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9. Find the value of the expressions -- -** 1- (1.6.opezet ishi? vode 135

jo (cos l_ cos'la – sin 17 sin la cos 11-12) where p = 4000 l=36° 45' 10"

L1=65° 27' 38" ** Visit I 19=72° 15' 1714

La = 15° 4' 22"

(* 4. : ** ใกล้ ** * * & 8. From each of two ships a mile apart the angle is observed which: 35, is subtended by the other ship and a mark on shore : these anglos are found to be 52° 25' 37" and 75° 9'.45" respectively: find the distances of the mark from each ship correct to five places of decimals.

9. Find a general expression for the area of the segment of a circle.

Find the area of the minor segment ont from a circle of radins lifti noi 3-in. by a chord which subtends at the centre an'angle of 23o 40.'

10. Prove that the volume of a frustum of a cone is ** (E+VÉT É, 1 Ej) cubic feet, where the thickness of the frastuin: is k feet, and the ends contain El and Eq sqpare feet respectively,

The volame of the frnstum of a cone is 407 cubic inches, and its thickness is 101 inches. If the diameter at one end is 8 inches, find the diameter of the other evid..

11. A solid “sphere fits closely into the inside of a closed cylindrical 40 box, the height of which is equal to the diameter of the cylinder. . Having given the rading of the sphere, write down the expression for the volume of the sphere, the snrface of the sphere, and the volame of the empty space between the spliere and the cylinder.

Find the values of the above expressions in cubic feet for a diameter assumed by yourself.

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ALGEBRA, TRIGONOMETRY "AND PLANE ANALYTI.

CAL GEOMETRY.
Examiner-c. LÍTTLE, Esq., M.A.
The figures in the margin indicate full marks.

1. Find the sum of n terms of a series in Arithmetical Progression. 30

If the pth term of the arithmetical progression is q, and the qth term is p, find the mth term,

din 2: Write down the general term and find the sum of the following: 30 series to n terms :

1+3+7+13+21 ........ 3. Prove the Binomial Theorem when the index is a positive integer. 40

If C CQ... ...c be the coefficients in the expansion of (1 + )», prove that

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10

008

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3n+1+.

4 Sbov that loge(*+a) -log("-a)

al

303 5n6 Having givon that,

Jogo 2-6931373 show how the above formula may bo used to find logo 10 and and is corroct to the oth dooimal placo. 5. Resolve

683 +682 + 50

02-1). (*+1) into partial frnotions.

6. . Prove De Moivre's theorem ; and 1188 the theorom to obtain an 30 expansion of sin ne in ascending powers of sin.

7. Find the snın of the cosines of a series of angles, the angles boing 40 in arithmetioni progression. sam to n terms the following series

37

51 2n +1 2n +1 8. Prove that

1) cosh ka-sinh 20 = 1

(2) cosh' (2-6) = coshi a cosh B-sinh a sinh B. 9. Find the equation of in straight line, in arig form ; 'And show how 30 the equation in this form can be best used for iracing the straight line.'

Find the equations of the sides of the triangle whose ungulir points are (1, 2), (2.3), anii (-3,-5) 10. Find the eqnation of the cir•le in its simplest form.

40 Find the eqnation of the circle, pussing through the points (9, 4), (0,3), (6,-4); "nd show th:t its radius is 5:1 aprroximately. 11. Prove that the polar of any point on ihe circle

web + % – 2a6-342=0. with respect to the oiccio

902 + y2 + 200 - 3u? =(.. will toach the parabola

y2 + 440 = 0. Draw the oiroles and the paçabola, and slow in the figare one of the poles and the corresponding prior 12. Find the equation of the ellipse in the form

30

-1=0.

Find the eqnation of tha noints in which the straight line 52+ 2y = 9 meets the ellipse 2584 +2y2 = 100, and the length intercopted.

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HATHEMATICS.

THIRD PAPER

STATICS AND DYNAMIC6.

Examiner-C. LITTLE, Esq., M.A.

The Agures in the margin indicate full marks.

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1. Bxpláin' bow aniform and uniformly accrlerated motion are, measurer).

Prove that when a point is moying with uniformly increasing velocity 46 its average pulocity for any given interval is equal to the velocity at the middle instant of that interval.

Hence or otherwise find the space passed over by a falling body in time t.

2. A point has 82 nnits of acceleration : it has at a certain point a 30 oert in velocito in the Omlosite dir .ction to its acceleration : it retorns to that point after an interval of 4 grconds; find how far it travels from that point and its velocitu at that point

8 Find the ringe the time of flight nud the greatest height of a 25 particle projected from a point in a horizontal pline.

4 From the font of an inclined plane whose rise in 7 in 25 A shot is pro- 30 jerted with a prlo•ity of 600 ft. ver we oond at an angle of 30' with the horizontal, (l) up the plane, (2) down the plane Find the range in each

5. Ronanciute Newton's First Law of Motion ; and state whit infer. 80 ence it on blog us 1.o draw in the case , if mwiform: umorior: in a circle.

6. Find the resultant of two parallel forc • scting on a rigid body. 40 A Quiforın beam 4 feet long. 18. ,map: vortei in at b. rizontal position by two prope, which are three feet aport. so that the beam projects one.. foot bevond one of the propx : show that the pressure on one prop is double that on the other.

7. Find the centre of muss of a triangulnr la vina of nniform density. 30

8 Find the conditions of equilibrium of a coplanır system of 30 forces.

9. A uniform ladder reste in limiting eqnilibrinm with one end, on a 40 rough floor, whose coefficient, of friition, in ys and with the other againsta smoth vertical wall': 'show that its inclination to the vertical is tonn - 1:2

10 Find the relation between the power und the wright in the case 30 of a girew, when frirtion is taken inio ucr.onnt.

11. A smooth sphere of R8 m impingen directly with velocity-u on 10 another smoor.no spiere, of mano mo moring in the same direction with velocity u'. If the spheres be perfecıly els-tic, find their velocities after impact.

iz, State the principle of work and energy : and show that it holds 20 in the case of a body falling under gravity.

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MATHEMATICS.

FOURTH PAPER.

DIFFERENTIAL AND INTEGRAL CALCULUS.

Eoaminer---. LITTLE, Es, M.A.

The figures in the margin indicate, full marks.

1. From the definition of a differential coefficient dednoe.its value for .45 the following fanctions :sin ,, 'exand im

19

;.. Differentiate

(1). sin:(1+x2).

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dx4

Apply Leibnitz's theorem to the above expression to find the relation

dty connecting and lower differential co-efficients of y with respect to's.

4. Enunciate Taylor's theorem, and by means of it expand logela"**):40 in ascending powers of .

ole) 5. Prove that if : (a)=0 and '\(a)=0, the limiting value of

*(*) i Olfal). when x is equal to a is

4/9) Find the value of

2-3-3002 + 1

when a = 1. 3x6-503 + 2

30

1

30

6. If y=f(x), show how maximum and minimum values of 3 can be determined and identified. Illustrate the theory by means of a diagram.

If f(x)=509 + 12x6, 1594.-4003 + 150? +600 + 17, find whether the values of the function corresponding to a values t1 and 2 are maximum or minimum values or neither.

7. Find the equations of the tangent and normal to the curve y-f(). 40 - Find the subtangent and sabnormal in the case of the parabola y2-4ax. · Trace the cprve aly.=(b). 8. Explain the connection between the series

40 +

ho{a+n-in}

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9. Prove the following formula for integration by parts :

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du

da. dwo

Apply it, to find the value of the integral

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30

,,11. Find the moment of inertią of a semi-circular lamina about an extremity of the bounding diameter.

Write down the moment of inertia,of a cirçalar lamina about a point in the circumference.

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· GEODESY,
Examiner. E. R. GARDINER, ESQ.

The figures in the margin indicate full marks. 1. Construct a scale of 2 miles to the inch to read furlongs. What is 20 its representative fraction ?

2. Dengrike the work of a road burmay witkai x prismátio compass; wid 30 note the checks you would apply to your work.

08. Pre survey of a large plőt is to be done with the chafi. Descrive 30 the process, and detail all the steps you would take to ensure its acouracy:

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