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SECOND PROFESSIONAL EXAMINATION,
[Full credit will be given for answering FIVE-SIXTHS of this Paper.]
If af+bg+ch = 0, prove that there is an infinity of solutions, and find them in a symmetrical form.
2. Borings are made at three points P, Q, R in a horizontal plane, and rock is found at depths 35, 57, 43 feet, respectively. The lengths PQ, QR are 100 and 150 feet, and the angle PQR is a right angle. Assuming the surface of the rock to be a plane, find its inclination to the horizon to the nearest minute.
3. Find the area of a spherical equilateral triangle whose side is 50°.
4. A circle touches a parabola at two points: prove that the chord of contact must be perpendicular to the axis of the parabola.
Prove also that the tangent to the circle from any point on the parabola is equal to the perpendicular from the same point on the chord of contact.
5. Find the shortest distance from the point (c, 0) to the ellipse
showing that there is a distinction, according as the numerical
GEOMETRY AND INTEGRAL CALCULUS.
7. If two circles are so placed that a triangle can be drawn inscribed in one and circumscribed to the other, find the relation connecting their radii with the distance between their centres.
8. Given a straight line and two points not all in the same plane, show how to find the point on the line such that the sum of its distances from the points is a minimum.
9. If y = cos3x, find
Expand y in powers of x up to 24, and write down an expression for the remainder of the series.
a3 cos 30. Find
10. A curve is given by the equation 3 its r and p equation, and also that of its first pedal. 11. Find a formula of reduction for fsin" Ode. Find
when n is odd, and when n is even.
12. Find the whole area of the curve
x2y2 = (a − x)(x – b),
a, b both being positive.
1. A solid hemisphere rests with its convex surface in contact with a perfectly rough sphere. Find the lowest position of the point of contact in which equilibrium is possible.
2. A solid cone is supported with its axis horizontal by a string which, after passing round a smooth pulley, is attached to the vertex, and to a point on the perimeter of the base of the cone. Find the length of the string in terms of r the radius of the base and h the height of the cone.
3. A rough hollow sphere of radius r is spinning with angular velocity w round a vertical axis through its centre. Deduce an equation for the determination of the highest position in which a particle will rest in contact with its concave surface, at which the coefficient of friction is μ.
4. A weight is suspended by two strings of the same length, making an angle with each other; if one of the strings be cut, find the sudden increase in the tension of the other.
5. Two balls, whose masses are as 3 to 1, moving in opposite directions collide. Find the ratio of their velocities if the first ball be brought to rest, the coefficient of elasticity being.
6. A hollow right cone, vertex upwards, is just filled with liquid of specific gravity σ, and set with its base horizontal into a vessel containing liquid of specific gravity p. Find the resultant vertical pressure on the curved surface if the outer liquid just covers the cone. Examine whether it is possible by changing the depth of the outer liquid to produce equilibrium between the inner and outer fluid pressures on the curved surface.
7. A cone of vertical angle 2a and height h floats in water with its vertex fixed in the surface and its axis horizontal.
Find what weight must be attached to a point on the circumference of the base to sink the cone until the surface of the water is a tangent plane, and examine whether the equilibrium is stable.
8. What is the equation of time?
Explain, by means of diagrams, the causes to which it is due, and point out how the amount due to each cause varies throughout the year.
9. How does the achromatism of the object-glass differ from that of the eye-piece in an astronomical telescope? Find the conditions for the achromatism of each.
10. Find an expression for the magnifying power of a compound microscope.
1. Set up the two lenses so that a parallel beam entering the combination will emerge parallel.
2. Find the humidity of the air, from observation of the wet and dry bulb thermometer.
3. Find how the given cells should be arranged to send the greatest current through the galvanometer as arranged.
[Special stress will be laid upon the written record of your work, and your attention is directed to the following points :
(a) Give a concise account of all the steps of the processes you employ, and of all the tests you use in searching for the different substances.
(b) If you find a metal capable of forming two series of compounds, ascertain, if possible, to which of these series the metal present in the substance you are examining belongs.
(c) In testing a solid, dry way tests, in addition to wet way tests, must be employed.
(d) In testing a solution, dry way tests should be employed in all cases where it is advisable to do so-in addition to liquid tests.
(e) Use confirmatory tests where it is possible to do so.
(f) At the end of your paper, give a statement of the constituents found in each solid or solution given you for examination.]
1. Detect two bases and two acids in the solid marked 1. 2. Detect acid and base in the solid marked 2.
1. In levelling across a narrow ravine, where only the level of the top of banks at each side is required, but it is impossible to avoid sights of very unequal lengths, how may error of collimation be avoided?
2. Find the formula for the time taken in discharging the contents of a given vertical cylindrical vessel through a given orifice in its bottom.
3. A main pipe, mile long, 3 square feet area of crosssection, is running full of water, the velocity being 12 feet per second. By shutting a valve at the lower end, the flow is stopped in 20 seconds, the rate of diminution of velocity being kept constant. Find the pressure on the valve (in addition to statical head) due to the inertia of the water in the main.
4. Find the formula for discharge of a V notch.
5. A cutting of uniform depth H is to be formed with retaining walls for part of its depth h, the remainder, above the walls, to be sloped at sides s horizontal to 1 vertical. The thickness t of the walls is a given proportion, say λ, of their height h. Given that the cost of earthwork is e per cubic yard, of the masonry is p, and the depth H, of the cutting find the most economical height for the retaining walls.