# University of St Andrews Mathematics Examinations 1884-5

At the University of St Andrews two Mathematics papers, one on Geometry and Trigonometry, the other on Algebra and Coordinate Geometry, were set in October 1884 and two papers on the same topics in April 1885.

In addition, two Honours Mathematics papers were set in 1884.

We give only a sample of these questions, omitting the questions that contain mathematical symbols which make them hard to display on the web.

#### MATHEMATICS: OCTOBER 1884. First Paper.

GEOMETRY AND TRIGONOMETRY.

(Few questions well answered will be more valuable than a greater number imperfectly answered.)
1. Prove that parallelograms on the same base and between the same parallels are equal to one another.

Equal parallelograms on the same base and on the same side of it are between the same parallels.
2. Divide a straight line into two parts, so that the rectangle contained by the whole line and one of the parts may be equal to the square on the other part.
3. Divide a given straight line into two parts such that the square on one of them may be double the square on the other.
4. Prove that if, from any point without a circle, two straight lines be drawn, one of which cuts the circle and the other touches it, the rectangle contained by the whole line which cuts the circle and the part of it without the circle is equal to the square on the line which touches it.

Show how to describe a circle which shall pass through two given points and touch a given circle.
5. Inscribe a circle in a given triangle.

Show that the lines joining the centres of the escribed circles pass through the angular points of the original triangle.
6. Prove that equal parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional.
7. Prove that every solid angle is contained by plane angles which are together less than four right angles; hence show that there can only be five regular solids.
8. Define a parabola, and prove that the locus of the feet of perpendiculars from the focus on the tangent is the tangent at the vertex.
9. Prove that $\cos (A - B) = \cos A \cos B + \sin A \sin B$, when $A$ and $B$ are each less than 90° and $B$ less than $A$.

Draw the figure applicable to the proof of the same proposition when $A$ lies between 180° and 270°, and $B$ is less than 45°.
10. Find the numerical values of sin 45° and sin 22.5°, and express the latter to four places of decimals.
11. Simplify$\cos^{4}A + 2 \sin^{2}A \cos^{2}A$.

Prove $2\tan 2A/(1+\tan^{2}A) = \sin 2A$

and simplify
$(\sin 3A - \sin A)/(\cos 3A + \cos A) + (\sin 3A + \sin A)/(\cos 3A - \cos A)$.
12. If $\cos 3A +\sin 3A = 1/√2$, prove that $A = n.120° + 15° ± 20°$, when $n$ is any integer.
13. Prove that in any triangle-
$\sin \large\frac{A}{2}\normalsize = √((s - b)(s - c)/bc)$
where $a, b, c$ are the sides opposite the angles $A, B, C$ respectively, and where $2s = a + b + c$.
14. Prove that in any triangle-
$(\sin A)/a = (\sin B)/b = (\sin C)/c$
15. The elevation of a tower is observed to be α from a point in the horizontal plane of its base, and to be β from another point a feet distant from the former in a direct line towards the tower. Find the distance of the tower from the first point.

#### MATHEMATICS: OCTOBER 1884. Second Paper

ALGEBRA AND CO-ORDINATE GEOMETRY.

1. Multiply $x^{4} + 4x^{2}(x^{2} - 2ax + a^{2}) - a^{4}$ by $x^{2} + 2x(a + x) + a^{2}$, and divide the first expression by $x - a$.
2. Simplify the following-

(1.) $(x^{3} + 2x^{2} + 2x + 1)/(x^{3} - 2x - 1)$.

(2.) $(a/(a + b) + b/(a - b))/(a/(a - b) - b/(a + b))$.

(3.) $(1/2a + 1/(2a - x))(1/3a - 1/(3a - x) - (x^{2} - 4ax)/(6a^{2}(2a - x)(3a - x))$.
3. Define $x^{n}$ in its original and extended significations, and from your definition prove that $(x^{m})^{n} = x^{mn}$.
4. Divide$-2a^{-8}x^{5} - 17a^{-4}x^{6} - 5x^{7} - 24a^{4}x^{8}$ by $2a^{-3}x^{3} - 3ax^{4}$.
5. Solve the following equations

(1.) $\large\frac{1}{2}\normalsize (x - 1) + \large\frac{1}{3}\normalsize (x - 1) + \large\frac{1}{4}\normalsize (x - 1) = 1$.

(2.) $√(x - 16) + √x = 8$.

(3.) $x/a - y/b = c, x/m + y/n = d$.

(4.) $\large\frac{1}{4}\normalsize (2x - 4) = 3 - (9 - x)/(x - 3)$.

(5.) $√(x + 1) - 2√(√(x + 1)) = 4$.
6. Prove the binomial theorem for positive integral exponents.
7. Write down the first four terms in the expansion in ascending powers of $x$ of the expressions-

(1.) $(2 - 3x)^{5}$.

(2.) $(1 + 2x)^{-3}$.

(3.) $(1 - x)^{1/2}$.
8. Show how to find the sum of $n$ terms of a geometrical series.

Explain what is meant by the sum of an infinite number of terms of such a series.

Sum to $n$ terms the series-

(1.) $1 + 2a + 4a^{2} + 8a^{3} + etc.$

(2.) $1 - 2a + 4a^{2} - 8a^{3} + etc.$
9. Prove that $\log_{b} N = \log_{c} N . \log_{b} c$.
10. Given that $\log_{10} 2 = 0.3010300$ and $\log_{10} 7 = 0.8450980$, find the numerical value of $\log_{10} 5$, $\log_{2} 5$, l$og_{10} 14$, $\log_{2} 8$.
11. A number, consisting of three digits, is such that when the digits are reversed, and the number thus formed subtracted from the original number the remainder is 198: the sum of the digits is 6, and they are in arithmetical progression. Find the number.
12. A person bought a certain number of oxen for 80 guineas, and if he had bought 4 more for the same sum, they would have cost a guinea apiece less. How many oxen did he buy?
13. Mark in a diagram the positions of the points (-1, 2), (-2, 1), and find the equation to the line joining these two points.
14. Find the equation to a circle of radius $a$ passing through the origin, and having its centre situated on the axis of $x$.

Find the values of $x$ and $y$, where this circle is cut by the straight line $y = mx$.
15. Draw the figure represented by the following equations, and obtain the equation to the straight line drawn from the intersection of the first two perpendicular to the third:-

(1.) $5x + 2y = 13$.

(2.) $3y - 4x = 7$.

(3.) $7y - 2x + 13 = 0$.

#### MATHEMATICS: APRIL 1885. First Paper.

GEOMETRY AND TRIGONOMETRY.

1. Give an axiom upon which the propositions respecting parallels are founded.

Hence prove that if a straight line falls on two parallel straight lines, it makes the two interior angles on the same side equal to two right angles.
2. To divide a straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part.

Show that in the figure there are other three straight lines divided in the required manner.
3. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.

If a circle be inscribed in a triangle, the straight lines which join the points of contact form an acute-angled triangle.
4. To inscribe a circle in a given triangle.

Give the construction for escribing a circle to a given triangle.

If the inscribed and escribed circles touch the base $BC$ of a triangle $ABC$ in $D$ and $G$ respectively, prove that $BG = CD$.
5. In a right-angled triangle if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another.

$ABC$ is a triangle; $AD$ is drawn perpendicular to $BC$, meeting it in $D$ (between $B$ and $C$); if $BA$ is a mean proportional between $BD$ and $BC$, the angle $BAC$ is a right angle.
6. Planes to which the same straight line is perpendicular are parallel.
7. Define a tangent, and show that the tangent to any one of the conic sections makes equal angles with the generating lines. Given a tangent to a parabola (at a given point) and the focus, find the directrix and vertex.
8. Prove for an acute angle $\sin^{2}A + \cos^{2}A = 1$. Given $\sin A = \large\frac{5}{13}\normalsize$ find $\tan A$ and $\sec A$.
9. Assuming $\sin (A + B) = \sin A \cos B + \cos A \sin B$, and $\cos (A+B) = \cos A \cos B-\sin A \sin B$ deduce values for $\sin 2A, \cos 2A, \tan 2A, \sin \large\frac{1}{2}\normalsize A, \cos \large\frac{1}{2}\normalsize A$, and tan $\large\frac{1}{2}\normalsize A$ in terms of the ratios of $A$.
10. Verify the formulae-
$1 - \tan^{2}A \tan^{2}B = (\cos^{2}B - \sin^{2}A)/( \cos^{2}A \cos^{2}B)$.

sin 2A = 2 tan A /(1 + tan2 A).
If $2 \tan^{2}\theta = \sec^{2}\theta$, find a general expression for θ.
11. In any triangle prove
$a/(\sin A) = b/(\sin B) = c/(\sin C)$.
Having given two angles and a side of a triangle, show how to find the other sides and angle.
12. The diagonals of a rhombus are $2a$ and $2b$; prove that the cosines of its angles are $±(a^{2} - b^{2})/(a^{2} + b^{2})$.
13. Define the cosine of an angle generally, and trace the variations of $\cos A$ as $A$ increases from 0° to 360°.
14. Define the two common units of angular measurement. Transform into equivalents in the other system angles which in either are respectively 20, 1/2, and π times the unit angle.

#### MATHEMATICS: APRIL 1885. Second Paper.

ALGEBRA AND CO-ORDINATE GEOMETRY.
1. Simplify $(x + y + z)(x + y - z)(x - y + z)(z + y - x)$, if $z^{2} =x^{2} + y^{2}$.
2. Divide $z^{2} + 1+ 1/z^{2}$ by $z - 1 + \large\frac{1}{z}\normalsize$.
3. Simplify-

(i) $(2x^{3} y + 2x y^{3} - (x^{2} + y^{2})^{2}/(x^{4} - y^{4})$.

(ii) $(1/2a + 1/(2a - x))(1/3a - 1/(3a - x)) - (x^{2} - 4ax)/(6a^{2}(2a - x)(3a - x))$.
4. Solve-

(i) $1/(x - 1) - 1/(x + 3) = 1/35$.

(ii) $a^{2}/x^{2} + y^{2}/b^{2} = 6$, and $a/x . b/y = 1$.

(iii) $x^{2} - y^{2} = 7$, and $x - y = 1$.

(iv) $x + y + z = 14, x^{2} + y^{2} + z^{2} = 84, xz = y^{2}$.

(v) $\large\frac{1}{2}\normalsize x^{3} + 7(x - 14√x) = 0$.
5. The sum of three numbers is $(p + 1)(q + 1) n$; the sum of the two larger is $p$ times the smallest, and the sum of the two smaller is $q$ times the largest: find the numbers.
6. If α and β be the roots of the equation ax2 + bx + c = 0, prove α + β = -b/a and αβ = c/a.

The roots of $8x^{2} - mx + 9 = 0$ are in the ratio of 2 : 1; find them.
7. Prove that the sum to $n$ terms of a G.P. whose first term is $a$ and common ratio $r$, is $a(r^{n} - 1)/(r - 1)$.

Hence deduce the sum to infinity when $r < 1$.

Explain clearly the meaning of the latter value.
8. If $a$ and $b$ are two unequal numbers, show that their geometrical mean is less than their arithmetical mean.

Find the sum of one of the following series (whose first three terms are given) to infinity, and of the other to $n$ terms:-

(i) $\large\frac{51}{15}\normalsize + \large\frac{14}{10}\normalsize - \large\frac{3}{5}\normalsize$.

(ii) $\large\frac{49}{15}\normalsize + \large\frac{14}{10}\normalsize + \large\frac{3}{5}\normalsize$.
9. In an election where every voter may vote for any number of candidates not greater than the number to be elected, there are six candidates and four members to be chosen. In how many ways may a man vote (i) When he may give not more than one vote to any candidate, whatever number he votes for? and (ii) When he may divide his four votes as he pleases - e.g., giving one to each of four, or two votes to one and one to each of two others?
10. Prove $\log_{a} b \times \log_{b} a = 1$.

State the advantages of taking the base 10 for a system of logarithms.

Given $log_10 3 = 0.4771<\latex>, find log_1000 243<\latex>.
11. Draw the figure whose vertices are the points (2, 6), (6, -2), (0, -8), and ( -4, 0); and determine by analytical methods its precise character.
12. What does the equation to any locus give? What do m$
13. and $c$ stand for in $y = mx + c$? Find the equation to the straight line through (5, -7), (i) parallel to 3x + 5y = 8 ; and (ii) perpendicular to the same line.
14. Prove that (2, 3) is the centre of one of the circles touching the lines $4x + 3y = 7, 5x + 12y = 20$, and $3x + 4y = 8$, and find the equation to the circle.
15. State two tests by which it can be proved that three straight lines are concurrent, and apply one of them to show that the line joining the middle points of two opposite sides of a parallelogram passes through the point of intersection of the diagonals.
16. Find the equation to the tangent to the circle $x^{2} + y^{2} = 10$ at the point (-3, 1).

#### HONOURS MATHEMATICS: 1884 First Paper.

1. Equilateral triangles are described on the three sides of any triangle, and the centres of the circles inscribed in these equilateral triangles are joined: prove that the triangle thus formed is equilateral.
2. Find the sum of the squares of the coefficients in the expansion of $(1 + x)^{n}$ when $n$ is a positive integer.
3. If $n$ be any prime number and $x$ any integer, prove that $x^{n}$ and $x$ when divided by $n$ will leave the same remainder.
4. Show that the following series is convergent, and find its sum:
$\Large\frac{1}{1.2}\normalsize + \Large\frac{1.3}{1.2.3.4}\normalsize + \Large\frac{1.3.5}{1.2.3.4.5.6}\normalsize + ...$, to infinity.
5. Define an ellipse, and from your definition deduce its equation in the form:-
$\Large\frac {x^2}{a^2}\normalsize + \Large\frac {y^2}{b^2}\normalsize = 1$.
6. Find the conditions that the equation
$ax^{2} + 2bxy + cy^{2} + dx + ey + f = 0$
should represent (1) a parabola; (2) two intersecting straight lines.
7. Solve either of the equations-
$\cos x \cos 3x = \cos 2x \cos 6x.$

$\sin 2x + \cos 2x + \sin x - \cos x = 0$.
8. State and prove Demoivre's theorem.

Exemplify it to find the three cube roots of √-1, and also to find expressions for all the values of $(a + b√-1)^{1/2}$.
9. Change the independent variables from $x$ and $y$ to $r$ and θ in the expression
$\Large\frac {d^2 V}{dx^2}\normalsize + \Large\frac {d^2 V}{dy^2}$, where $x = r \cos \theta; y = r \sin \theta.$
10. Integrate the expression $\int \sin^{4}\theta \cos^{3}\theta d\theta$.
11. Prove that $\int udv = uv - \int vdu$.

Show how to extend this theorem, and apply it to find the value of $\int e^{ax}\cos bx dx$.
12. Find the integral with respect to $x$ of the expression-
$\Large\frac 1 {(Ax + B)\sqrt{a + bx + cx^{2}}}$

### HONOURS MATHEMATICS: 1884 Second Paper

1. Show how to resolve $x^{m} - 1$ into real quadratic factors, where $m$ is any even integer.
2. Prove that in an equation with real coefficients, imaginary roots occur in pairs.
3. Prove that in any equation the number of positive roots cannot exceed the number of changes in the signs of the coefficients.

Prove that the equation $x^{7} - 3x^{4} - 4x^{2} + x - 1 = 0$ cannot have more than three real roots.
4. Find the positive root to 3 places of figures of $x^{3} - 6x - 13 = 0$.
5. If α = 0, β = 0, γ = 0, be the equations to the sides of a triangle $ABC$ opposite the angles $A, B, C$, prove that $\alpha \sin A - \beta \sin B$ is the equation to the straight line bisecting $AB$ from $C$.
6. Find the equation to the tangent to the hyperbola in terms of the tangent of the angle which it makes with the axis of $x$.

Hence show that the locus of the intersection of a tangent to an hyperbola with a perpendicular upon the tangent from the centre is $(x^{2} + y^{2})^{2} = (a^{2}x^{2} - b^{2}y^{2})$.
7. A floor is ruled with equidistant parallel lines; a rod shorter than the distance between each pair is thrown at random on the floor: find the chance of its falling on one of the lines.
8. Prove that the polar formula for the volume of a solid is
$\int \int \int r^{2} \sin \theta \, d\theta\, d\phi\, dr$,
and hence find the mass of a sphere whose density varies inversely as the square root of the distance from the centre.
9. Prove by quaternions that the bisectors of the sides of a triangle meet in a point which trisects them.
10. Find by quaternions the locus of a point the ratio of whose distances from two given points is constant.

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