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9. Describe the floral envelopes in Phoenix, Nelumbiun, Mussaenda.

10. Describe the fructification of Funaria, Adiantum, Marchan.

tia.

BOTANY, II.

Examiner-DR. Geo. KING. 1. Give a general account of the influence of temperature on vegetation.

2. Describe the sexual reproduction of Ferns.

3. What are the advantages of cross-fertilization ? Give examples of the various ways in which it is secured in Nature.

4. What is meant by Cleistogamy? Give two examples of its occurrence in Indian plants.

5. Give an account of the manner in which Coal has been formed, and enumerate the chief fossils found in Indian Coal.

6. Give the geographical distribution of the Natural orders to which Nymphæa, Magnolia, Nephelium, Melia, Cinchona, Jatropha and Cyperus belong.

7. What is Aleurone? Where does it occur, and how may it be recognised?

8. Describe the objects mumbered 1, 2, 3.

BOTANY, III.

Examiner-Dr. Geo. King. 1. A copy of Roxburgh's Flora Indica and three plants numbered 4, 5, and 6 are herewith given to you. Find out the names given to these plants by Roxburgh, and state to what Natural orders they belong.

2. Define the terms Monopodial, Sympodial, Helicoid, Scorpioid as applied to inflorescences.

3." What is meant by the term Carnivorous as applied to plants. Give examples of carnivorous plants; state how they are supplied in nature with animal food, and how they assimilate it.

4. Describe the changes which occur in a phanerogamous ovule. after fertilization.

5. Define the term Bark and describe the Histological changes which lead to the production of some of its most characteristic forms.

6. Describe the chemical and morphological changes that occur in the passage from the herbaceous to the woody condition in the shoot of a dicotyledonous tree.

7. Give a full account of the processes of Assimilation and Metastasis.

8. Give a brief description of the fossils named Stigmaria, Lepidodendron, Calamites, and state in what strata they are found.

9. What is the oldest known fossil vegetable ? State in what strata it occurs.

Honor Examination.

1881.

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ALGEBRA, TRIGONOMETRY, THEORY OF

EQUATIONS, CONIC SECTIONS.

Examiner—MR. Eliot, M. A. 1. Expand am in powers of x. Shew that if n be greater than 3 n.n-1

2) + 1. 2. 3. 4 (n — 4)3 + &c., = n2 (n + 3) 2" — 4 2. Shew how to convert a surd into a continued fraction, and prove that the last quotient before the quotients occur is twice the first.

3. Prove that the distance between the centres of the circum. scribing and inscribed circles is ✓R--- 2 Rr, where R and r are the radii of these circles.

From a point P perpendiculars PL, PM, PN are drawn to the sides of a triangle ABC : shew that twice the area of the triangle LMN is equal to (R? — da) sin A sin B sin C, where R is the radius of the circle circumscribiog the triangle ABC and d the distance of its centre from P.

4. Investigate an expression for cos a in exponentials ; and prove that the sum of n terms of the series

cos a cos 2 cos 3a
1 +
+ - + -

+ ......
cos a cos' a * cos a
is equal to 0 if cos a = t.

5. Resolve o 21 – 23" cos @ + 1 into real quadratic factors.

6. Investigate Newton's method of approximating to the real roots of an equation.

Prove that if c be very near a root of the equation f (x) = 0 the root is

6-9 (c) _f (c)2f" (c) ofic) - 2 fc)}+

+ &c.

7. If Sr represent the sum of the oth powers of the roots of the equation x" + P2.201—? + ... + pn = 0

prove if myn

Sm + Pi Sm-1 + ... + PSm-n = 0 Shew also that

6 (aby)" = $3,– 352m Sm + 253m 8. Form the equation the roots of which are the squares of the differences of the roots of the equation 23 + 9% + g = 0. Hence or in any other way find the condition that this equation may have a pair of imaginary roots.

Find whether the equation 23 - (a* + 12 + c3 ) x + 2abc = 0 has imaginary roots.

9. Obtain an expression for the area of the trangle contained by the three lines Ax + By + C = 0, A'x + B'y + C' = 0 and A" X + B"y + C" = 0. 10. When the equation to a conic is given in the form

axo + 2 c'xy + bya + 2 a'y + 20'x + c = 0 find the co-ordinates of the centre. Also prove that the eccentricity is given by the equation

et, (a + b — 2c cos w) - 4

1-e (ab - 02) sino w w being the angle between the axes.

11. Find the equation to a hyperbola referred to its asymptotes as axes.

Shew that the equation to the chord joining two points may be put in the form

cos a cos B + y sin a sin B = c sin (a + b). where c tan 0, c cot are the co-ordinates of any point on the curve.

12. If Ø (xy) = (ax + By)2 + 2gx + 2fy + c = 0 be the equation to a parabola, the equation to its axis is

dx * dý=0.

ALGEBRA. Examiner-MR. W. GRIFFITIIS, M. A. 1. Shew that the equation to a conic circumscribing the triangle of reference is

2,m

+2=0.

12

' B Shew also that the condition that the conic may be a rectangular hyperbola is

I cos A + m cos B + n cos C = 0.

If a right-angled triangle be inscribed in a rectangular hyperbola prove that the perpendicular from the right angle on the hypothenuse is a tangent to the curve. 2. Having given the locus of one focus of the conic

V ła + ✓ MB + v ny = 0, shew how to find the locus of the other focus.

Prove also that if the above curve be a parabola the co-ordinates of the focus are proportional to

I'm ñ 3. When a point is in motion in a carve, find its accelerations along, and perpendicular to, the tangent at any instant.

A circle is described by a particle under the action of two forces to the ends of a diameter. If i, po be the distances of the particle from the two centres of force ; P, P the forces, prove that

ар «Р БР.

ar + 1 = are 4. A particle is projected from a given point in a given direction and with a given velocity, and moves under the action of a central attraction varying inversely as the square of the distance. Find the orbit and also the conditions that it may be an ellipse, parabola or hyperbola, and find the values of the major axis, eccentricity, and area described in a unit of time.

If the particle move in a resisting medium, find the general effect on the major axis and eccentricity.

5. A particle is constrained to move on a given smooth plane curve, under the action of given forces in the plane of the curve, find the velocity and the pressure on the curve.

If the curve be a parabola whose axis is vertical and vertex uppermost, and gravity the only force acting shew that the pressure

on the curve at any point = (

-1), sin

; where r is the

focal distance of the point, o the inclination of r to the vertical, and h its vertical distance from a fixed line.

Hence shew that a particle projected freely will describe a parabola. 6. A heavy particle is projected vertically upwards in air, the

resistance of which is measured by (velocity)? ; prove that, if U, V be the velocities with which the particle leaves and returns to the point of projection.

1 1 1

V2 - U2 = 7. Investigate the general equation of continuity.

If the motion be symmetrical in all directions about a fixed centre, write down the equation in its simplest form.

8. Define steady motion, and write down the equation of motion of a steadily moving stream. Also find the pressure on a plane lamina immersed perpendicular to the direction of the stream.

9. Find an expression for the cosine of an angle of a spherical triangle in terms of the sides. Shew that if the triangle be equilateral tan 4 =1--2 cos A.

sec or

10. Shew that there can be only five regular polyhedrons.

11. Describe the nature of the errors to which a Transit Instru. ment is liable. If a, 8, y, be the errors of collimation, level, and deviation respectively, shew that the error in time of transit of a star whose declination is 8 is

a + B cos (1 - 8) + y sin

15 l where l is the latitude of the place.

12. Determine the interval from sunrise to sunset at a given place.

If the increise of the sun's declination from noon to noon be A°, and t, t' be the times from sunrise to noon and from noon to sunset respectively, shew that

A

sin 1 sec o
ť -t=? 180 ✓cos (I — ) cos (1 + 0)

- nearly where l is the latitnde of the place, and ò the sun's declination at sunrise.

13. How is the apparent position of a heavenly body affected by parallax ?

Explain what is meant by horizontal parallax. if he is the parallactic change of the hour-angle h, prove that

sim 12

sin c sin P

sin (h + ) sin A wbere c is the co-latitude of the observer, A and P the north polar distance and horizontal parallax of the observed body.

1, the

STATICS, HYDROSTATICS AND OPTICS.

Examiner-MR. ELIOT, M. A. 1. Investigate formula for determining the position of the centre of mass (gravity) of any solid.

Find the position of the C. G. of the solid bounded by the co-ordi. nated planes and the surface

6 radicals being all positive.

2 Find the attraction of a circnlar lamina on an external particle situated on a straight line drawn at right angles to the lamina through its centre.

A right circular cone of given volume attracts a particle at its vertex. Find the form of the cone so that the attraction may be a maximum.

3. A rough heavy body rests on a fixed rough surface. Deter

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