Resolve into factors \[ (y-z)^2(y+z-2x)+(z-x)^2(z+x-2y)+(x-y)^2(x+y-2z). \] If \[ cy+bz=az+cx=bx+ay=ax+by+cz, \] prove that \[ (b+c)(c+a)(a+b) = a^3+b^3+c^3+6abc, \] and \[ (b-c)x/a+(c-a)y/b+(a-b)z/c=0. \]
Shew that there are in general two values of \(\lambda\) for which \[ ax^2+2bx+c+\lambda(a'x^2+2b'x+c') \] is a square and deduce that \(ax^2+2bx+c\), \(a'x^2+2b'x+c'\) may be put in the forms \[ p(x-\alpha)^2+q(x-\beta)^2, \quad p'(x-\alpha)^2+q'(x-\beta)^2. \] Express in these forms \(7x^2+2x+4\) and \(4x^2-28x-5\).
Find the sum of the squares and cubes of the first \(n\) odd integers. Shew that the sum of the products two at a time of the first \(n\) odd integers is \[ \tfrac{1}{3}n(n-1)(3n^2-n-1). \]
Sum the series \[ 1+\frac{m}{1!}\frac{1}{2^2}+\frac{m(m-2)}{2!}\frac{1}{2^4}+\frac{m(m-2)(m-4)}{3!}\frac{1}{2^6}+\dots \] to infinity when \(m\) is an odd integer. Find the coefficient of \(x^{2n}\) in the expansion of \(\frac{1+x}{(1+x+x^2)^2}\) in ascending powers of \(x\).
Differentiate (i) \(\log \sin x\), (ii) \(\tan^{-1}\frac{4x(1-x^2)}{1-6x^2+x^4}\). Find the \(n\)th differential coefficient of \(\frac{x^2-3x+1}{x^2-4x+3}\).
Prove that, if \(y\) is an implicit function of \(x\) satisfying the equation \(f(x,y)=0\), then \[ \frac{dy}{dx} = -\frac{\partial f/\partial x}{\partial f/\partial y}. \] If \(A,B,C\) are the angles of a triangle and \(\sin^2 A+\sin^2 B+\sin^2 C=k\), prove that, \[ \frac{\partial A}{\partial B} = \frac{\tan C - \tan B}{\tan A - \tan C}. \]
Find the equation of the tangent at a point of the curve given by \[ x:y:3a = t^3:t^2:1+t^3. \] The tangent at \(P(x,y)\) meets the curve again in \(Q\); prove that \(PQ\) subtends an angle \(\tan^{-1}\left(\frac{3a}{x-y}\right)\) at the double point.
Prove with the usual notation that \(\tan\phi = r\frac{d\theta}{dr} = \frac{p}{\sqrt{r^2-p^2}}\). If \(PT, PT'\) are two perpendicular tangents from \(P\) to the cardioid \(r=a(1+\cos\theta)\), prove that the chord of contact \(TT'\) subtends an angle of 60\(^\circ\) at the pole and that the minimum length of \(TT'\) is \((2-\sqrt{3})a/2\).
Trace the curve \(a^3y^2=x^4(b+x)\), and find the area of the loop.
Evaluate the integrals