The middle point of a rod \(AB\) moves uniformly with given velocity in a circle, centre \(O\), and the end \(A\) moves in a straight line through \(O\) in the plane of the circle; shew graphically the velocity of \(B\).
Solve the equations \[ x (x-a) = yz, \quad y(y-b) = zx, \quad z(z-c) = xy. \]
A uniform solid hemisphere is placed with its curved surface in contact with a rough inclined plane. Shew that for equilibrium to be possible, the inclination of the plane to the horizontal must be less than \(\sin^{-1}(3/8)\). Shew, also, that there are two positions of equilibrium if the inclination of the plane lies between \(\tan^{-1}(3/8)\) and \(\sin^{-1}(3/8)\). Discuss the stability of the equilibrium, and shew that when one position only exists it is stable.
From the angular points \(A, B, C\) of an equilateral triangle, whose side is 3 inches, lines \(AP, BQ, CR\) are drawn, of lengths 1, 2, 4 inches respectively, perpendicular to the plane \(ABC\) and on the same side of it. Find where the plane \(PQR\) meets \(CB\) and \(BA\), and prove that the angle between the planes \(ABC\) and \(PQR\) is a little greater than \(45^\circ\).
Prove that the equation of a conic which touches the axis of \(x\) at the origin is of the form \[ ax^2+by^2+2hxy=2y; \] and that, near the origin, the curve lies on the positive or negative side of the axis of \(x\) according as \(a\) is positive or negative. \par Obtain the conditions that two conics, \[ ax^2+by^2+2hxy=2y, \quad a'x^2+b'y^2+2h'xy=2y, \] (i) should intersect in two real points, (ii) should have two real common tangents other than the axis of \(x\); and shew that the conditions are identical when \(a\) and \(a'\) are of the same sign.
Find the relation between the ``Watt'' and the ``Horse-power,'' given that 1 inch = 2.54 cms., and that 1 lb. = 453.6 grammes. \par An electric motor costs \pounds100 and runs for 1000 hours per annum: interest on the capital cost, depreciation and upkeep amount to 15\% of the cost per annum. If the average load be 10 H.P., if the average motor efficiency be 80\% and if electric energy costs 2d. per kilowatt-hour, find the total cost per horse-power-hour.
Prove that, if \[ u_2 = u_1^2 - 1, \quad u_1u_3 = u_2^2 - 1, \quad u_2u_4 = u_3^2 - 1, \quad u_3u_5 = u_4^2 - 1, \dots \] then \[ u_1+u_3 = u_1u_2, \quad u_2+u_4 = u_1u_3, \quad u_3+u_5 = u_1u_4, \quad u_4+u_6 = u_1u_5, \dots. \]
\(ABCD\) is a quadrilateral of smoothly jointed rods, having the angles at \(A\) and \(B\) equal to \(60^\circ\), the angle \(ACB\) equal to \(90^\circ\), and \(AB\) parallel to \(DC\). The framework is kept in shape by a rod joining \(A\) and \(C\). It is supported at \(A\) and \(B\) with \(AB\) horizontal, having loads \(X\) and \(Y\) placed at \(C\) and \(D\). Draw a diagram giving the stresses in the rods, and shew by use of it that the stress in the rod \(AC\) is \(\frac{1}{2}(X-Y)\).
State any rules you know for determining whether a number is divisible by 2, 3, 4, 5, 8, 9, and 11. \par Find a number of three digits, not necessarily different, such that (i) all its digits are prime, (ii) all numbers that can be formed by taking two of its digits are prime, (iii) all numbers that can be formed by taking all three of its digits are prime.
Prove that a function which vanishes with \(x\), is continuous, and has a differential coefficient positive for all positive values of \(x\), is itself positive for all positive values of \(x\). \par Prove that each of the functions \[ 1-\cos x, \quad x-\sin x, \quad \tfrac{1}{2}x^2-1+\cos x, \quad \tfrac{1}{6}x^3-x+\sin x, \quad \tfrac{1}{24}x^4-\tfrac{1}{2}x^2+1-\cos x, \] is positive for all positive values of \(x\). Hence obtain expansions of \(\cos x\) and \(\sin x\) in powers of \(x\) valid for all real values of \(x\).