COLLEGE OF PRECEPTORS.

(Incorporated by Royal Charter.)

PROFESSIONAL PRELIMINARY EXAMINATION.—MARCH, 1884.

WEDNESDAY, March 12th-Morning, 9.30 to 11.30.

ALGEBRA.

Examiner-J. MCDOWELL, M.A.

1. Given x-y= 5, 3x-5y = 8; find the numerical value of

2. Find the L. C. M. of 2æ3—x2+x+4 and 4x3−x+12.

3. Solve the simple equations:

and calculate the two values of x in (ii.) true to 4 places of decimals; also show that the sum of these two values is exactly 3. 5. A certain sum of money put out at Simple Interest for two years, at a certain rate per cent. per annum, will amount to £88, but the same sum at one per cent. less would amount to £96 in five years. Find the sum and the rates of Interest.

6. There are two numbers (excluding zero), such that nine times the cube of either number exceeds three times its square by ninety times the number. Find the two numbers.

7. Investigate the two real and rational quadratic factors of x2+4a2x2+16a*.

8. Investigate, partly by actual division, what real values of a and b will make a1+2x2+9 exactly divisible by x2+ax+b, and write down the quotients.

9. State and prove a rule for finding the L. C. M. of two compound algebraical expressions, such as those in question 2.

COLLEGE OF PRECEPTORS.

(Incorporated by Royal Charter.)

PROFESSIONAL PRELIMINARY EXAMINATION.

-MARCH, 1884.

WEDNESDAY, March 12th-Morning, 9.30 to 11.30.

1. When x =

ALGEBRA.

Examiner J. McDowELL, M.A.

=

3, and z = 0; find the numerical value of 4x2-9y2+ √(64x2-27y2+12xyz2+12).

2. Find the quotient and remainder, when

6x+x+5x2+5x+1 is divided by 22-x+1.

3. Find the L. C. M. of x3-1 and x*+x2+1.

4. Reduce to its simplest form

+

b+c

(a-b) (b-c) (b-c) (a-c)' (c-a) (a−b)'

7. Divide the number 20 into two parts, such that the difference of their squares shall exceed their difference by 38.

8. A grocer buys two sorts of teas at ls. and 1s. 6d. per lb respectively, mixes them and sells P lbs. of the mixture at 2s. per lb., thus gaining 32s. on the prime cost, but, if he had reversed the proportions of the teas in the mixture, he would have gained only 28s. by selling P lbs. at 2s. per lb. What were the quantities of teas in P lbs. of the first mixture?

9. Explain why aa3= a*, and state clearly the general rule of which this is a case.

1. Define the terms-right angle, angle in segment of a circle, rectilinear figure described about another rectilinear figure.

2. Define a parallelogram, and prove that the opposite sides of parallelograms are equal. State the converse of this Proposition.

3. In any right-angled triangle, the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

4. If a straight line be divided into any two parts, the squares on the whole line and on one of the parts are together equal to twice the rectangle contained by the whole line and that part, together with the square on the other part.

[Enunciate (without proof) a subsequent Proposition in the course of the proof of which appeal is made to this Proposition.]

5. If, from a point within a circle which is not the centre, three straight lines be drawn, one of them passing through the centre, and the other two both on the same side of that line; then the line through the centre shall be the greatest of the three, and, of the other two, that nearer to the greatest shall be greater than that more remote.

6. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

7. If in equal circles, angles (whether at the centres or the circumferences) be equal, the arcs on which these angles stand shall be equal.

8. To describe an isosceles triangle having each of the angles at the base double of the third angle.

In the figure for this Proposition, produce DA and DC to meet the other circumference at G and H, and prove that H is the middle point of the arc BHG.

COLLEGE OF PRECEPTORS.

(Incorporated by Royal Charter.)

PROFESSIONAL PRELIMINARY EXAMINATION.

-MARCH, 1884.

WEDNESDAY, March 12th-Morning, 11.30 to 1.

EUCLID.
BOOK I.

Examiner-W. J. REYNOLDS, M.A.

1. Define the terms-parallel straight lines, plane surface, figure. [State the three cases of the following Proposition, but demonstrate the second case only.]

2. On the same base, and on the same side of it, there cannot be two triangles such that they have those sides equal which are terminated at one extremity of the base, and have likewise equal those sides which are terminated at the other extremity.

3. Any two angles of a triangle are together less than two right angles.

4. If a straight line fall on two parallel straight lines, it shall make the alternate angles equal to one another.

5. If two triangles have two angles of the one equal to two angles of the other, each to each, and have one side equal to one side, viz., either the sides adjacent to the angles which are equal, each to each, or sides which are opposite to equal angles in each triangle (demonstrate this latter case only); then shall the other sides be equal, each to each, and also the third angle of the one triangle shall be equal to the third angle of the other.

[Write the enunciation (without the proof) of the subsequent Proposition in proving which this one is made use of.]

6. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

7. The angle BCA of a triangle ABC is a right angle, and the straight line bisecting the angle BAC cuts BC at D. Through any point P in this bisector, a straight line is drawn perpendicular to AB and cuts BC (produced if necessary) at E. Prove that two of the sides of the triangle PDE are equal.

1. ABCD is a square board having a side 12 inches in length. Along AB a force of 5 lbs. acts, along DC a force of 7 lbs. acts, and along BC a force of 5 lbs. acts: find the direction and magnitude of the resultant of the three forces.

2. A horizontal straight bar, 8 feet long and weighing 12 lbs., is supported at each end, and a weight of 36 lbs. is hung 3 feet from one end: find the pressure on each of the supports.

3. Explain the Triangle of Forces, and use it to construct an inclined plane on which a horizontal force of 5 will support a weight of 12. Find the resistance of the plane.

4. A wheel and axle is used to raise a bucket from a well. The radius of the wheel is 30 inches, and while it makes seven revolutions the bucket rises 11 feet. What is the smallest force that will raise the bucket, supposing its weight to be 30 lbs.? (You may assume that the circumference of a circle is 22 of its diameter.)

5. Two bodies are let fall from the same point at an interval of one second: find how far they will be apart after a lapse of four seconds from the fall of the first.

6. State Newton's Second Law of Motion, and mention some illustrations of its truth.

7. How far will a body of 12lbs., which starts with a velocity of 10 feet a second, move against a uniform retarding force equal to one-third of its weight?

8. With a delicate balance, how can we determine the specific gravity of a given fluid?

9. State the rule by which the amount of pressure may be found that is exerted by a fluid against a plane area.

10. A vessel of water has one side vertical. A circle 7 feet in radius is drawn on this side: when the water just covers this circle, find the whole pressure exerted by the water on the portion of side enclosed by the circle, supposing a cubic foot of water to weigh 1,000 ounces, and assuming the area of a circle to be 22 of the square of its radius.