)\theta\}, \quad y = \frac{a}{n}\{(n-1)\sin\theta - \sin(n-1)\theta\}. \] Prove that the total length of S is \(8(n-1)a/n\).
Evaluate the integral \[ I = \int_{-a}^{a} \frac{1-k\cos\theta}{1-2k\cos\theta+k^2} d\theta, \] where \(0 < k < 1, 0 < a < \pi\). Prove that