SECTION V. 1. Find the discount upon a given sum, which is due at a given period. What party gains by the usual way of reckoning interest for the given time, instead of the correct discount? 2. Find the amount of a given sum for a given number of years at compound interest. A merchant borrows £P for n years at r per cent. interest, and by trading makes it produce r+s per cent. What will be the amount of his gain upon this money, accumulated at the end of n years? GEOMETRY. SECTION I. 1. Define a circle, a triangle, an isosceles triangle, and and an equilateral triangle. Prove that the angles, which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles. 2. The angles at the base of an isosceles triangle are equal to each other. State the axiom which you assume respecting such lines; and explain where, and for what purpose, that axiom is introduced in the theory of parallels. State also the corresponding axiom in any other true theory of parallels with which you may be acquainted; and point out the equivalence of the two assump 3. Define parallel straight lines. tions. SECTION II, 1. Make a triangle of which the sides shall be equal to three given straight lines. State the condition which must be fulfilled by these given lines; and point out how the construction fails when that condition is not fulfilled. 2. Prove that parallelograms upon the same base and between the same parallels are equal to one another. Show hence that the area of a parallelogram is properly measured by the product of the numbers that represent its base and its height. SECTION III. 1. Show how to find the centre of a given circle, and prove that the point found is the centre. State the most convenient way of finding the centre of a circle traced out on a plane surface. 2. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle. 3. Show that if two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. From this proposition deduce the equation to the circle referred to rectangular co-ordinates. SECTION IV. 1. The areas of triangles and parallelograms of the same altitude are to one another as their bases. 2. Show that the areas of similar triangles are to one another in the duplicate ratio (or as the squares) of their homologous (or corresponding) sides. A portion of a triangle, next to one angle, is cut off by a line parallel to the opposite side, and cutting off one-third of each of the other sides. What portion of the area of the whole triangle is cut off? 3. Prove that the areas of equiangular parallelograms are to each other as the product of their sides. If the sides about the equal angles be proportionals, what will be the ratio of the parallelograms to each other? MECHANICS. SECTION I. 1. A labourer working with a wheel and axle 8 hours a-day can yield at the rate of 3600 units of work per minute. How much must he charge per ton for raising coals from a depth of 25 fathoms, in order that he may earn 2s. 6d. per day? 2. An engine of 5-horse power raises 30 cwt. of coals per hour from a pit whose depth is 240 fathoms, and at the same time gives motion to a forge hammer which makes 25 lifts per minute, each lift being 3 feet; it is required to determine the weight of the hammer. 3. Two men undertake to dig a drain 500 feet long, and to carry the materials in barrows to a heap at the end of it. Into what two parts must they divide the work, so that one-half of the labour may fall to the share of each? SECTION II. 1. How many cubic feet of water must descend a river every minute to drive a wheel of 4 effective horse power, by means of a fall of 16 feet, the wheel yielding '68 of the work of the fall. 2. How would you calculate the velocity with which a ball leaves the muzzle of an air-gun, knowing the dimensions of the condensing chamber and barrel, the pressure of the condensed air, and the weight of the ball? 3. A square reservoir is to be dug capable of containtaining 62,500 cubic feet of water. The material is to be carried an average distance of 10 feet from the edge of the reservoir. What must be its dimensions that the work of raising and transporting the material may be minimum? SECTION III. 1. Two cords, one 8 feet, and the other 10 feet long, are attached to a weight of 100 lbs. and fastened at their other extremities to two points in the ceiling 12 feet apart. What is the strain on each cord? 2. A barge, 40 feet long and 10 feet wide, is to be constructed with sheet-iron, each square foot of which weighs 20 lbs. What must be its depth that it may just carry 60 tons lading? 3. A train, weighing gross 80 tons, is allowed to descend freely an incline of 100 to 1, 500 yards in length. How far will it run along the horizontal line at the bottom of this incline with its acquired velocity, resistance being assumed throughout to be 8 lbs. per ton? 4. The wall of a reservoir is 15 feet high, and of the uniform thickness of 3 feet, each cubic foot of the material weighing 120 lbs. Shores (props) are placed to support it 10 feet apart; each shore is 15 feet long, and its foot rests at a point 8 feet from the base of the wall. What is the thrust on each shore when the reservoir is full? TRIGONOMETRY. SECTION I. 1. Trace the sign of the tangent of an arc through the four quadrants. 2. Show that cot A = cos A sin A. 3. Cosec (2 n + 1) π = 1. 4. Cot 2A cos 2A cot 2A cos 2A. 5. Sec (+A.) Sec (A) ( a + +A 4 ) tan ( π 4 cos A cos B sin A 3. 1 + sin 2 A = (cos A+ sin A). 1. Show that in any plane triangle = sin B. 3. I wish to determine the distance from one another of two inaccessible objects, C and D, by observations made at stations A and B, from which C and D are visible. How must I proceed, the distance from A to B being supposed to be known? |