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Or Give an expository and critical account of the psychological doctrines of 1, Plato, 2, Aristotle, 3, The Stoics, 4, The Neo-platonists, pointing out in each case the theory held regarding the nature of the Soul and the origin and limits of Cognition.
4. Give an acconnt of the Scholastic philosophies of the 11th and 12th Centuries as regards the questions at issue, the methods of Speculation, and the systems of Roscellinus, William of Champeaux, Anselm and Abelard. Shew how the questions at issue were dealt with by William of Occam, and estimate his influence upon the history of philosophy.
Give an account of the Arabian Philosophy, as regards its origin method, leading expounders and systems, and permanente contributions to Philosophy.
5. Give a summary of the philosophical system of Spinoza, including its sources, method, psychology, ethics and theology. Give an account of his life and works, quote any of his principal definitions, refer to the criticisms of Leibnitz and others upon his system and trace its influence upon modern German speculation.
Or Trace the development of Berkeley's philosophy through his various works, summarising the last of them. Discuss the validity of Reid's rendering of Berkeley's relations to Locke and Hume; and compare his idealism with that of Fichte.
6. Write a short paper on “ The development and principles of Greek Scepticism and modern Agnosticism.”
[5 questions only to be taken.] 1. Define or describe Ethics, mention its leading questions and the divisions of a complete system, and distinguish it from Æsthetics, Jurisprudence and Politics. Point out any important differences between the ancient and the modern Ethical systems, and account for these. Discuss briefly the Logic of Ethics, and criticise the methods and principles of Egoistic Hedonism, Intuitionism and Utili. tarianism.
2. Write concise historical and critical notes on the following expressions : Selfishness; Self-love; Benevolent affections; Moral Sense; Sympathy; Secondary Desires; Principles of action ; Casuistry; Determinism ; Liberty of Indifference; Heteronomy of the Will; Volitional Characters of feelings; Deontology; Duties of perfect and of imperfect obligation.
3. Give a brief sketch of the pre-Aristotelian attempts to form a theory of Virtue. State Aristotle's theory of Virtue, with some of bis illustrations as given in his Ethics. Compare Aristotle's theory with those of the Stoics and Epicureans; and reproduce Kant's Criticism upon it and give your own.
4. Write a summary of the politico-ethical system of Hobbes referring to his works ; and consider how far it was influenced by the historical conditions of the time. Trace the controversy which it originated, down to Butler ; and point out any new principles which it raised or established in Ethical philosophy.
Or Contrast carefully the moral systems of Bentham and Kant, in regard to the essential principles of Ethics. Consider the causes of their differences, the validity of their methods, and their influence upon subsequent Ethical speculation.
5. Write a short historico-critical dissertation on any of the following subjects :
(1.) The immutability of moral distinctions in the light of empirical variations.
(2.) Recent applications and the applicability of the theory of Evolution to Ethics.
(3.) Free-Will in relation to physical, logical, and historical Law.
PURE MATHEMATICS, II.
Examiner-MR. A. M. NASH, M. A. 1. The circle circumscribing the triangle formed by three tangents to a parabola passes through the focus.
2. Determine the equation and lengths of the axes of the conic axa + 2hxy + bya + 2gx + 2fg + c = = 0.
3. The locus of the poles of a given straight line, L, with respect to a series of confocal conics is a straight line, M, perpendicular to L. If a parabola be described having its focus at one of the given foci and touching L and M, its directrix will pass through the other focus.
4. The locus of the centres of all the rectangular hyperboles which can be described about a given triangle is the nine points circle of the triangles.
5. Find the equation of the asymptotes of the conic By = kaʻ, and obtain the condition for a rectangular hyperbola.
6. The product of the perpendiculars from any point of a conic upon the sides of an inscribed triangle ABC bears a constant ratio to the product of the perpendiculars from the same point upon the tangents to the conic at the vertices ABC.
In the circle this ratio is unity: hence obtain a property of the parabola by reciprocation.
7. What is meant by "inversion ?” The inverse of a circle with respect to a point is either a straight line or a circle.
Obtain a property of the circle by inverting the theorem-if A, B, C be three points in order upon a straight line AC = AB + BC. 8. Investigate the loci
(1) a2 + 2xy + ya – 2ax + bay + 5a = 0.
a straight line;
4xy - ax + 2ay + llaz a.
9. Determine the locus of a point from which three tangent lines, mutually at right angles, can be drawn to an ellipsoid.
10. Show that there are only five regular solids.
11. Define the osculating plane, and the binormal of a tortuous curve, and find their equations.
12. If the equation to a surface be z = f (wy), prove that the principal radii of curvature are given by the equation,
P ✓1 + p + q + (1 + pa + 9)? 0. The points of the surface 3a : = 203 + y3 + 3 axy at which the principal curvatures are equal and opposite lie upon the surface
axy + a® (2 + y)
PURE MATHEMATICS, I.
Examiner-MR. A. M. Nasi, M. A. 1. Find the number of homogeneous products of r dimensions that can be formed out of n letters.
2. Determine whether the following series is convergent or divergent:
(a + b)P (a + 20P 3. Find the number of integers which are less than a given number and prime to it.
If n be a prime number 73, n° + n7 - 13 - n is divisible by 5040.
4 Prove the following rule for determining the remainder when a very large number is divided by 37:
Divide the given number into sets of three figures beginning from the right, as in cube root: add together all these sets, and divide the sum by 37: the remainder is the number required. 5. Sum to n terms the series 1 3 7 13
21 1. 3. 5 2. 4. 6 3. 5. 7 4. 6. 8 5. 7.9 6. ABCD is a quadrilateral inscribed in a circle ; AB, BC meet
sin E AD - BC in E, AD, BC in F; show that
sin F AB? CD2 7. Eliminate 0 and p from the equations
1, (a® + 6%) cos (0 - 0) = (a? - 12) cos (0 + ). Interpret these equations geometrically.
2n 8. Resolve a 2x" cos 0 + 1 into factors.
2 + @
4 + 0 Hence prove that cos + COS
+ COS n
to 7 terms 0. 9. Sam to n terms the series whose nth term is nel
1-1 sin 4
n-1 0 (2 + 3 cos
N-1 cos 4
O cos 2:4 10. Explain Newton's method of determining the limits to the roots of an equation.
Apply it to the equation r5 4x4 + 743 -- 15x2 + 20x - 300 = 0.
11. The equation 2* — 4x3 + 8.x2 - 9x + 3 = 0 has one root between 1 and 2, find it to two places of decimals.
12. Show that the product of two determinants can always be expressed as a determinant. Write down (in the form of a determi. nant) the product of
an, A2, A3,
а, а bz, ba, bg, and Buy B2,
C1, C2, C3, 13. Any symmetric function of the differences of the roots of the equation
+ &c, = 0 must satisfy the differential equation do do
dф + (n - 1) P1
dpg Hence find for the cubic x3 - P. 2* + Po X — P3 = 0, whose roots are a, b, g the value of 3 (a - b)” (a – 79 in terms of the coefficients P1, P2, P3.
+ (n − 2) P2 dps
PURE MATHEMATICS, IV. Examiner-MR. A. M. Nash, M. A. n (n − 1)
n (n-1 (n - 2 (n-3) xn-2 +
1. If un
= R +
n (n − 1)
un - 4 &c.
2. 4 2. Prove that the problem of finding the nth convergent to a continued fraction in which the quotients recur can always be reduced to the summation of recurring series.
Find the nth convergent to
1 1 1 1
n (n − 1) COS
1 + n cos O +
3. Prove that (2 cos
(2 cos g)"
n in 1) (n 2) cos 20 +
cos 30 + to (n + 1) terms.
13 4. If a, 8, 7, 8 be the roots of the equation ** - p33 + qos + 8 = 0, prove that 1, 1, 1, 1
1, 1, 1, 1 a”, B2,
B, 7, a?, B3, yo,
a>, ga, g?, 82 a, b, c, do
ao, B3, go, 83 5. The tangents at three points of a parabola make angles a, b, with the axis : prove that the area of the triangle formed by these tangents is to the area of the triangle formed by the norwals as
1: (cot a + cot B + cot ap). 6. Prove that the radical axis of the circles of curvature at the extremities of any ordinate to a given diameter of a parabola passes through a fixed point. If the diameter be variable, the locus of this point is a parabola.
7. The normal to any curve at a point of inflexion is an asymptote of the evolute of the curve.
8. From a point P four normals are drawn to the semi-cubical parabola aya = x3, so that the sum of the angles which they make with the axis is constant ( = a): prove that the locus of P is a straight line making an angle a with the axis, and passing through the focus of the curve.
9. A hypocycloid is generated by a circle of radius a rolling inside a circle of radius na, n being an integer. Determine the asymptotes of the curve which is the polar reciprocal of the hypocy. cloid with respect to the fixed circle.
If the given curve have three cusps the equation of the reciprocal is 4x3 = 3 *** + y) (x −3 a). Trace it.
10. A variable plane cuts off a given volume from a given ellipsoid, show that it envelopes one of three ellipsoids similar to the given ellipsoid. 11. Find the locus of the foci of the sections of the surface
4 ax made by planes passing through the origin.
13. Find the mean area of sections of the ellipsoid
62 ca made by planes parallel to the plane of øy.