Prove that, if the sum of the resolutes in a given direction of the external forces on any number of particles is zero, the sum of the momenta of the particles in that direction is constant. Two particles \(A\) of mass \(m\) and \(B\) of mass \(2m\) are connected by a light inextensible string of length \(a\) which is stretched at full length perpendicular to the edge of a smooth horizontal table. The particle \(B\) is drawn just over the edge of the table and is then released from rest in this position. Describe the nature of the subsequent motion and shew that after \(A\) leaves the table the centre of mass of the two particles describes a parabola of latus rectum \(8a/27\).
Prove that \(v^2/\rho\) is the acceleration along the normal inwards of a point moving with velocity \(v\) in a curve, where \(\rho\) is the radius of curvature at the point. A circular cylinder of radius \(a\) is placed in a fixed position with its axis horizontal on a smooth horizontal plane. A perfectly elastic particle is placed on the highest generator of the cylinder and being slightly displaced slides down the cylinder. Prove that the distance between consecutive points at which it strikes the horizontal plane is \(40a\sqrt{2}/27\).
Define simple harmonic motion and shew how to find the period of oscillation when the acceleration at any distance from the centre is given. A light elastic string of unstretched length \(l_0\) is suspended from a fixed point and has a mass \(m\) attached to its other end. In equilibrium its stretched length is \(l\). A blow \(B\) is applied vertically upwards to \(m\). Find \(B\) so that the string just becomes slack in the subsequent motion. If the blow applied had been twice this value of \(B\), shew that \(m\) would have risen to a height \(l-l_0+3B^2/2gm^2\) above its equilibrium position.
Shew that the value of a determinant is zero if two of its rows or two of its columns are identical. Deduce that \[ \begin{vmatrix} a_1+pb_1+qc_1, & b_1, & c_1 \\ a_2+pb_2+qc_2, & b_2, & c_2 \\ a_3+pb_3+qc_3, & b_3, & c_3 \end{vmatrix} \] is independent of \(p\) and \(q\). Use the equations \(x^3+px^2+qx+r=0, x^2+bx+c=0\) to express \[ (r+bc-pc)^2 - (cq-c^2-br)(bp-b^2-q+c) \] as a determinant of the fifth order.
If \(\dbinom{n}{r}\) denotes the number of combinations of \(n\) things taken \(r\) at a time, shew that \[ \binom{n}{r} = \frac{n}{r}\binom{n-1}{r-1} \text{ and derive an interpretation for } \binom{n}{0}. \] Prove that \[ \binom{m+n}{r} = \binom{m}{r}\binom{n}{0} + \binom{m}{r-1}\binom{n}{1} + \binom{m}{r-2}\binom{n}{2} + \dots + \binom{m}{1}\binom{n}{r-1} + \binom{m}{0}\binom{n}{r}. \] By means of the identity \(1-x^6 = (1-x^3)(1+x+x^2)\), or otherwise, prove that \[ \binom{6m}{n}\binom{6m}{n} + \binom{6m-1}{n}\binom{6m-2}{n} + \binom{6m-2}{n}\binom{6m-4}{n} + \dots + \binom{3m}{n}\binom{0}{n} \] \[ = \binom{n+6m-1}{6m-1}\binom{n}{0} - \binom{n+6m-4}{6m-3}\binom{n}{1} + \binom{n+6m-7}{6m-6}\binom{n}{2} - \dots + (-1)^m \binom{n+1}{0}\binom{n}{2m}. \]
Obtain the cube roots of unity and establish their principal properties. Express in terms of the exponential function the sums of the infinite series
If \(\phi(x)\) is a polynomial of degree not greater than that of a polynomial \(f(x)\), shew that \[ \frac{\phi(x)}{(x-a)f(x)} \equiv \frac{\phi(a)}{(x-a)f'(a)} + \frac{\text{a polynomial in }x}{f(x)}, \] provided \(f(a)\neq 0\). Discuss the case \(f(a)=0\). Expand \(\dfrac{2x^8-5x^4+2x^3+6x^2-2}{(x+2)(x^2-1)^3}\) in a series of ascending powers of \(x\), stating carefully the general term.
Prove Taylor's theorem for a function \(f(x)\), in the range \(a \le x \le b\), stating the necessary restrictions on the behaviour of \(f(x)\) and its derivatives. Find the first four terms in the expansion of \(\log_e(1+\sin^2 x)\) in increasing powers of \(x\), and also the first three terms of the expansion of \(x\) in increasing powers of this function.
Explain the application of the Calculus to the discussion of inequalities, giving simple illustrations. Hence or otherwise prove that if \(0<\theta<1\), \[ \theta + \frac{\theta^3}{3} < \tan\theta < \theta + \frac{2\theta^3}{3}. \]
Trace the curve \(r\cos\theta+a\cos 2\theta = 0\). Shew that the area of the loop is \(a^2(2-\frac{\pi}{2})\), and that the area enclosed between the curve and its asymptote is \(a^2(2+\frac{\pi}{2})\).