If \(\alpha, \beta\) denote the roots of a given quadratic equation \(Ax^2+Bx+C=0\), find the quadratic of which the roots are \(\frac{a\alpha^2+b\alpha+c}{a'\alpha^2+b'\alpha+c'}\) and \(\frac{a\beta^2+b\beta+c}{a'\beta^2+b'\beta+c'}\). Prove that, if \(x\) be restricted to be real, \(\frac{kx^2+kx+1}{x^2+kx+k}\) can have all values in case \(k\) is negative and not numerically less than \(\frac{1}{4}\); that there are two values between which it cannot lie when \(k\) is negative and numerically less than \(\frac{1}{4}\), or also when \(k>4\); and that there are two values between which it must lie in case \(k\) is positive and less than 4, these two values being coincident when \(k=1\).
Find the sum of \(n\) terms of the series \(1^3+2^3+3^3+\dots\). Find also the sum to \(n\) terms of the series \[ \frac{x}{1+x} + \frac{x^2}{1+x+x^2+x^3} + \dots + \frac{x^{2^{n-1}}}{1+x+\dots+x^{2^n-1}}; \] and shew that the sum to infinity is \(x\) or 1 according as \(x\) is numerically less than or greater than 1.
Prove, by means of the identity \(\frac{p}{1-px} - \frac{q}{1-qx} = \frac{p-q}{(1-px)(1-qx)}\), or otherwise, that, if \(n\) be an even integer, \begin{align*} (-1)^{\frac{1}{2}n} \frac{p^{n+1}-q^{n+1}}{p-q} = (pq)^{\frac{1}{2}n} &- \frac{(n+1)^2-1^2}{2.4}(pq)^{\frac{1}{2}n-1}(p+q)^2 + \dots \\ & + (-1)^r \frac{\{(n+1)^2-1^2\}\{(n+1)^2-3^2\}\dots\{(n+1)^2-(2r-1)^2\}}{2.4.6\dots4r} (pq)^{\frac{1}{2}n-r}(p+q)^{2r} + \dots. \end{align*}
Prove that a simple periodic continued fraction \[ a_1 + \frac{1}{a_2+} \frac{1}{a_3+} \dots \frac{1}{a_n+} \frac{1}{a_1+} \frac{1}{a_2+} \dots \] is the positive root of a quadratic equation with coefficients rational in \(a_1, a_2, \dots a_n\). Shew also that the negative root is \(-\left(\frac{1}{a_n+} \frac{1}{a_{n-1}+} \dots \frac{1}{a_1+} \frac{1}{a_n+} \frac{1}{a_{n-1}+} \dots\right)\). Prove that the \(2n\)th convergent of \(\frac{1}{1+} \frac{1}{k+} \frac{1}{1+} \frac{1}{1+} \dots\) is the \(n\)th convergent of \(\frac{k}{k+1-} \frac{1}{k+2-} \frac{1}{k+2-} \dots\).
Prove that \(\frac{\sin\theta}{\theta}\) diminishes steadily from 1 to \(\frac{2}{\pi}\) as \(\theta\) increases from 0 to \(\frac{\pi}{2}\). If \(\tan(\phi-\theta)=(1+\lambda)\tan\phi\), where \(\lambda\) is very small, prove that one value of \(\tan\phi\) is \((1-\frac{1}{2}\lambda)\tan\frac{1}{2}\theta\), approximately.
Prove that the area of the triangle formed by joining the feet of the perpendiculars from the corners of an acute-angled triangle \(ABC\), of area \(\Delta\), on the opposite sides is \(2\Delta \cos A \cos B \cos C\). Shew that for any such triangle this area cannot exceed \(\frac{1}{4}\Delta\).
Shew that the problem of determining the \(n\)th roots of 1 is equivalent to that of inscribing a regular polygon of \(n\) sides in a circle. If \(n\) denote an even integer, shew that the product \[ \left(x^2+\cot^2\frac{\pi}{2n}\right) \left(x^2+\cot^2\frac{3\pi}{2n}\right) \left(x^2+\cot^2\frac{5\pi}{2n}\right) \dots \left(x^2+\cot^2\frac{(n-1)\pi}{2n}\right) \] is equal to \(\frac{1}{2}\{(1+x)^n+(1-x)^n\}\).
Explain what is meant by the limit of \(\frac{f(x+h)-f(x)}{h}\) as \(h\) converges to zero, and illustrate your explanation geometrically. Determine the limit in the case \(f(x)=x^n\), (1) when \(n\) is a positive integer, and (2) when \(n\) is any rational fraction, positive or negative. Find the \(n\)th differential coefficient of \(\frac{x}{x^2+1}\).
Prove that \(f(x+h) = f(x)+hf'(x+\theta h)\), for some value of \(\theta\) between 0 and 1, provided \(f(x)\) and its differential coefficient \(f'(x)\) satisfy certain conditions to be stated: and give a geometrical illustration of the theorem. If \(f(x)=e^x\), shew that \(\theta = \frac{1}{2} - \frac{h}{24} + \frac{h^2}{48}\) approximately, when \(h\) is small.
If \(N, T\) be the points in which the ordinate and the tangent at a point \(P\) of the curve \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\) cut the axis of \(x\), prove that \(PT.OT=a.NT\), when \(O\) is the origin of coordinates. Shew also that the radius of curvature at \(P\) is three times the length of the perpendicular from \(O\) on the tangent at \(P\).