Problems

Find the centre of mass of a thin uniform wire of length \(l\) bent into an arc of a circle of radius \(a\). \(ABC\) is a uniform semi-circular wire, of weight \(w\) per unit length, and rests in a vertical plane with \(AC\) horizontal and \(B\) in contact with the ground. Find expressions for the bending moment and the shearing force at any point of the wire.

Shew that if a focus be taken as pole, then the polar equation to a conic may be written in the form \[ \frac{l}{r} = 1 + e \cos \theta. \] A circle through the focus meets the conic in four points whose distances from the focus are \(r_1, r_2, r_3, r_4\). Prove that \[ \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_4} = \frac{2}{l}. \]

If the roots \(x_1, x_2, x_3\) of the equation \[ x^3 = 3p^2x + q \] are all real and distinct, prove that \(4p^6 > q^2\). Obtain the equation whose roots are \(x_1^2 - r, x_2^2 - r, x_3^2 - r\), where \(3r = x_1^2 + x_2^2 + x_3^2\). Given \(x_1^2 < x_2^2 < x_3^2\), prove that \[ x_2^2 - x_1^2 < x_3^2 - x_2^2 \] if and only if \(2p^6 < q^2\).

Prove that if \[ \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} = 0, \] there exist three numbers \(x, y, z\), not all 0, satisfying simultaneously the three equations \begin{align*} ax + by + cz &= 0, \\ a'x + b'y + c'z &= 0, \\ a''x + b''y + c''z &= 0. \end{align*} Hence (or otherwise) prove that if \[ |a| > |b|+|c|, \quad |b'| > |c'|+|a'|, \quad |c''| > |a''|+|b''|, \] the above determinant cannot vanish. [\(|a|\) denotes the absolute value of \(a\).]

A projectile is fired in a fixed vertical plane with maximum velocity \(u\). Shew that all points which it can reach lie within or on a parabola, which is the envelope of those trajectories for which the velocity of projection is \(u\). Obtain its equation and shew that the point of projection is a focus of the parabola. \(A\) and \(B\) are two similar guns each capable of firing a shell with muzzle velocity 1000 feet per second. \(A\) is at the top of a tower 100 feet high and \(B\) is on the ground. Find the area of ground in which \(B\) may be placed so that \(B\) may be hit by \(A\) but \(A\) may not be hit by \(B\).

If \(\alpha, \beta, \gamma\) are the eccentric angles of three points \(P, Q, R\) on an ellipse, the normals at which are concurrent, prove that \[ \sin(\beta+\gamma) + \sin(\gamma+\alpha) + \sin(\alpha+\beta) = 0. \] \(A\) is a vertex of the ellipse, and \(P', Q', R'\) are points on the ellipse such that \(AP'\) is parallel to \(QR\), \(AQ'\) parallel to \(RP\), \(AR'\) parallel to \(PQ\). Prove that the centre of gravity of the triangle \(P'Q'R'\) lies on the major axis.

A card is drawn at random from an ordinary pack and is then replaced. A second card is then drawn at random and afterwards replaced, then a third, and so on. Prove that the chances in favour of all the four suits having turned up during the first seven draws are very slightly better than 21 out of 41.

Perform the following integrations: \[ \int \frac{e^{\sin^{-1} x}}{\sqrt{1-x^2}} dx, \quad \int \sqrt{\frac{e^x+a}{e^x-a}} dx, \quad \int \cosh mx \sin nx dx. \]

A motor car of mass 1 ton exerts a constant force of 100 lb. weight and has a maximum speed of 50 miles per hour on the level. Assuming that the frictional resistances are proportional to the square of the velocity, find the distance required for the car to accelerate from 10 to 20 miles per hour up an incline of 1 in 100.

Prove either of the two following theorems and deduce the other:

1. [(i)] The circumcircle of a triangle circumscribed to a parabola passes through the focus.
2. [(ii)] If \(ABC\) is a triangle circumscribed to a parabola with focus \(S\), then the perpendiculars at \(A, B, C\) to \(SA, SB, SC\) respectively are concurrent.