CONTROL PAPERS.

LIV.

1. Reduce 4 tons 7 cwt. 1 qr. 13 lbs. 12 oz., to ounces. 2. If 13 lbs. of sugar cost 4s. 10d., what will be the cost of 213 lbs.

3. Find (by Practice) the cost of 63 cwt. at £5 12s. 6d. a

ton.

4. Find the simple interest on £106 13s. 4d. for 15 months at 4 per cent. per annum.

9. Add together 84.006, 002604, 845 and 70.0015603. 10. Subtract 73.4698 from 108.30125.

11. Multiply 302-468 by 400.

12. Divide 78 by 361.059 to 4 places.

13. Reduce 3.8375 ac. to roods and perches.

14. In 164723 pints how many quarters, bushels, pecks, &c.

15. If 15 men reap 20 ac. in 6 days, working 14 hours a day, how many men must be employed to assist 10 other men to reap 6 ac. in 12 days of 8 hours each?

16. Find (by Practice) the dividend on £721 13s. 6d. at 14s. 2d. in the pound.

17. Find the amount of £8900 in 3 years at 2 per cent. compound interest, neglecting fractions of a penny. 18. Add together 14, 33, 155, 27.

22. Add together 84 6912, 001567, 1-0056, 549.2.

23. Subtract 23.69428 from 50.012.

24. Multiply 40-061 by 0054.

25. Divide 055757592 by 009207.

26. Reduce 33 yds. to the decimal of a mile.

27. A person takes a railway return-ticket for a month, paying 25 per cent. for it more than he would have paid for a single ticket. At the end of the month he obtains an extension of the time for a week by paying 5 per cent. on the monthly ticket. The whole sum paid is £2 12s. 6d. ; find the price of the single ticket.

28. Extract the cube root of 9555119848.

29. When the three per cents. are at 87, and shares paying 5 per cent. are at 1304, which is the more profitable investment? What sum does a person invest when the difference of the incomes arising from the two investments is £561 ? 30. Reduce 2671875 to a vulgar fraction, and if the unit be £6 reduce the fraction to shillings, pence and decimals of a penny.

31. Find the prime factors of 111540, 42336, 67392: and thence write down the least number which they will all divide without a remainder.

32. If the weight of 1 cubic ft. of water is 62.35 lbs. avoirdupois, find the error in calculating the weight of 1000 cubic ft. on the approximate assumption that 1 cubic ft. weighs 1000 oz.

LV.

1. If the angles at the base of an isosceles triangle be bisected, the angle contained by the bisecting lines is equal to the exterior angle at the base.

2. Divide a circle into two segments such that the angle contained in one shall be five times that contained in the other.

3. Show that the straight lines joining the centres of the escribed circles of a triangle pass through its angular points.

4. If O is the centre of the circle inscribed in a triangle ABC, and P the centre of the circle which touches the side BC and the other two sides produced, show that the angles OBP, OCP are right angles.

5. (1.) Divide x6+4x-3x-16x3+2x2+x+3 by x3+ 4x2+2x+1.

(2.) a +√ab+b by √a+1√ ab+√ō.

1

1

−3a)+

6. (1.) Simplify (x—a)(x—2a)(x—a) (x—3a)

7. Find the L.C.M. of

x2+5x+4, x2+2x−8, x2+7x+12.

8. Solve the equations:

(1.) (x+1)2=(x+2) √x2+2.

(3.) x(x+1)+3√√/2x2+6x+5=25-2x.

9. Sum the series :

(1.) 2—22+23—24+ to 32 terms.

4 8

(2.) 22+ +to infinity.

3 9 27

(3.) The series whose rth term is 2r-1 to 10 terms.

10. Write down the first 4 terms and the middle term of

2n+1

the expansion of (1+x) ; show that the difference of the

coefficients of x

r+1

and x in this expansion is equal to the

difference between the coefficients of x

pansion of (1+x)".

r+1

r-1

and x

in the ex

11. Define the cotangent of an angle and trace its change in sign and magnitude as the angle increases from 0° to 360°.

Find the value of cot 30°.

12. Prove the following relations:

(1.) Cos (A-B)=cos A cos B+ sin A sin B.

Ex. sin 210°=-1, find sin 105°.

14. A man wishing to find the distance between two signal staffs which are inaccessible, plants a staff on a spot from which one of the staves eclipses the other: he next measures 100 yds. in a line perpendicular to that which passes through the staves, and then observes that the angles between the staff which he has planted, and the two signals are 45° and 60° respectively. Find the distance required. 15. Given the three sides of a triangle; find the expressions for the radii of the inscribed and circumscribed circles. b2+c2-a2

16. Assuming cos A="

2bc

prove that

greatest angle in the triangle whose sides are 5, 6, 7 ft. respectively. Having given

log 6=7781514 L cos 39°14' 9.8890644 and the difference for 60"-1032.

LVI.

1. Reduce 5 tons 10 cwt. 3 qrs. 14 lbs. 7 oz. to ounces.

2. The proportion of the diameter of a circle to its circumference is very nearly 113 to 355. Find the circumference of the circle whose diameter is 132 ft.

3. Find (by Practice) the cost of 157 cwt. at £2 16s. 8d. per cwt.

4. Find the simple interest on £533 6s. 8d. for 146 days 3 per cent. per annum.

1 13 5

5. Add together 53⁄4, 428, 21, 96•

6. Subtract 6 from 103.

9. Add together 216-005, 002604, 845, 6914.02.

10. Subtract 36.978 from 500-32105.

11. Multiply 216-79 by 750.

12. Divide 34.015 by 701.5 to 4 places.

13. Reduce 1.0165 ac. to square yards.

14. In 237164 ft., how many miles, furlongs, poles, &c? 15. How long will 40 men take to build a wall 10 ft. high, if 11 men require 17 days to build one of the same length, but only 7 ft. high?

16. Find (by Practice) the dividend on £1816 13s. 4d. at 15s. 33d. in the pound.

17. Find the compound interest on £4500 in 3 yrs. at 6 per cent. compound interest, neglecting fractions of a penny.

22. Add together 38.9126, 005169, 3·0012. 648.03.

23. Subtract 623-92842 from 1005. 6123.

24. Multiply 319-12 by 0016.

25. Divide 485976·5 by 20·165.