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6. Give the necessary formula for the complete solution of a triangle in which two sides and the included angle are given, so as to find first the remaining angles and afterwards the remaining side.

If b = 6, c = 4, A = 49° 22′ find B and C.

Given log 2 = .3010300, L cot 24° 41′ = 10·3376235.
I tan 23° 31′ = 9.6386473, L tan 23° 32′ = 9.6389925.

2. Mensuration and Surveying.

7. A rectangular building 48 feet long and 24 feet wide has a roof, which is symmetrical, with a ridge 24 feet long running lengthwise, and height above the wall-plates 9 feet: find the area of the roof.

8. A cask, 3 ft. 6 in. long, in the form of two conic frustums of equal height joined at their bases, has a bung-diameter of 24 inches and head-diameters of 18 inches each: find how many gallons, omitting fractions, it will contain, if a gallon be assumed to measure 277 cubic inches, and the ratio of the circumference of a circle to its diameter be 37.

9. Plan and find the area of a field from the following

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10. Find the equations of the straight lines which pass through the point (hk), and are respectively parallel and perpendicular to

a

y

+ = 1.

15. If the rim PinGpse be produce the atlary re in Q. shew that

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16. The fine joining the hers of an ellipse t the perpendienar from the other to the tangent a passes through the middle print of the normal

17. If the ordinate NP of a hyperbola be meet the asymptotes in & and r, prove that RP.

SATURDAY, JUNE 9, from 2 to 4.30

C. Mathematics.

[N.B. No credit will be given for any answer, t of which is not shewn.]

1. Trigonometry.

1. Investigate an expression for all the an a given tangent.

If tan (2a-3,3)=cot (3a-2,3), and ta: =cot (3a+28), then both a and 3 are multi;

2. Prove that

(1) tan 34-tan 24-tand = tan 3A ta (2) (sin A—cos A)2 + (sin A + cos A)* = 3. Solve the equations:

(1) 2sing-sin 2 x = 2(1 + cos x)2;
(2) tan-1(ax+b)+tan−1(ax—b) =

4. Find the radii of the escribed circles prove that

4

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0, 01, 02, 0, being the ce

circles.

5. Prove that in any

is 28

(1)

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11. Find the equation of a circle referred to polar coordinates.

A circle passes through two points on the initial line whose distances from the pole are a, ; and also through

c2

a

c2

two points on a line through the pole perpendicular to the initial line, whose distances from the pole are b, ; prove that the polar equation of the circle is

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b

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12. Find the condition that x cos 0 + y sin 0 = p should

22 y2
+ = 1.
b2

touch the ellipse a2

If P1, P2, be the perpendiculars from the centre on tangents at the extremities of a pair of conjugate diameters, prove that + is constant.

I

I

P12 P22

13. Obtain the equation to the hyperbola in the form xy = c2.

Shew that the equation to a chord of this hyperbola meeting it in the points (x,y), (272) is

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14. A line is drawn from the origin to meet the curve ax2+2hxy+by2+gx+fy + c = 0,

find the length of the intercept cut off by the curve; and hence deduce the equation to the tangents from the origin.

FRIDAY, JUNE 8, from 5.30 to 8 P.M.

C. Mathematics.

1. Mechanics and Mechanism.

1. Enunciate the parallelogram of forces; and, assuming it true for the direction, prove it for the magnitude of the resultant.

Forces are represented by the lines OA, OB, OC, OD: shew that their resultant is 40G, G being a point which remains fixed so long as A, B, C, D are fixed.

2. Find the centre of gravity (1) of particles placed at the angular points of a triangle, whose weights are proportional to the opposite sides, (2) of the portion of a square lamina from which a triangular piece has been cut off by a line through an angular point and the middle point of one of the opposite sides.

3. In the common balance, the weights being unequal, find the position in which it will rest. Hence determine the

stability and sensibility of the balance.

4. Find the relation between P and W in that system of pulleys where each pulley hangs by a separate string, neglecting the weight of the pulleys.

If in this system W be raised or depressed, shew that the centre of gravity of P and W remains in the same position.

5. Define acceleration.

How is it measured?

Prove that the space described by a body uniformly accelerated from rest is equal to half the space it would describe in the same time with the last acquired velocity continued uniform.

6. A heavy particle is projected from a given point with a given velocity, what is the nature of its path?

A particle after sliding down an inclined plane moves in a parabola of given latus rectum; find the length of the plane in order that the focus of the parabola may be vertically below the point from which the particle started.

7. Draw any two vertical circles touching each other at P, their highest point, and let a chord through P cut them in A and B. Let T and T2 be the times of descent from P to A and P to B respectively. Prove that T2-T2 is proportional to the difference of the diameters of the circles.

8. Describe exactly what takes place when two elastic bodies impinge upon one another.

An elastic particle drops from a height h on a smooth and hard plane of inclination a to the horizon: if e be the coefficient of elasticity, prove that the distance between the first and second points where the particle meets the plane is

4e(1+e)h sin a.

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