Problems

Prove that the sum of the odd coefficients in the binomial expansion is equal to the sum of the even coefficients, and each is equal to \(2^{n-1}\), where \(n\) (a positive integer) is the index of the expansion.

Develope ab initio the principal properties of Determinants. Include in particular the proof of the theorem that, if \[ D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}, \quad \Delta = \begin{vmatrix} \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \end{vmatrix}, \text{ then} \] \[ D\Delta = \begin{vmatrix} a_1\alpha_1+b_1\beta_1+c_1\gamma_1 & a_1\alpha_2+b_1\beta_2+c_1\gamma_2 & a_1\alpha_3+b_1\beta_3+c_1\gamma_3 \\ a_2\alpha_1+b_2\beta_1+c_2\gamma_1 & a_2\alpha_2+b_2\beta_2+c_2\gamma_2 & a_2\alpha_3+b_2\beta_3+c_2\gamma_3 \\ a_3\alpha_1+b_3\beta_1+c_3\gamma_1 & a_3\alpha_2+b_3\beta_2+c_3\gamma_2 & a_3\alpha_3+b_3\beta_3+c_3\gamma_3 \end{vmatrix}. \]

In considering the size and speed of a merchant ship for a given service, the following assumptions are made:

1. [(i)] The receipts per voyage and the cost of casual labour and maintenance per voyage are proportional to the ``displacement.''
2. [(ii)] The cost of personnel per annum and the depreciation in value per annum are proportional to the ``displacement.''
3. [(iii)] The resistance to motion is proportional to the square of the speed and also to the area of the submerged surface.
4. [(iv)] The engine efficiency, and the interval in dock between successive voyages, are constant.

For similar and similarly loaded ships, express the annual profit in terms of the displacement \(\Delta\), the speed \(V\), and constants, and derive an equation to give the speed for maximum annual profit for any assigned displacement. Also show that, theoretically, the larger the size, the greater the profit per ton displacement without limit.

A car rests on four equal weightless wheels of diameter 3 feet, which can rotate about central bearings of diameter 2 inches. If the angle of friction between each wheel and its bearing be 18\(^\circ\), show that the car will not rest on a rough inclined plane, if the inclination of the plane to the horizontal be greater than 1\(^\circ\), approximately, assuming that a wheel and its bearing are in contact along a single generator.

Show that, if \(x\) and \(y\) are positive, \(m\) and \(n\) positive integers, and if the greatest term of the Binomial expansion of \((x+y)^m\) is the \(p\)th, and the greatest term of that of \((x+y)^n\) is the \(q\)th, then the greatest term of the expansion of \((x+y)^{m+n}\) is the \((p+q)\)th or the \((p+q-1)\)th.

Obtain the real solutions of the equations \[ x^3 + \frac{7}{3}xy^2 = y^3 + \frac{7}{3}yx^2 = 1 \] to three significant figures.

Starting from the definitions of sine and cosine by means of the infinite series, \[ \sin x = x - \frac{x^3}{3!} + \dots, \qquad \cos x = 1 - \frac{x^2}{2!} + \dots, \] give an outline, without detailed proof, of the steps by which the chief results of the trigonometry of a real angle can be established.

Simultaneous values of the speed and the acceleration are observed, during the run of a train, to be as follows:

Find by means of graphs the distance travelled in the first 5 minutes from rest.

A uniform heavy beam \(AB\) of weight \(3W\), loaded with equal weights \(W\) at \(A\) and a point of trisection \(D\), rests in a horizontal position on two supports at \(B\) and a point of trisection \(C\), where \(AC=CD=DB\). Sketch diagrams showing the distributions of shearing stress and bending moment along the beam.

Prove that, if \(S_r\) denotes \(1^r + 2^r + 3^r + \dots + n^r\), then \[ S_5 + S_7 = 2S_1^2. \quad [sic] \]