6. If y varies as the sum of two quantities, one of which varies as x, and the other varies as x2, and if when x= 1,y=2, and when x=2, y=3, find the equation between x and y.

7. Find a number which when multiplied by 9 exceeds 50 as much as 50 exceeds the original number.

8. A starts from a place C, and travels towards a place D at the rate of six miles an hour. Two hours afterwards B starts also from C, and, travelling 10 miles an hour, arrives at D four hours before A. Find the distance between C and D.

10. A person spends £20. 8s. in purchasing a certain number of bottles of sherry and claret. He pays as many shillings per dozen for each kind of wine as he buys bottles of that wine. If he had paid as many shillings per dozen for each as he bought bottles of the other he would have spent £18. How many bottles of each did he buy?

11. Define arithmetic, geometric, and harmonic progression.

Insert two arithmetic means, two geometric means, and two harmonic means between 24 and 81.

12. Having given the first and second terms of an arithmetical progression, find the nth term.

1

In an arithmetical progression the first term is 2'

and the

constant difference Find the number of terms whose sum is 48.

1 3'

13. Assuming the Binomial Theorem for a positive integral index, prove the theorem for a positive fractional index.

If a be a positive integer, find the sum of the coefficients in the expansion of (1+x)", and prove that the sum of the squares of

these coefficients is equal to

2n

INDIAN PUBLIC
WORKS AND

TELEGRAPH
DEPARTMENTS,

1878.

INDIAN PUBLIC 14. Find the sums of each of the following series, continued to infinity:

WORKS AND

TELEGRAPH

DEPARTMENTS, 1878.

Tuesday, 16th July 1878. 2 P.M. to 5 P.M.

1. If two triangles have two sides of the one equal to two sides of the other each to each, but the angle contained by the two sides of the one greater than the angle contained by the two sides equal to them of the other, the base of that which has the greater angle shall be greater than the base of the other.

2. If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles.

3. Triangles upon the same base and between the same parallels are equal to one another.

Let ABC, ABD, be two equal triangles upon the same base AB, and on opposite sides of it; if CD be joined meeting AB in E, show that CE=DE.

4. A straight line BC is divided in G, and on BC and BG on opposite sides of BC are described the squares ABCD, BEFG. A point P is taken between A and B, so that E P=AB, and PF is joined. The parallelogram FPDN is drawn: show that it is a square equal to the sum of the other two, and that the triangles PAD, DCN, are equal in all respects.

5. If a straight line be bisected and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected, and of the line made up of the half and the part produced.

6. The angle at the centre of a circle is double of the angle at the circumference on the same base.

If a chord be drawn through one of the two points of intersection of two circles, to cut the two circles, its whole length will subtend a constant angle at the other point of intersection. 7. If from any point without a circle two straight lines be drawn, one of which cuts the circle and the other meets it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it.

8. Inscribe a circle in a given square.

If a straight line is drawn touching the circle at any point, show that the segment intercepted upon it by any two opposite sides of the square subtends a right angle at the centre of the circle.

9. Describe a square which shall be equal to a given rectangle.
10. Find a mean proportional between two given straight lines.

WORKS AND
TELEGRAPH

11. If the vertical angle of a triangle be bisected by a straight line INDIAN PUBLIC which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the DEPARTMENTS, segments of the base, together with the square on the line that bisects the base.

1878.

5. Add together 27.170, 002090, 381 9340, and ⚫0721653. 6. Subtract 376.9429 from 810 0371.

7. Multiply 31 079 by 00426.

8. Divide 75 049 by 251.

9. Express 15s. 8d. as the decimal of 17. 3s. 4d.

14. Add together 5·031 of a mile and 375 of a furlong, and give the answer in yards.

15. Subtract 11 079 of an ounce from 3 101 of a pound troy.

16. Multiply 9.20407 by 0927.

17. Divide 36 154 by 0073 to three places of decimals.

18. Express 2.07831 cwt. as cwts. qrs. lbs. ozs.

N.B.-The first eighteen questions should be answered before the others are attempted.

Use a separate book for these questions.

19. Multiply by duodecimals 3 ft. 9 in. 3 pts. by 4 ft. 6 ins. 7 pts., and express the product in square feet, square inches, and the fraction of a square inch.

20. The sides of an oblong field containing 6 acres are as 4 to 3. Find, within an inch, the length of the diagonal.

12

169

21. Find the square root of 501 and the cube root of 629 422793.
22. A cubical block of stone weighs 325 tons 8 cwt. 3 qrs. 20 lbs., and
each cubic foot weighs 125 lbs.; what are the dimensions of
the block?

23. If I lend a friend 4007. for 9 months when the current rate of
interest is 3 per cent., for how long ought he to lend me in
return 2201., when the rate of interest is 4 per cent.?

24. A rectangular grass plot is 20 yards long and 15 yards wide and has a gravel path round it of uniform width.

The area of the grass is to that of the path as 6 to one. Find the width of the path.

I 447.

D D

INDIAN PUBLIC 25. The extreme length of an oblong tent with semi-circular ends is 60 yards and its width 20 yards. Find the number of square feet of ground covered by it.

WORKS AND TELEGRAPH DEPARTMENTS, 1878.

26. The diameter of the base of a circular cone is 3 feet, and its slant height 7 feet, find the area of the convex surface.

1. If the parallelogram of forces hold for the direction of the resultant of P and Q, and also of P and R, prove that it will hold for the direction of the resultant of P and Q+R.

Define the resolved part of a force in any direction. If the resolved part of P in the direction OA be m times that of Q in the same direction, and if the resolved part of P in another direction OB be m times that of Q in this other direction, prove that P=m Q.

2. Find the resultant of two parallel forces in the same direction.

A man seated in a boat, with its stern towards the shore, pulls the handle of each of a pair of sculls with the force P. Find the force with which a man on the shore must pull a horizontal chain fastened to the stern of the boat in order to keep it at rest, each scull being divided at the rowlock in the ratio of m : n. 3. Prove that every system of forces in one plane on a rigid body may be replaced by a couple and by a force acting through any point in the plane rigidly connected with the body.

If a system of three such forces be fully represented by the lines OA, OB, and OC, and if L, M, and N, be the couples according as the resultant force acts at A, B, or C respectively, prove that

L+M+N=0.

4. Find the centre of gravity of any system of heavy particles.

A body consists of two parts A and B. If the part B be moved while A remains fixed, prove that the centre of gravity of the whole body moves parallel to the line of motion of the centre of gravity of B.

5. If three forces in one plane keep a body at rest, prove that they must either be parallel or meet in one point.

The square ABCD is supported with the angle A in contact with a smooth wall by the string BE equal in length to AB attached to the point E vertically above A.

Prove that tan BAE =

1

3'

6. Find the power P which will support the weight W by means of a system of weightless pulleys each hanging by a separate string.

If the pulleys have each the weight w, and if P1 be the power required to support W in this case, prove that P1- -w will support W-w by means of a system of the same number of weightless pulleys.

DYNAMICS.

INDIAN PUBLIC
WORKS AND

1. Find the space described from rest in time t by a particle moving TELEGRAPH with a uniform acceleration.

16 25

A stone dropped from the top of a tower falls through of
the height in the last second of its full. Find the height of the
tower, g being 32.

2. A bird rises and flies horizontally with the velocity u at an angle of
135 with the line joining the bird and a sportsman, and at a height
from the ground equal to that of the sportsman's shoulder. In
what direction should the sportsman fire if the shot have a
velocity=(3+1) u, supposing the shot to move horizontally?
3. The path of a projectile is a parabola.

v

If the velocity and angle of projection of a projectile be v and respectively, prove that after the time the projectile will be moving perpendicularly to the direction of projection.

g sin a

4. Find the time of descent from rest down a smooth inclined plane, and the proportion in which the time is increased when the plane is rough, the coefficient of friction being μ.

5. Find the acceleration when two unequal weights are connected by a string which passes over a smooth pulley.

Two inclined planes AB and AC have a common altitude AD and are each inclined to the horizon at the angle 30°.

Any number of trucks each weighing 5 cwt., and each containing one ton of ballast, are placed upon the plane AB and connected by a rope passing over the point A with the same number of similar empty trucks on the plane AC.

If there be no friction, find the time of passing over the first 30 yards from rest.

6. Find the velocities of two spheres of given masses and elasticity
after direct impact.

Prove that in such a case there is, in general, loss of kinetic
energy.
What becomes of the energy thus lost?

DEPARTMENTS, 1878.

PLANE TRIGONOMETRY.

Thursday, 18th July 1878. 2 P.M. to 5 P.M.

1. Prove that the circumferences of circles vary as their radii. Explain the advantages of measuring angles by the English, French, and circular methods respectively.

The angles of a triangle are in geometrical progression, and the number of grades in one is to the circular measure of another as 50: ; find the two solutions.

2. Find the other five trigonometrical ratios of an angle in terms of

its co-tangent.

any

Prove that if two angles have the same sine, and also of the other five trigonometrical ratios (with one exception) the same, they will differ by a multiple of 2.