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Set to candidates for the Admiralty who selected Algebra as a subject
No. 1. 1. Find (1) the sum, (2) the difference, of the two expressions1
#4 -5.68 +4x3+3x+9 (a)
4x3 – 15x2 +8x+3 1 - XN2x + x2
+ (B) N
1-x+ 2x+x2 1- x + 2x + x2 1 - x - 2x + x2 3. Prove the rules for pointing in the multiplication and division of
decimals. 4. Solve the following equations
+y = 5. Find a number of two digits, which is three times the sum of its
digits, and such that the difference between the digits is 5. 6. If any number of fractions be equal, show that each of them
sum of any multiples of the numerators
sum of the same multiples of the denominators
7. Prove that a'=1, and a”="/ā.
Is any assumption necessary in order that this may be true? 8. Investigate a rule for finding the sum of n terms of an arith
In the series of }-+*-&c., find s (1) when n=5,
(2) when n = infinity. 9. When does one quantity vary (1) directly, (2) inversely, as
If x varies as y, prove that 22 + y2 will vary as x2—42. 10. Expand
in a series of ascending powers of x by means (a3 — 23) of the Binomial theorem, writing down the first 4 terms and
the rth term. 11. Express the numbers 957 and 23.125 in the septenary scale.
2. Add together 4 (2-*-2y) and P7 (---)
aax - a2y + x3 1. Find the value of axy
when a=2, x=3, y=5. a– y2
5 3. Divide 4.25 – x3+4x by 2x2+3x+2. 4. Investigate a rule for finding the square root of an algebraical
quantity, and explain how the method of extracting the
square root of a numerical quantity may be deduced. 5. Find the square root of (a+b)2 - 2 +a+c)2 – 12+(6+c)2 - a?. 6. Find the greatest common measure of 4x4 – 1208 +5x2 + 14x-12
and 6x4 – 11x: +9x-13x+6. 7. Solve the equations
4 12 10
8. Investigate an expression for the sum of a geometrical series ;
and find the sum of 30 terms of the series 2+3+7+&c.
product diminished by 4, the digits are inverted; and if 10
as a subject of examination.
A. -PURE MATHEMATICS. 1. To a given straight line apply a parallelogram, which shall be
equal to a given triangle, and have one of its angles equal to
a given rectilineal angle. 2. A segment of a circle being given, describe the circle of which
it is the segment.
show that similar polygons may be divided into the same
another that the polygons have.
3x4 +14x3 +9x+2 5. Reduce to its lowest terms the expression
2x4 + 9x3+14x +3 6. Solve the following equations
y + 1 /
(2) ax2+bx+c=o. (1)
22 - 2x +3 ** + 2.1 +4
may be two positive integers.
sion of (a−b)an.
8. Find the sum of n terms of the following series
1 5 4 3 6 3 1 1 3
+ + + 3
+ 2 4
18+ 23 + 38 + 9. Given two sides of a triangle and the angle opposite one of
them: show how to solve the triangle, and point out when the case is ambiguous.
If a=5, b=7, and A=sin -14, is there ambiguity? 10. Given log 1071968=4.8571394; diff. for 1=60, find the value
of ♡0719686 to seven places of decimals. 11. If the three sides of a triangle are x2+x+1, 2x+1, and 22-1,
show that the greatest angle is 120°. 12. A right cone is cut by a plane which meets the cone on both
sides of the vertex; show that the section is a hyperbola. Under what condition is it possible to cut an equilateral
hyperbola from a given cone. 13. In an ellipse prove that the line drawn through the focus at
right angles to the focal distance intersects the tangent and the directrix in the same point.
B.-MIXED MATHEMATICS. 1. Assuming the truth of the parallelogram of forces as to direction,
prove its truth as to magnitude. 2. Given the sum of two forces and their resultant, and also the
angle which one of them makes with the resultant. Determine
the forces, and the angle at which they act. 3. State the result of any experiments made with reference to
friction. A body weighing 12,000 tons, placed on a plane whose inclina
tion is 1 in 12, and acted on by two chains (each capable of sustaining a strain of 200 tons) in the direction of the plane, is just on the point of moving when the chains break. Find
the coefficient of friction between the body and the plane. 4. An area is cut from one angle of a triangle equal to half the area
of the triangle by a line parallel to the base. Find the centre of gravity of the remainder.
5. Enunciate the first and second laws of motion, and mention any
experiments which seem to suggest their truth. How is their
truth finally established? 6. A body moving uniformly in a straight line is suddenly acted on
by a constant force always acting in a given direction.
Determine the subsequent motion. 7. A body of given elasticity is projected vertically upwards with a
given velocity, and strikes against a horizontal plane. Deter
mine the velocity with which it reaches the ground. 8. Find the line of quickest descent from the focus of a parabola,
whose axis is vertical and vertex upwards, to the curve. 9. Define “specific gravity," and show that when a solid is im
mersed in a fluid, the weight lost is to the whole weight of the
body as the specific gravity of the fluid is to that of the solid. 10. Explain the principle of the hydraulic press, and find the mecha
nical power in a machine of given dimensions. 11. A particle moves in a circle under the action of a central force
resident in an external point. Find the law of force. Is the force attractive or repulsive?
QUESTIONS IN PRACTICAL GEOMETRY,
BUILDERS' WORK, &c. Set to Candidates for the Situation of Clerk of the Works in the
Engineering Branch of the War Department. A.-PRACTICAL GEOMETRY AND MENSURATION. 1. How much paper, 4 yard wide, will be required for a room that
is 22 feet long, 14 feet wide, and 9 feet high, if there be 3
windows and 2 doors, each 6 feet by 3 feet? 2. How many square feet are contained in a plank whose length is
10 feet 10 inches, and breadth at the two ends 34 feet and
21 feet? 3. What would be the cost of paving a semicircular alcove with
marble at 2s. 6d. a foot, if the length of its semicircular arc .
was 22:42 feet? 4. A stone 18 inches long, 17 broad, and 7 deep, weighs 278 lbs. how many
cubic feet of this kind of stone will freight a vessel of 230 tons burthen?