ELPHINSTONE'S HISTORY OF INDIA. Morning Paper. 1. Draw a Map of India shewing (1.) The basins of the principal rivers. the Moohummudan invasion. 2. What regulations in the code of Menu with respect to war and the treatment of women shew a humane disposition on the part of the Hindus ? 3. Describe an Indian township, and the duties of its principal officers. 4. How does Elphinstone sum up the arguments on the question of right of property in the soil” ? 5. Give a sketch of the evidence on which it is asserted that the names Sandracoptus and Chandragupta refer to the same person. 6. Describe the steps by which Mr. Prinsep was enabled to decipher the inscription on Firuz Shah's column. Afternoon Paper. 7. What is the general character of the Hindu drama ? 8. Point out the extent of the commerce of the Hindus in early times. 9. On what grounds does Tod suppose that some of the Rajput tribes are of Scythian descent ? 10. Shew by a table the states which existed in India before the Moohummudan conquest, and where they are first and last mentioned. 11. Give the date and some of the circumstances of the invasion of India by Cassim. 12. In which of his expeditions did the Sultan Mahmud plant the first permanent garrison of Moohummudans beyond the Indus. account of the storming of Somnauth. Give an Mathematics. FIRST CLASS. DIFFERENTIAL AND INTEGRAL CALCULUS. Morning Paper. (1.) Having given u = 0 a relation between x and y, shew how to find the differential co-efficient of with y respect to x, find dy dac n and if y d? n y = cos mx find doc" (2.) Apply the method of limits to find the equation to a straight line which touches the curve at a given point. Find the asymptotes to the curve y ax2 + 203 (3.) Shew how to determine the value of a vanishing fraction in all cases va-N + va when x = a ✓ 2ax x2 (1) a T Sin mo COS m a no) (2) when 0 =om be a whole numsin no + cos ( + mo) 2 ber of the form 4 p + 1 and n of the form 4 p + 3. (4.) Shew how the maximum and minimum values of a function of one variable may be determined. If this method be applied to find those conjugate diameters in an ellipse, of which the sum is a maximum or a minimum, it appears to fail in the latter case-- Explain the cause of this. (5.) Explain the transformation of the independent variable and trans dạy form the equation 2 dy + y = 0, where x is the independent variable, into one where is the independent variable, being equal to cos—10 If R represent the radius of curvature, it may be proved equal to dS? ✓ (dox) + (døya) where x and y are co-ordinates of a point in the curve, and S the length of it. (6.) If AP be any curve referred to a pole S; find the differential expression for the area: and if u be the solid generated by the revolution of the area ASP, about AS, SP =r and the angle ASP = 0. du dᎾ (7.) Shew that is the polar subtangent of a curve, u being the du reciprocal of the radius vector; aud there is generally a point of inflexion dź u where u + 0. do? Find the asymptotes and points of inflexion in the curve whose equa ө tion is r = a and trace the curve. sino (8.) Integrate the following expressions : dx xdx x?dx (az + x2)} (x2 + ax) (x2 + 5a2)' (1 + x2) (log x)} Find the value of Som dx (tan x ju (9.) What is meant by integration between limits? When the function to be integrated changes its sign between the limits, how is the true value of the definite integral to be found ? Trace and find the area of the curve r = a (2 cos 6-1). xxx, dx SECOND CLASS. GEOMETRY OF TWO DIMENSIONS, AND NEWTON. Morning Paper. (1.) Shew how to draw a straight line by means of its equation both when the co-ordinate axes are rectangular and oblique, the angle between the axes being 108°—draw the straight lines. V 5+1 1 V5 (1) y = (x-a), (2) y x + a, (3) x + y = 2 2 3 + v 5a 2 and shew that if (1) and (3) be produced to meet the axes, and (2), the lines intercepted between the origin and the suc ssive points of i ersection, will form a rectangular pentagon. X2 (2.) If y2 + a = 1 be the equation to an ellipse (hk) a given point, a? 62 hx ky what does the equation + a2 62 = 1 represent, (1) when (hk) is in the circumference of the ellipse, (2) when without it, (3) when within it. (3.) Define an hyperbola and thence find its polar equation, the centre being pole. If the transverse axis be indefinitely increased the hyperbola passes into a parabola. (4.) If a right cone be cut by a plane, find the equation by the section, and shew that it will be an ellipse, hyperbola, or parabola. (5.) Shew that the equation to the parabola referred to any two tan y gents as axes may be put under the form + 1 where a b and b are portions of the tangents between the curve and their intersection. (6.) Explain the mode of reasoning by which Newton determines the ratio of quantities which vanish together; and prove that the ultimate ratio of the arc, chord and tangent to each other is one of equality. (7.) Enunciate Lemma XI. What is meant by saying that every curve of finite curvature is ultimately a parabola ? How is this proved ? (8.) If a body revolve round a fixed centre of force, the areas described by lines drawn from the body to the centre of force lie in one plane, and are proportional to the times of describing them. Point out the laws of motion assumed in the proof of this proposition. (9.) State Kepler's laws; and enunciate the various propositions in Newton by means of which they may be deduced from the theory of universal gravitation. (10.) A body moves in a parabola, to find the law of force tending to the focus, and compare its velocity at any point with that of a body moving in a circle radius SP and described round the same centre of force.. none. +1 (2.) An equation must have an even number of impossible roots, or How far is this true of irrational possible roots? Having given one root of the equation x4 6x2 - 48x — 11 = 0 is 2 + 15 solve the equation. (3.) If f(x) be a rational and integral function of x; Explain the formation of its successive derived functions ; Shew that an odd number of roots of f'(x) = 0 lies between every two possible roots of f (x) = 0 and that if f(x) = 0, f'(a) = 0, $" (a) = 0, -f(r) (a) = 0 f(x) is divisible by (x—a)" + (4.) Investigate a method for finding the commensurable roots of an equation whose co-efficients are rational. If the constant term have many divisors, how may the operation be shortened ? Solve the equation 29 - 3x4 9x3 + 21x2 - 10x + 24 = 0. (5.) An equation of m dimensions has n equal roots, shew how to find them; Solve the equation 34 + 13x3 + 33x2 + 31x + 10 = - 0 which has 3 equal roots. (6.) Give Waring's method of separating the roots of an equation ex. 23 11x + 11 = 0. (7.) If from either extremity QVQ of a parabola a perpendicular QD is let fall on the diameter then (QD2 4AS. PV.) (8.) In the ellipse the rectangle under the abscissæ of the axis major is to the square of the semiordinate, as the square of the axis major to the square of the axis minor (AN. NM: PN2 = AC2: BC) (9.) The rectangle under the perpendiculars drawn from the foci of an hyperbola on the tangent is equal to the square of the semi-axis minor (Sy. Hz BC). (10.) In the hyperbola parallelograms formed by the tangents at the vertices of pairs of congregate diameters have all the same area. = FOURTH Class. EUCLID AND ALGEBRA. Morning Paper. (1.) In any right angled triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the other two sides which contain the right angle. Is this proposition included in any more general one ? D |