Section 4. 1. Find the sum (S) of n terms of the series whose first term is a and common difference b, and apply it to find the sum of 13 terms of the series whose first term is 4 and common difference lì. 2. Show that in a series in geometrical progression if a be the first term, I the last term, and S the sum of the terms, then a 1 a S= In3. Given P the pli term, and Q the quh term of an arithmetical progression, find N the nth term. 4. Having given the first and the last terms and the sum of the terms in a geometrical progression, investigate an expression for the number of terms. TRIGONOMETRY. Section 1. Prove the following theorems : 1 Cos. A. 1. Sin. 'A + Cos. 'A = 1., Cot. A = Tan. A. Cosec. A = 1 2. Sin. (v + 0) Sin. 6., Sin. 30° Tan. 30° 2's 3 1 1 Sin. A Sin. 18° = 5 (N5–1). 3. Sin. (A + B) = Sin. A. Cos. B + Sin. B Cos. A. Sin. 3 A = 3 Sin. A - 4 Sin. 3A. 1 1 2 + . 2 . A ) (– B). 08. A a Section 2. Prove the following relations of the sides and angles of plane triangles :1. In a right angled triangle c = v(a-b)(a + b); and in any other plane ; -b с atb 2 3. Sin. A = NS (8 - a) (S – b) (S. - c). triangle: ī 2. Tan. (A – B) Cot.com bc Section 3, 1. Explain what is meant by the base of a system of logarithms, state what is the logarithm of 10,000 in the system whose base is 10, and show generally that log. M ~ N = log. M + log. N. M and Jog. N log. M - log. N. 2. What does the characteristic of a logarithm denote when positive; and what, when negative? How may a negative logarithm be changed into an equivalent positive one having a negative characteristic ? 3. Having given two sides of a triangle and the included angle, show how the third side may be found, by logarithmic calculation. 4. Investigate a formula for determining the numerical value of a logarithm in a converging series. Section 4. 1. To find the height of an object, the foot of which is inaccessible. 2. To find the distance from one another of two distant and inaccessible objects. 3. The distances from one another of three abjects in the same plane being known, and the angles which these distances subtend from a third abject in that plane, being observed, to determine the distance of that third object from either of them. a = Tan. . C. HIGHER BRANCHES OF MATHEMATICS, Section 1, Cos. c = Cos. a Cos. b + Sin, a Sin. 6 Cos, C. -1 + a 3. Prove Demoivre's formula in the case in which the index is a positive integer, and apply it to determine the values of Sin, n o and Cos. n o in series ascending by powers of Sin 6 and Cos). 4. Show that in any spherical triangle 1 Cos. # (@-6) 1 . Cot. Section 2, 1'- 3x + 2.4 the expansion of in a series ascending by powers of t. 1++37 3. Prove the Binomial Theorem in the case in which the index is a positive integer. 4. Expand as in a series ascending by powers of z; and investigate a general expression for the logarithm of any number n to the base a. Section 3. Ņ1 + 2*, Sin. -'x, Sin. 3 x Cos. 2 x. 4. Show that when a function of x is a maximum or a minimum, the value of its first differential co-efficient is zero; and determine the sides of a rectangular enclosure containing a given area (A), and separated into smaller enclosures by n equidistant divisions parallel to one of its sides ; so that the whole length of fencing may be a minimum. Section 4. 1. Trace the line whose equation is y = - ax + 6, and find the inclination 1 of this line to that, whose equation is y= * + a 2. Investigate the equation to a Parabola, and also that to a tangent to it, at any point. 3. Trace the curve whose polar equation is r = a (1 + Cos 6). Section 5. 1. Investigate a general equation to the tangent to a curve, and a general expression for its substangent. 2. Find the differential co-efficient of a solid of revolution, and apply the expression to determine the volume of a sphere. 3. Find a general expression for the radius of curvature of a plane curve, and apply it to determine the radius of curvature of a parabola. Section 6. Perform the following integrations :1. d (a x" + b) dx, (a + x) Cos. I "6 Statiu ) S cos. Sa S Set 2. 2x - 5 Tan. Ida, (2 + 1) (x + 3) dr do Naš + ** Tan. 1 3. Find the area of the segment of a circle. 4. Find thc volume and the surface of a groin whose horizontal sections parallel to its base are squares, and whose vertical section is a quadrant of a circle. Section 7. 1. Investigate an expression for the number of combinations which can be formed with n things taken r together. 2. Show that in any equation the co-efficient of the second term is equal to the sum of the roots with their signs changed. 3. Show that in a series of converging fractions the terms are alternately less and greater than the fraction toward which they converge. 4. State Des Cartes' rule of signs, and prove it. MENSURATION. ( The questions marked with an asterisk may be solved either by calculation or cove struction.) Section 1. 1. Explain what is meant by a unit of surface, and show that the number of units of surface in a rectangle is equal to the product of the numbers of units of length in its sides. 2. Investigate a rule for determining the area of trapezoid. 3. Find the area of a rectangle whose sides are 9 ft. 7 in. and 8 ft. 5 in. by the rule of duodecimals, and explain fully the different denominations in the Section 2. 1. Explain the construction and use of the diagonal scale. 2. *The height of a wall is 40 feet, at what distance from the base of it must the foot of a ladder 50 feet long be placed so that it may just reach the top of it. 3. I took a point in the continuation of the diagonal of a square, and found its distances from the nearest corners to be 40 and 60; required the side of answer. the square. 4. *The distances from one another of three objects A, B, C, are A B = 12 miles, BC = 7.2 miles, and AC = 8 miles. From a station D, I took the angles B D C = 25°, and C DA = 19o. Required the distance of the station D from C. Section 3. 1. Describe Gunter's Chain. 2. Required the rent at 30s. per acre of a triangular field whose sides are respectively 469, 427-8 and 512-8 links. Required the number of square feet in the surface of a circular arch, the length of the exterior arc of which is 20 feet, and its radius 15 feet, and the radius of the lesser arc 12 feet. 4. Show that the deviation to be made in levelling, for the curvature of the earth, is nearly 8 inches per mile. Section 4. 1. A box open at the top is 2 feet long, 1; feet broad, and 3 feet deep; how many cubic feet of deal 2 inches thick will it require to make it. 2. Investigate a rule for determining the solidity of a pyramid having a triangular base. 3. Investigate a rule for determining the solidity of a rectangular prismoid. 4. Investigate a rule for determining the surface of a sphere. a GEOMETRY. Section 1. 1. Upon the same base, and upon the same side of it, there cannot be two triangles having their two sides terminated at one extremity of the base equal, and likewise their two sides terminated at the other extremity. 2. To a given straight line to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. Section 2. 1. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half, and the part produced. 2. If a point be taken within a circle from which there fall more than two straight lines to the circumference, that point is the centre of the circle. 3. If two straight lines cut one another in a circle, but not at right angles, one of which passes through the centre, and the other does not, the rectangle contained by the segments of one of them shall be equal to the rectangle contained by the segments of the other. Section 3. 1. To describe a circle about a given triangle. 2. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other two sides of the triangle proportionally; and conversely , if two sides of a triangle be cut proportionally, the straight line which joins their points of section shall be parallel to the third side of the triangle. 3. Similar triangles are to one another in the duplicate ratio of their homologous sides. Section 4. 1. If one angle of a triangle equal the sum of the other two, the greatest side is double the distance of its middle point from the opposite angle. 2. The area of a rhombus is equal to half the rectangle contained by its diagonals. 3. If two chords in a circle intersect each other at right angles, the sum of the squares described upon the four seginents is equal to the square described upon the diumeter of the circle. a INDUSTRIAL MECHANICS. Section 1. 1. How many bushels of coals must be expended in a day of 24 hours, in raising 150 cubic feet of water per minute, from a depth of 100 fathoms—the duty of the engine being 60 millions ? 2. The piston of an engine is 3 feet in diameter, the length of the stroke is 6 feet, and 6 strokes are made per minute, what must be the resistance upon the piston that the engine may yield there 75 horse power ? 3. "There is a water-wheel which is worked by a stream whose section is 2 feet by 3, and its mean velocity 2 feet per second. The fall is 15 feet, and the modulus of the wheel 6; it is used to raise water from the upper level of the stream to a height of 40 feet above it. How many cubic feet will it raise per minute ? 4. Steam is admitted into the cylinder of an engine whose stroke is 10 feet, at a pressure of 34 lbs. per square inch, and cut off at one-fourth the stroke. a How many units of work will it.do per stroke on each square inch of the piston ? Section 2. 1. A rod 16 feet long, is of uniform thickness, and weighs 13 lbs. ; a weight of 25 lbs. is suspended from one extremity, and one of 9 lbs. from the other extremity. On what point will it balance ? 2. A block of granite 55 feet long, 2 feet wide, and I foot thick, cach cubic foot of which weighs 164 lbs. ; is supported in an inclined position, resting on its end, by means of a rope 60 feet long fastened to a point distant 3 feet from its top, and fixed to the ground at a distance of 25 feet from the point on which it rests. What is the tension on the rope ? 3. Show that the centre of gravity of a triangle is situated in the line drawn from the vertical angle to the bisection of the base, at a distance from the base equal to one-third of that line. 4. The traction of a waggon, weighing, gross, W lbs. up a hill, is P lbs.; and the traction down the hill p lbs.; what is the inclination of the hill, and what the proportion of the resistance of the road to the gross load ? Section 3. 1. What is the pressure upon the circular plug of a water-main 2 inches in diameter, situated 100 feet beneath the surface of the reservoir which supplies the main ? 2. Investigate a rule for determining the pressure of a fluid upon a vertical rectangular plane, and calculate the total pressure upon a food-gate 36 feet high, and 12 feet wide, when the water reaches to its surface, and the pressure upon the lower half of the gate ? 3. Give a method of determining the specific gravity of solid body and illustrate it by an example. 4. Show how the stability of the wall of a reservoir may be determined. Section 4. 1. A body falls freely by the force of gravity ; find the space fallen through in a given time. 2. Show that the number of units of work which a body of a given weight, moving with a given velocity, is capable of yielding is represented by half its vis viva. 3. The traction of the engine upon a train, whose gross weight is W lbs., 1 is P. Ibs., and the resistance of the rail is th part of the gross weight; investigate an expression for the space which will be traversed in the first t seconds of the motion of the train. а m |