(2.) To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, shall be equal to the square of the other part. Can this be solved arithmetically? if so, find approximately into how many parts the given line must be divided. (3.) Prove that the opposite angles of any quadrilateral figure which can be inscribed in a circle are together equal to two right angles. (4.) If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides or those produced proportionally, and if the sides or the sides produced be cut proportionally, the straight line which joins the point of section shall be parallel to the remaining sides of the triangle. Hence shew how a line may be drawn on the ground through a given point, parallel to a given straight line by means of a piece of string. (5.) Every solid angle is contained by plane angles, which together, are less than four right angles. (6.) A person who had a 91 anna share in an Indigo factory, made his younger brother a present of 75 per cent. of his share, and sold the remainder to his cousin, who soon after purchased 7 of the younger brother's share, but now offers to dispose of half his interest in the factory for Rs. 7000. Estimating at the same rate what was the value of the whole factory, and each brother's share? (7.) If a and b be two integral numbers prime to one another and the product a X c be divisible by b, shew that c must be divisible by b. Find the form of the denominator of a vulgar fraction in its lowest terms when it is reducible to a terminating decimal, Is it so reducible ? (8.) To extract when possible the cube root of a binomial surd, one of whose terms is a rational quantity, and the other a quadratic surd. Ex. - 4 — 10 2 (9.) Solve the equations a3 + 23, ai – 2013 = 4 a* : • . .. (1) In equation (3) explain the result when the values of x and y assume the form of (10.) Insert m harmonical means between a and b. The distance between Calcutta and Barrackpore is 14 miles, now if a single stone were laid upon every yard of that distance, and the first one was a yard from the basket, what distance would a man travel in bringing the stones one by one to the basket. (1).) Write down the number of variations of m things taken r and r together. . Find the greatest term in the expansion of (1 + x)m without regard to sign m being positive and a a proper fraction. Will the same investigation hold when m is negative ? (12.) Find the amount of an annuity, left unpaid for m years, at simple interest. Explain why it is not consistent with the principle of simple interest to consider the amount of an annuity, to be sum of the present values due at the periods 1, 2, 3, .. .. m, years. (13.) Investigate a rule for forming the consecutive converging fractions. How may converging fractions be employed to find the logarithm corresponding to any number? (2.) Within a given parabola inscribe the greatest parabola, the vertex of the latter being at the bisection of the base of the former. (3.) Investigate a differential expression for the radius of curvature, and shew that it is identical with Newton's expression, (arc) I limit subtense I to the tangent In the curve y = -(7 + - •) the ordinate at any point is a mean proportional to the radius of curvature there and at the point x = 0. (4.) Define the evolute of a curve. Investigate the property on which it depends; find the evolute to the cycloid. (5.) Determine the nature of the curve whose equation is y3 + x3 — ax? = ), find the maximum ordinate, and point of inflexion. Trace and find the area of the curve whose equation is 24 + y4 - axy = 0 (6.) If in the radius vector SP of a parabola, (the vertex of which is A, and Sy the perpendicular from the focus S upon the tangent at P) a point Q be taken, such that SA : Sy = SQ : SP, find the equation to the curve which is the locus of Q; trace the curve and shew that the areas of the curve and parabola between the vertex and the latus rectum of the parabola are as 3 : 4. (7.) Shew how to find the length of a curve referred (1) to rectangular co-ordinates, (2) to polar co-ordinates. Prove that the length of the curve whose equation is 2* + y} = aš intercepted between the axes of x and y is (8.) Find the volume of the solid generated by the revolution, about the axis of x, of the lemniscata the equation of which is (x2 + y2)2 = a? (x? — y) SECOND CLASS. Afternoon Paper. (1.) State the steps in the reasoning by which it is shewn that f (x + h) admits of development in a series proceeding by ascending positive and integral powers of h. (2.) If (w) be a function of y, y a function of «, om= .Employ this proposition to differentiate by substitution the function. XN ---1 a” + x2 (3.) Define a multiple point, and shew from the definition that if hy be obtained from the equation to the curve made free of radicals, the coordinates of the multiple point will make it assume the form Take as an example the curve xạy?= a(22 - y2) and determine the direction of its branches at the multiple point. (4.) A curve is convex or concave to the axis of x, according as 9"} has, or has not, the same sign as the ordinate. Determine the minimum value of (x — a)m being odd. (5.) Find the differential expression for the radius of curvature, and shew that it agrees with Newton's. If y and æ be functions of a third variable 0, the expression for the radius of curvature is du doc da dy do doz do dó? determine what this expression becomes when is the arc of the curve. (6.) Trace the curves defined by the equations y = x V x2 – az V r t a? Vaz + x3 2? + a2, 9 m2 = a, Y = X a ? — a? (7.) Investigate the differential expression for a surface of revolution ; and find the surface generated by the revolution of the lemniscata, the polar equation of which is p2 = a? cos 20. (8.) Find the locus of the intersection of the perpendicular, drawn from the vertex, and tangent to any point of a parabola. Trace the curve and find the area between the curve and its asymptote. • 23 (9.) Integrate for J x (a + bx3) ) x (a? + x2)n Jo sin mo cos no. Make the integral of Sata? + 2*)n depend on that of Secat txa)n-1 (10.) Obtain the integral of Sovat box + call Third Class. Afternoon Paper. (1.) Shew how to transform an equation into one which shall want the second or third term ; under what circumstances can both be made to disappear at one operation ? Form an equation of six dimensions having the co-efficients of the 2nd and 3rd term so related that they can both be taken away at one operation. (2.) The limiting equation must always have as many possible roots as the original wanting one. Hence prove that if m consecutive terms be wanting in an equation, it cannot have more than (n—2m.) possible roots. How many possible roots can the equation an — ax2 + b = o have. (3.) Give Cardan's method for the solution of a cubic equation. Shew that it fails when all the roots are real, and succeeds when two roots are imaginary, or when all real but two equal. Ex. 23 — 3 x2 – 3 x —7=0. (4.) If several roots of an equation lie between two consecutive integers, how may Sturm's Theorem be applied to find an approximation to each ? Find by this method an approximate value of a root of the equation 2013 - 22 — 5= 0. Correct to three places of decimals. (5.) Explain Newton's method of approximating to the roots of an equation, and shew when it may safely be applied. Obtain an approximate value of a root of 2 + 4 22 – 1=0. Cor.. rect to two places of decimals. (6.) Define the asymptotes of an hyperbola. If any straight line Qq perpendicular to either axis of an hyperbola meet the asymptotes in Q and y and the curve in P the rectangle Q P. Pq is invariable. (7.) In the Ellipse the sum of the squares of the conjugate diameters is constant (C P + C D = A C2 + B C2.) If the normals at P and D intersect in K shew that K C is perpendicular to P D. (8.) If any chord AP through the vertex of an hyperbola be divided in Q so that A Q: QP= A C: B C'>, and Q M be drawn perpendicular to the foot of the ordinate M P shew that Q O at right angles to Q M cuts the transverse axis in the same ratio. FOURTH CLASS. EUCLID AND ALGEBRA. Afternoon Paper. (1.) 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. Hence prove that AB is parallel to CD. |