Ungeduldig einmal, die Kutsche begierig Die uns sollte hinaus zum Brunnen führen der Doch sie kam nicht; ich lief, wie ein Wiesel, Treppen hinauf und hinab, und von dem -GOETHE. PURE MATHEMATICS.-Part I. The Board of Examiners. 1. On a given straight line describe a segment of a circle which shall contain an angle equal to a given angle. Construct a triangle, having given the base, the vertical angle, and the altitude. 2. If a straight line be drawn parallel to one side of a triangle it shall cut the other sides, or those sides produced, proportionally; and conversely, if the sides or the sides produced be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle. If any two straight lines are cut by three parallel straight lines they are cut proportionally. 3. If two triangles be equiangular to one another, the sides about the equal angles shall be proportionals, those sides which are opposite to equal angles being homologous. If two parallel straight lines are cut by three straight lines which pass through the same point, they are cut proportionally. 4. If two straight lines are parallel, the straight line which joins any point in one to any point in the other is in the same plane as the parallels. If a straight line is parallel to a plane, shew that any plane passing through the given straight line will have with the given plane a common section which is parallel to the given straight line. 5. Shew how to solve two equations which contain two unknown quantities, one equation being of the first degree, and the other of the second degree. shew that each of these ratios is also equal to la + mb + no la' + mb' + no'' Prove also that (a3 + b3 + c3) (a'3 + b'3 + c'3) = (a2a' + b2b′ + c2c′) (aa'2 + bb'2 + cc'2). 7. Find the sum of any number of terms of an arithmetical progression. Find the sum of all the odd numbers between 2m and 2n where m, n are integers. 8. State and prove the binomial theorem for a positive integral exponent. Find the middle term in the expansion of (x + y)2m. 9. Define the trigonometrical ratios of an angle, and shew that they are always the same for the same angle. 10. Prove that cos (A + B) = cos A cos B sin A sin B, A, B being positive angles whose sum is less than a right angle. 12. Shew how to solve a triangle having given two sides and the angle opposite one of the sides. If a = 1, b = √3, A = 30°, find c, B, C. PURE MATHEMATICS.-PART II. TO BE USED ALSO AS FIRST HONOUR PAPER. The Board of Examiners. 1. Find the equation to a straight line in terms of the intercepts it makes on the axes. A straight line moves parallel to the base BC of a given triangle ABC and cuts the sides AB, AC in P, Q; BQ, CP are joined; find the locus of their point of intersection. 2. Find the equation of a circle referred to any rectangular axes. Find the equation of the circle which passes through the three points a, b; a, b' ; a', b. 3. Find the equation of the chord joining any two points on the parabola y2 = 4ax. If the chord subtends a right angle at the vertex shew that it meets the axis at a fixed point. 4. Find the locus of the middle points of a system of parallel chords of an ellipse. Shew that the tangent at an extremity of a diameter is parallel to the chords bisected by that diameter. 6. Prove that under certain conditions f(a + h) = f(a) + hf' (a + 0h), and state the conditions. Expand secx in ascending powers of x as far as the term in a1. 7. Shew how to find the value of an expression which takes the indeterminate form 1°. Find the values when x = 0 of sin x (sin 1 x ) = ( tan x ) = ( cos x) 1⁄2. x 8. Investigate a rule for finding maximum and minimum values of a function of one independent variable. An open tank is to be constructed with a square base and vertical sides, so as to contain a given quantity of water. Shew that the expense of lining it with lead will be least if the depth is made half of the width. |