Find the conditions that \(ax^2+2hxy+by^2+2gx+2fy+c\) should
Shew how to express \(ax^2+2bx+c\) and \(a'x^2+2b'x+c'\) simultaneously in the forms \(p(x-\alpha)^2+q(x-\beta)^2\) and \(p'(x-\alpha)^2+q'(x-\beta)^2\). Apply your method to \[ 7x^2-22x+28 \quad \text{and} \quad 27x^2-62x+68. \]
Prove that \[ \log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots. \] Sum the series \[ \sum_{n=0}^{n=\infty} \frac{(3n+2)}{n^2+3n+2}x^n. \]
Prove the rule for finding successive convergents of the continued fraction \[ \frac{1}{a_1+} \frac{1}{a_2+} \frac{1}{a_3+} \dots. \] In the recurring continued fraction \[ \frac{1}{1+} \frac{1}{2+} \frac{1}{3+} \frac{1}{1+} \frac{1}{2+} \frac{1}{3+} \dots, \] prove that \(3p_{3n}=2q_{3n}+q_{3n-3}\).
Prove Wilson's theorem that \((n-1)!+1\) is divisible by \(n\) when \(n\) is a prime. Prove that \(\frac{72!}{(36!)^2}-1\) is divisible by 73.
State Maclaurin's theorem on the expansion of a function \(f(x)\) in ascending powers of \(x\). If \(y=e^{\sin^{-1}x}\) and \(y_n\) denote the \(n\)th differential coefficient of \(y\) with respect to \(x\), prove that, when \(x=0\), \(y_{n+2}=(n^2+1)y_n\), and hence expand \(y\) in ascending powers of \(x\).
Explain how the maxima and minima values of a function \(f(x)\) may be obtained. A right circular cone with axis vertical is drawn to circumscribe a sphere of radius \(a\) resting on a horizontal plane, the base of the cone also resting on the plane. Prove that the minimum volume of the cone is twice the volume of the sphere. Shew also that the sum of the areas of the curved surface of the cone and of its plane base is a minimum at the same time.
Prove that in polar coordinates \(r\frac{d\theta}{dr}\) is the tangent of the angle between the radius vector and tangent to a curve. In the case of the curve \(r^n=a^n\cos n\theta\), prove that \(a^n\frac{d^2r}{ds^2}+nr^{2n-1}=0\).
Evaluate \[ \int_0^\infty x^2\sin x, \quad \int_0^\infty \frac{xdx}{(1+x)(1+x^2)}, \quad \int_a^b \frac{dx}{(a+b-x)^2\sqrt{(x-a)(b-x)}} \quad (b>a). \] Find a formula of reduction connecting \[ \int \cos m\theta \cos^n\theta d\theta \quad \text{with} \quad \int \cos m\theta \cos^{n-2}\theta d\theta. \]
Trace the curve \[ x(y^2-\frac{1}{2}a^2) - y(x^2-\frac{1}{2}a^2) = a^3, \] and shew that the radius of curvature where it meets either axis is \(\frac{25}{12}a\sqrt{2}\).