Problems

Prove that, if \(\alpha + \beta + \gamma = 360^\circ\), \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma - 2\cos\alpha\cos\beta\cos\gamma = 1. \] If the cosines of the angles of a plane quadrilateral are \(c_1, c_2, c_3, c_4\), prove that \[ \Sigma c_1^4 - 2\Sigma c_1^2 c_2^2 + 4 c_1^2 c_2^2 c_3^2 - 4c_1c_2c_3c_4 (\Sigma c_1^2 - 2) = 0. \]

Find \[ \lim_{x\to 0} \frac{(1+x)^{1/x}-e}{x}. \]

A solid cylinder of weight \(w\) and of radius \(R\) rests with its axis vertical on a rough horizontal plane. If the coefficient of friction between the surfaces in contact be \(\mu\), shew that the couple required to rotate the cylinder about its axis is \(\frac{2}{3}\mu w R\), assuming that the normal pressure on the base of the cylinder is uniformly distributed over the area of the base. Find also what couple will be required if it be assumed that the normal pressure per unit area of the base varies inversely as the distance from the axis of the cylinder.

Prove that two conics which intersect in four distinct points have one and only one common self-polar triangle. Two parabolas touch the sides of a triangle \(ABC\) and intersect one another in \(P, Q, R, S\). Prove that the line joining any two of the points \(P, Q, R, S\) passes through one of the vertices of the triangle formed by the lines through the vertices of \(ABC\) parallel to the opposite sides.

Prove that \[ \cos\alpha + \cos(\alpha+\beta) + \cos(\alpha+2\beta) + \dots + \cos\{\alpha+(n-1)\beta\} = \cos\left(\alpha + \frac{n-1}{2}\beta\right) \sin\frac{n\beta}{2} \text{cosec}\frac{\beta}{2}. \] Shew that, if \(\alpha = \frac{\pi}{17}\), then \[ 16 \cos3\alpha \cos5\alpha \cos7\alpha \cos11\alpha = -1. \]

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

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.

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.

Find

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

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.