Solve the equations: \begin{align*} y^2+z^2-x(y+z) &= a \\ z^2+x^2-y(z+x) &= b \\ x^2+y^2-z(x+y) &= c \end{align*}
Determine the range of values for which the two infinite series \[ 1+x+\frac{x^2}{2!}+\dots+\frac{x^n}{n!}, \] \[ x-\frac{x^2}{2}+\frac{x^3}{3}-\dots+(-1)^{n-1}\frac{x^n}{n} \] are respectively convergent. Prove that in the approximate formula used by Napier for calculating logarithms \[ \log_e \frac{a}{b} = \frac{a-b}{2}\left(\frac{1}{a}+\frac{1}{b}\right), \] where \(\dfrac{a-b}{a}\) is small, the error is of the order \(\dfrac{1}{6}\left(\dfrac{a-b}{a}\right)^3\).
Explain the method by which the \(n\)th term and the sum of \(n\) terms of a recurring series \(u_1+u_2+u_3+\dots\) of which the scale of relation is \(u_n+pu_{n-1}+qu_{n-2}=0\), can be found when the first four terms of the series are known. Apply the method to the two recurring series
The base \(a\) of a triangle and the ratio \(r(<1)\) of the sides are given. Prove, geometrically or otherwise, that the altitude \(h\) cannot exceed \(ar/(1-r^2)\). Prove also that, when \(h\) has this critical value, the vertical angle of the triangle is \(\frac{\pi}{2}-2\tan^{-1}r\).
Express the radii of the inscribed and escribed circles of a triangle in terms of the radius of the circumcircle and the angles of the triangle. Shew that two circles can be drawn within the angle \(BAC\) of a triangle to touch \(AB\) and \(AC\) and to cut off a chord of given length from \(BC\), provided the given length does not exceed twice the geometric mean of the radii of the inscribed circle and the escribed circle opposite \(A\).
Shew how to determine the four fourth roots of a complex expression of the form \(a+ib\).
Find \(\dfrac{dy}{dx}\) in the case where (i) \(y = \sin^{-1}\left(\dfrac{b+a\cos x}{a+b\cos x}\right)\), (ii) \(y=x^{x^{\sin x}}\), (iii) \(y(y^2+x^2)=x+y\).
A conical vessel is being filled with water at the rate of 2 cubic ft. per second; the semi-vertical angle of the cone (whose axis is vertical) is \(30^\circ\). Find the rate of rise (in feet per second) of the water level when the level above the vertex is 9 feet.
Shew how to find the points of inflexion of the curve \(y=f(x)\). Find the maximum point and the inflexions of the curve \(y=x e^{-x}\), and trace the curve.
Integrate: \(\int x \sin x dx\), \(\int \frac{(x+1)dx}{x^2+x+1}\), \(\int \sin^2 x \cos^3 x dx\). Find by integration the area common to the parabola \(y^2=ax\) and the circle \(x^2+y^2=4a^2\).