Problems

Find

1. [(i)] \(\displaystyle\int \cot^3 x \sin^5 x \, dx\),
2. [(ii)] \(\displaystyle\int \frac{dx}{x^4+x}\),
3. [(iii)] \(\displaystyle\int_0^\infty \frac{dx}{(x+1)^2\sqrt{x^2+1}}\).

Prove that the joins of mid-points of opposite edges of a tetrahedron meet in a point. Shew that, if the three joins are mutually perpendicular, then the four faces of the tetrahedron are congruent to each other, and conversely.

Obtain an expression for the potential energy stored in a stretched elastic string. A catapult consists of a light elastic string, of modulus \(\lambda\) and unstretched length \(2l_0\), connected to fixed points \(A\) and \(B\) at the same level. At the centre of the string is attached a small socket of mass \(m_1\) for holding the projectile. The socket is pulled down along a line bisecting \(AB\) at right angles until the string has a total length \(2l_1\) and projects vertically a stone of mass \(m_2\). Shew that, apart from air resistance, the stone would rise to a height \[ \frac{\lambda(l_1-l_0)^2}{(m_1+m_2)gl_1} \] above the point of release, assuming that it leaves the socket.

State the energy test of stability of equilibrium. A uniform rod of length \(l\) is attached by small rings at its ends to a smooth wire in the form of a parabola of latus rectum \(a\) placed with its axis vertical and vertex downwards. Prove that, if \(a

Find the solution of the equation \[ (x+2)^2 y'' - 2(x+2) y' + 2y = 0 \] which represents a curve touching the parabola \(y = x^2 + 1\) at the point \((1, 2)\).

If the normals at four points of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) are concurrent, and if two of the points lie on the line \[ lx + my = 1, \] prove that the other two lie on the line \[ \frac{x}{a^2l} + \frac{y}{b^2m} + 1 = 0. \] Hence, or otherwise, shew that if the feet of two of the normals from a point \(P\) to the ellipse are coincident, then the locus of the middle point of the chord joining the feet of the other two normals is \[ \left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right)^2 \left(\frac{a^6}{x^2} + \frac{b^6}{y^2}\right) = (a^2-b^2)^2. \]

A uniform thin chain, 20 feet long and weighing 10 lb., rests in a small space on the ground. One end of it is given a constant vertical acceleration of 5 feet per second per second by a force applied to that end. Determine the work which has been done by this force when the whole chain is just clear of the ground.

A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\) is such that it will break when stretched to a tension \(T_0\), its length then being more than doubled and less than trebled. A particle of mass \(m\) is fastened to the middle point of the string which is stretched between two fixed points at a distance \(2a\) apart on a smooth horizontal plane. Prove that the least impulse applied to the particle in the direction of the string that will cause the string to break is \((T_0 - \lambda)\sqrt{(ma/\lambda)}\). Point out in what way the limits of extension corresponding to \(T_0\) affect the problem.

Find the coordinates of the node of the curve \[ (x+y+1)y + (x+y+1)^2 + y^3 = 0, \] and the area of the loop at the node.

Shew that in general two triangles can be inscribed in the hyperbola \(xy=k^2\) with sides parallel to the lines \[ y+lx=0, \quad y+mx=0, \quad y+nx=0. \] State under what conditions the triangles will be real. Shew that if \(l, m, n\) vary, the area of either triangle is proportional to \[ \frac{(m-n)(n-l)(l-m)}{lmn}. \]