Problems

Show that the equation of a conic may be put in the form \(zx-y^2=0\), when homogeneous coordinates are used, and that the triangle of reference may be chosen in a doubly infinite number of ways. Given \(Y=x-(a+b)y+abz\), determine \(X\) and \(Z\), so that \(ZX-Y^2=0\) and \(zx-y^2=0\) may represent the same conic.

Resolve into factors:

1. \((bc+ca+ab)^3 - abc(a+b+c)^3\);
2. \((bcd+cda+dab+abc)^2 - abcd(a+b+c+d)^2\).

Find the condition that \[ (b-c)^n + (c-a)^n + (a-b)^n \] should be divisible by \(\Sigma a^2 - \Sigma bc\).

Find the conditions that

1. \(ax^2+2bx+c\) may be positive for all real values of \(x\);
2. \(ax^2+2hxy+by^2+2gx+2fy+c\) may be positive for all real values of \(x\) and \(y\).

Sum the series:

1. \(\dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + \dots\) to \(n\) terms;
2. \(1^2+2^2x+4^2x^2+5^2x^3+7^2x^4+8^2x^5+\dots\) to infinity.

Prove that \(\log(1+x)=x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dots\), if \(|x|<1\). Prove that \((1+x)^{1/x} = e - \dfrac{ex}{2} + \dfrac{11e}{24}x^2 - \dfrac{7e}{16}x^3 + \text{etc.}\)

Find the conditions that the roots of the cubic \(a_0x^3+3a_1x^2+3a_2x+a_3=0\) should satisfy the relation (i) \(\alpha\beta + \beta\gamma = 2\alpha\gamma\), (ii) \(\alpha^2\beta + \beta^2\gamma = 2\alpha^2\gamma\).

Define the differential coefficient of a function. Has \(x\sin\dfrac{1}{x}\) a differential coefficient at \(x=0\)? Find the \(n\)th differential coefficient of \(x\tan^{-1}x\).

State McLaurin's theorem on the expansion of a function of \(x\) in ascending powers of \(x\). Prove that, if \(a_0+a_1x+a_2x^2+\dots\) is the expansion in ascending powers of \(x\) of \(\{\cosh^{-1}(1+x)\}^2\), \((n+1)(2n+1)a_{n+1} + n^2 a_n=0\).

Find the equation of the tangent at any point of the curve given where \(x=f(t), y=\phi(t)\). Prove that, if \(\theta_1, \theta_2, \theta_3\) are the vectorial angles of three points on the curve \(r(\cos^3\theta+\sin^3\theta)=3a\cos\theta\sin\theta\) at which the tangents are concurrent, then \(\Sigma\cot\theta_1=0\).

Integrate: \(\sec x, \quad \dfrac{1}{(x^2-x-6)\sqrt{1+x+x^2}}, \quad \dfrac{\sqrt{a^2+b^2\cos^2\theta}}{\cos\theta}\). Find a formula of reduction for \(\displaystyle\int_0^{\frac{\pi}{2}} x^n \cos^m x dx\).