\(ABC\) is a triangle and \(AD\) the perpendicular on \(BC\). Obtain a formula for \(\cos A\) in terms of the lengths \(AD, BD, AC\).
Prove that, if the coefficients in the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] are real, and \(a, h, b\) are not all zero, the real part of the locus represented by the equation is either (1) an ellipse, (2) a hyperbola, (3) a parabola, (4) two intersecting straight lines, (5) two parallel straight lines, (6) one straight line (counted doubly), (7) a point, or (8) non-existent; and state the conditions in each case. Determine the different types of locus represented by \[ 2x^2 + 2\lambda xy + \lambda y^2 + \lambda(\lambda^2-1) = 0, \] as \(\lambda\) changes from a large negative to a large positive value.
A conical buoy 4 ft. high with a base 3 ft. in diameter floats with its axis vertical and point downwards in a smooth sea. The buoy weighs 300 lbs., and sea water weighs 64 lbs. per cu. ft. If the buoy be slightly depressed, find the time of a small vertical oscillation.
A smoothly jointed framework of light rods is loaded at the joints and supported as shown in the figure below. Give a diagram determining the stresses in the various members, and find the value of \(\alpha\) so that the stress in the rod \(PQ\) shall vanish. [A diagram of a loaded framework is shown. A horizontal rod PQ has a joint below its midpoint from which a weight 2W is suspended. A weight of 4W is suspended from joint Q. The framework is supported by two vertical upward forces. Various angles are given as 30\(^\circ\), 90\(^\circ\), and \(\alpha\).]
Four real or complex numbers (other than zero) are such that their squares are the same numbers in the same or a different order; prove that each number is a root of unity.
A triangle whose angles are 47\(^\circ\), 71\(^\circ\), and 62\(^\circ\) is inscribed in a circle of radius 4 inches. Calculate the area of the triangle, and the area of the segment of the circle cut off by the longest side.
Discuss the chief properties of a system of confocal conics. Deduce properties of a system of coaxal circles, or conversely. Shew that the envelope of the polar of a fixed point with regard to a system of confocal conics is a conic; and indicate five or more tangents of this conic.
The pendulum of an electric clock terminates in an electro-magnetic pole which swings in a circular arc of 30 centimetres radius. Another electro-magnetic pole projects from the base of the clock to within 1 cm. of the centre of the path of the first pole. In every 10th oscillation, the magnets are simultaneously excited to a pole strength of 10 \textsc{cgs}. units from the moment the pendulum makes an angle 5\(^\circ\) with the vertical until it is vertical. If it be assumed that while the magnets are excited they retain a constant pole strength, that at other times they are completely de-magnetized, and that all work done by the attraction of the magnets is expended in maintaining the motion, find the average energy lost per oscillation. Criticise the assumptions made above.
A shot is fired through three screens placed at equal distances 200 feet apart and the times taken to pass between the first pair and the second pair are observed to be \(\cdot\)2 sec. and \(\cdot\)21 sec. Show that the retardation, assumed to be uniform, is 232 feet per sec. per sec. If the error in reading the time intervals may be as much as 0\(\cdot\)5 per cent. for each interval, show that the actual retardation may be as small as 185 feet per sec. per sec. approximately.
Prove that, if any two of \[ \sin(B+C) + \sin(C+A) + \sin(A+B) \] and the three similar functions obtained by permuting \(A, B, C, D\) are equal, then all four are equal, provided that no two of \(A, B, C, D\) differ by a multiple of \(\pi\).