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The Student is here presented with a number of such examination papers as have actually been set before the candidates for prizes and honours at the annual and other examinations. By the first fifteen of them Freshmen, at the termination of their noviciate, were examined ; the next fifteen were given to such students as had completed their second year, that is, to Junior Sophs; the five following ones to Third-Year Men, or Senior Sophs; the two next to candidates for Scholarships; and the three last were given at Fellowship Examinations. The value of such papers to the aspirants after honours at Cambridge is too evident to be insisted on for a moment. They will also be exceedingly useful to such students as wish to be taught on the Cambridge Plan, without incurring the expenses of an University Education ; and it is presumed, there are but few who would not be so įnstructed. Moreover, these papers, together with the Senate-House Problems, supply the reader with specimens of every kind of Mathematical Examination held at Cambridge. They completely develop the nature of the studies pursued there.
1. Describe a parallelogram equal to a given rectilinear figure, having an angle equal to a given rectilinear angle.
2. If a straight line be divided into two equal, and also into two unequal, parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.
3. If two straight lines within a circle cut one another, the rece tangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.
4. Inscribe an equilateral and equiangular hexagon in a given circle.
5. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents.
6. If magnitudes taken jointly be proportionals, they shall also be proportionals when taken separately.
7. If four magnitudes of the same kind be proportionals, the greatest and least of them together are greater than the other two together.
8. Similar triangles are to each other in the duplicate ratio of their homologous sides.
9. In any parallelogram, the sum of the squares of the diagonals is equal to the sum of the squares of the sides.
10. If two circles touch each other externally, any line drawn through the point of contact will cut off similar parts of their circumferences.
11. Determine a point in the base of a triangle, from which if a line be drawn to the vertex, it shall be a mean proportional between the segments of the base.
12. If the sides of a trapezium be bisected, and the points of section be joined, the inscribed figure is a parallelogram, and equal half the area of the trapezium.
13. The sum of the opposite sides of a quadrilateral figure described about a circle are equal.
14. Describe a triangle that shall be equal in area to a given equilateral and equiangular pentagon.
15. Divide a triangle into two equal parts; 1st, by a line drawn parallel to one of its sides ; and 2dly, by a line passing through a given point in that side.
16. Through two given points in the diameter of a circle, one of which is the centre, to describe another circle that shall touch it internally.
17. AB, DC, are two diameters of a circle perpendicular to each other; and the arc AEB is described with C as centre, and CA as radius. The lune ADBE equals the triangle ABC.
18. Given the base, the vertical angle, and the sum of the two remaining sides of a triangle ; to construct it.
19. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides.
1. How many sides has that polygon whose interior angles equal ten right angles ?
2. In any right-angled triangle, the square whch is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.
3. If E be any point in a rectangle, AES + ED = EB* + EC%.
4. Bisect the four sides of the trapezium ABCD in a, b, c, d: join ab, be, cd, da, and shew that the trapezium abcd equals 1 ABCD
5. Upon stretching two chains AC, BD, across a field ABCD, I find that BD and AC make equal angles with DC, and that AC makes the same angle with AD that BD does with BC, from these data prove that AB is parallel to CD.
6. To find the side of an equilateral and equiangular dodecagon inscribed in a circle.
7. In equal circles, angles, whether at the centres or circumferences, have the same ratio which the circumferences on which they stand have to one another.
8. Prove that the area of any trapezium, whose opposite sides are parallel, is found by multiplying the arithmetic mean between the two parallel sides into their perpendicular distance.
9. Given the sines, cosines, and tangents of two arcs to find the sines, cosines, and tangents of their sum and of their difference.
10. There is a triangular piece of ground, whose area equals 525 square yards, and two of whose sides measure 30 and 42 yards respectively; required the length of its remaining side.
il. At the top of a tower which is 30 yards high, I find a certain obelisk makes an angle of 20° 55'; also, the foot of the tower is horizontally distant 40 yards from the foot of the obelisk ; required the height of the obelisk.