a 4. If a house, a pond, or a swamp occurs in measuring a line, how is such an obstacle to be overcome, so as to complete the line and to continue its measurement beyond it ? 5. If it is impossible to carry the chaining to the end of a side of a triangle of a survey, owing to the angular point being fixed on a large building, bow might the distance to this inaccessible point be determined ? 6. Dispose the lines on the accompanying rough diagram (H) of a form so as to survey the ground in the most advantageous manner. 7. Give an illustration of the mode in which the field book of a chain survey should be kept from any one of the triangles into which you have divided the accompanying rough diagram () in answer to 6. 8. Make a neat plot from your own entries of the triangle selected on a scale of 4 chains to the inch. Ink in and letter the plot. 9. Supposing you have been required to plot the whole of the work. included in the rough diagram, as surveyed by means of the lines you have marked upon it, in what order would you have laid down these lines ? 10. What are the different methods of ascertaining the horizontal length of distance measured op a slope ? 11. Ascertain the contents of the enclosure represented by the plot (Question 8) by drawing fine pencil lines across it both longitudi nally and transversely, and counting the squares. 12. Calculate the contents of the triangle supposed to have been employed in the survey of the above inclosure from the length you have assigned its sides in your imaginary field book, (Question 7). · 13. Explain the mode in which you would conduct traverse surveying, supposing ordinary chain surveying to be impossible. 14. The measured distance on a slope is 729 links; the angle of inclination from one end to the other of this distance is 1° 51', find the horizontal and vertical distance in feet between its extremities, entering your calculation in a convenient form for computation. 15. Explain how to adjust the level attached to the telescope of a 5 inch theodolite. 16. What do you mean by the line of collimation of the telescope of a theodolite ? 17. In what manner, in levelling with a spirit level, may the necessity of correcting for curvature and refraction be avoided. 18. Give an illustration of the mode in which you would keep a field in levelling for sections. a EUCLID. I. Specimen Paper. 1. If two triangles have two angles of one equal to two angles of the other, each to each, and one side equal to one side, viz., the side adjacent to the equal angles in each ; then shall the other sides be equal, each to each, and also the third angle of the one to the third angle of the other. 2. To a given straight line apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. 3. 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 balf the line bisected, is equal to the straight line which is made up of the balf and the part pro duced. 4. The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote : and the greater is nearer to the centre than the less. 5. A segment of a circle being given, describe the circle of which it is the segment. 6. If the outward angle of a triangle made by producing one of its sides be divided into two equal angles by a straight line which also cuts the base produced, the segments between the dividing line and the extremities of the base have the same ratio wbich the other sides of the triangle have to one another. 7. Equiangular parallelograms have to one another the ratio wbich is compounded of the ratios of their sides. 8. Trisect a given straight line. 9. Construct a rectangle which shall be equal to a given square (1) when the sum, and (2) when the difference of two adjacent sides is equal to a given line. 10. A tangent to a circle at the point A intersects two parallel tangents B and C, the points of contact of which with the circle are D, E, respectively : show that if BE, CD intersect in F, AF is parallel to the tangents BD, CE. II. 1. Define a circle and a rhombus, and give Euclid's 12th axiom (relating to parallel lines). 2. If from the ends of a side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. 3. What is meant by a corollary? State and prove the corollaries to the proposition in which it is proved that the three angles of a triangle are together equal to two right angles. 4. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. 5. Divide a given straight line into two parts, so that the rectangle con tained by the whole and one of the parts shall be equal to the square of the other part. 6. Draw a straight line from a given point, either without or in the cir cumference, which shall touch a given circle. 7. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle ; and the angle in a segment less than a semicircle is greater than a right angle. ALGEBRA. راه 20 C m n I. (Time, 8 hours.) 1 (3) the quotient when the former is divided by the latter. 2. Reduce to their simplest forms the expressions 24 — 5X3 +423 + 3x +9 - x + 2x+x2 + V1- x - -2x tä? 3. Prove the rules for pointing in the multiplication and division of decimals. 4. Solve the following equations 1 (a) + 1 y 24 + y4 82 4 5. Find a number of two digits, which is three times the sum of its digits, and such that the difference between the digits is 5. 6. If any number of fractions be equal, show that each of them sum of any multiples of the numerators y a + b 1 Is any assumption necessary in order that this may be true ? + C n m (») { c+årő a tö_c; show that C C 8. Investigate a rule for finding the sum of n terms of an arithmetical progession. In the series of stat &c., find s (1) when n= - 5, (2) when n= infinity. 9. When does one quantity vary (1) directly, (2) inversely as another ? 1f x varies as y, prove that x2 + y2 will vary as 22 – y? a3 10. Expand (a:-) in a series of ascending powers of x by means of the Binomial theorem, writing down the first four terms and the pth term. 11. Express the numbers 957 and 23.125 in the septenary scale ? у? ху ху 22 5 3. Divide 435 23 + 4x by 2x2 + 3x + 2. 4. Investigate a rule for finding the square root of an algebraical quantity, and explain how the method of extracting the square root of a numerical quantity may be deduced. 5. Find the square root of (a + b)2 – (2 + (a + c)2 – 52 + (b + c)— a?. 6. Find the greatest common measure of 4x4 - 1223 + 5x2 + 14. — 12 and 6x4 - 1123 + 9x — 13.0 + 6. 7. Solve the equations 1_4 1 10 1 (w) x + 0x2 + 11 = 11. 8. Investigate an expression for the sum of a geometrical series ; and find the sum of 30 terms of the series 2 + + 7 + &c. 9. A number consists of 2 digits ; if it be multiplied by 2, and the pro duct diminished by four, the digits are inverted ; and if 19 be subtracted from it, the remainder is equal to 3 times the sum of the digits ; find the number. 2 Pure Mathematics. 1. To a given straight line apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a rectilineal angle. 2. A segment of a circle being given, describe the circle of which it is the segment. 3. Give Euclid's definition of proportion and of similar figures, and show that similar polygons may be divided into the same number of similar triangles having the same ratio to one another that the polygons have. pression. 3x4 + 14x3 + 9x + 2 2x4 + 933 + 14x + 3 6. Solve the following equations Y + õ (1) (2) ax2 + bac +c=0. = m; C + y = n. d + (2) 21. 22 - 23 + 3 In equation (2) investigate the conditions that the values of x may be two positive integers. of (a - b)2n. 5 4 6 4 13+23 +33 + 9. Given two sides of a triangle and the angle opposite one of them : show how to solve the triangle, and point out when the case is ambiguous. If a=5,6=7, and A=sin -1 ], is there ambiguity ? 10. Given log1671968=4.8571394 ; diff. for 1=60, find the value of N-0719686 to seven places of decimals. 11. If three sides of a triangle are x2 + x + 1, 2x + 1, and 22-1, show that the greatest angle is 120°. 12. A right cone is cut by a plane which meets the cone on both sides of the vertex; show that the section is a hyperbola. Under what condition is it possible to cut an equilateral hyperbola from a given cone. |