INDIAN CIVIL ENGINEERING COLLEGE

EXAMINATION.

July, 1873.

ARITHMETIC AND MENSURATION.

1. Add together 5, 3, 5, 1, and .

2. Subtract 87 from 101.

3. Multiply together 51, 2, 4, 116, and 2.

4. Divide 7 by 91.

5. Add together 19.735, 000786, 4732-02, and

*375799.

6. Subtract 876.93387 from 974.3216.

7. Multiply 9.238 by 65'4.

8. Divide 2.890721925 by 03645.

9. Express 2s. 71⁄2d. as the decimal of £7.

10. Add together 4, 71, 91%, 13%.

11. Subtract 3411 from 100%.

12. Multiply together 83, 4534, 115, 1, and 21.

13. Divide 84 by 54.

14. Add together 145 of a furlong and 7.36 of a yard, and give the answer in feet and the decimal fraction of a foot.

15. Subtract 2·32 of an hour from 325 of a week.

16. Multiply 380·72 by 0725.

17. Divide 7036 by 73 to 4 places of decimals.

18. Express 1 cwt. 1 qr. 21 lbs. as the decimal of four tons and a half.

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

19. Find (by Practice) the dividend on £731. 14s. 6d. at 14s. 2d. in the pound.

20. In 5,462,764 square feet how many acres, roods, &c. are there?

21. If 40 men can mow a field of 19 acres in 81 days of ten hours each, how many acres can 17 men mow in 50 days of 8 hours each?

22. Find the cost of painting the 4 walls of a room at 9d. a square yard, the length of the room being 20 ft. 7 in., the breadth 15 ft. 4 in., and the height 12 ft. 4 in.

23. Multiply by duodecimals 8 ft. 2 in. 4 pts. by 4 ft. 6 in. 9 pts. and the product by 3 ft. 7 in. What does the product become when expressed in cubic feet, cubic inches, and a fraction of a cubic inch?

24. A cubic foot of a certain substance weighs 54 lbs. avoirdupois; find the length of the side of a cube of another substance which weighs 91 lbs. 8 oz. 13 drs., whose specific gravity is to that of the former as 3 to 2.

25. The area of a rectangular piece of ground is 28 acres and 22 perches. What is its length, its breadth being 625 links?

26. Find the number of gallons of water which pass in 10 minutes under a bridge 17 ft. 8 in. wide, the stream being 10 ft. 11 in. deep, and its velocity 8 miles an hour. [A gallon contains 277-72 cubic inches].

27. The diameters of the top and bottom of a frustum of a cone are 18 inches and 27 inches respectively, and the height is 30 inches; find its volume.

28. If the weight of one cubic foot of water is 62.35 pounds avoirdupois, find the error in calculating the weight of 1,000 cubic feet on the approximate assumption that one cubic foot weighs 1,000 oz.

29. Two opposite angles of a quadrilateral field are together equal to two right angles, and the sides measure respectively 4 chains, 3 chains 20 links, 2 chains 40 links, and 1 chain 90 links; find its area in acres, roods, perches, &c.

EUCLID, Algebra, and TRIGONOMETRY (1).

1. Draw from a given point a straight line equal to a given straight line.

2. Prove that if a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles.

Trisect a given finite straight line.

3. If a straight line be divided into two equal and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section.

4. If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.

Describe a circle of given radius which shall touch a given circle and a given straight line.

and

5. Resolve into factors the expressions:

(a+b-c) - (a-2b+ 2c), x2 - 19x + 88,

x* + x2y2+y*.

2n-1

6. If n be a positive integer, prove that "+1 is divisible by x+1, and that x2 + x2-1+1 is divisible by x+x+1.

7. Determine the highest common factor of

3x3- 4x2-x-6 and 2x-5x3 + 13x2 - 13x-18. Find the relation between a, b, c, and d, when the expressions x ax + b and x2 + cx+d have a common factor.

8. Solve the equations:

(1) (x−a)3+(x−b)3+(x−c)3=3(x− a)(x—b)(x − c). (2) 16x2 - 192x + 551 = 0.

9. Find an expression for the sum of n terms of an arithmetic series, having given the first two terms. Sum the series:

and

1 + 4 + 7 + 10+ ... to n terms,

1+2+3+...+n3,

22 × 1 + 32 × 2 + 4* × 3 +...+n2 × (n − 1).

10. Define the tangent of an angle, and trace the variations in its sign and magnitude as the angle increases from zero to four right angles.

Trace the changes in sign of the expression +tan 30 tan 40, as 0 changes from 0 to π.

11. Prove by means of geometrical figures the formulæ :

cos (AB) = cos A cos B+ sin A sin B,

and

tan (A+B)

=

tan A+ tan B

1- tan A tan B'

12. Find an expression for all the angles which have a given tangent.

Having given tan A, find tan

A

and explain, by

means of a figure, why two values are obtained. Prove that tan (37° 30′) = √/ (6) + √(3) − √(2) − 2.

13. Find expressions, in terms of the sides of a triangle for the radiir and R, of the inscribed and circumscribed circles.

Prove that the area of the triangle is equal to

Rr (sin A + sin B+ sin C).

Prove also that the distance between the centres of the inscribed and circumscribed circles is equal to

EUCLID, ALGEBRA, AND TRIGONOMETRY (2). 1. Inscribe a circle in a given triangle.

Also describe a circle touching one side and the other two sides produced.

Two sides of a triangle of given perimeter are given in position, prove that the third side always touches

a certain circle.

2. Inscribe a circle in a given equilateral and equiangular pentagon.

3. If the exterior angle of a triangle, made by producing one of its sides, be bisected by a straight line which also cuts the base produced, the segments between the dividing straight line and the extremities of the base shall have the same ratio which the other sides of the triangle have to one another.

Find the locus of a point at which two given circles subtend equal angles.