Shew that \[ 5\{(y-z)^7 + (z-x)^7 + (x-y)^7\} = 7\{(y-z)^5+(z-x)^5+(x-y)^5\}\{x^2+y^2+z^2-yz-zx-xy\}. \] Hence, or otherwise, find all the factors of \[ (y-z)^7 + (z-x)^7 + (x-y)^7. \]
Five numbers \(x, y, z, b\) and \(c\) are connected by the following three relations: \begin{align*} x+y+z &= 0, \\ x^2+y^2+z^2 &= b, \\ x^3+y^3+z^3 &= c. \end{align*} Find a relation connecting \(b\) and \(c\), the satisfying of which is a necessary and sufficient condition that two of \(x, y\) and \(z\) shall be equal.
(i) The radii of the escribed circles of a triangle are \(r_1, r_2\) and \(r_3\), the radius of the inscribed circle is \(r\). Shew that \[ \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{1}{r}. \] (ii) If the sides of a quadrilateral are given, shew that the area of the quadrilateral is greatest when it can be inscribed in a circle.
If \(a_n+a_{n-1}+a_{n-2}=0\), for \(n > 2\), shew that \[ a_1\cos\theta + a_2\cos 2\theta + \dots + a_n\cos n\theta = \frac{a_1+(a_1+a_2)\cos\theta - a_{n-1}\cos n\theta + a_n\cos(n+1)\theta}{1+2\cos\theta}. \]
A curve is given by the equation \[ ax+by+cx^2+dxy+ey^2=0. \] Find the values \(y', y''\) and \(y'''\) of \(\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\) and \(\dfrac{d^3y}{dx^3}\) respectively, at the origin. Substitute \(y=y'x + \dfrac{y''x^2}{2} + \dfrac{y'''x^3}{6} + \eta x^3\) in the above equation and shew that \(\eta\) tends to zero as \(x\) tends to zero. Find the limit of \(\dfrac{\eta}{x}\) as \(x\) tends to zero.
Shew that there are three points of inflexion on the curve \[ y = \frac{x}{x^2+x+1}. \] Shew that these three points of inflexion lie on a line.
Give an account of the application of the differential calculus to the investigation of the maxima and minima of a function of a single variable, explaining how to distinguish between maxima and minima. Investigate completely the maxima and minima of the distance of a variable point \(P\) on an ellipse from a fixed point \(Q\) on the major axis.
Shew that \[ \lim_{x\to 0} \frac{\cos(\sin x) + \sin(1-\cos x) - 1}{x^4} = -\frac{1}{6}. \]
Integrate \[ x^2\sqrt{(1+x^2)}, \quad \frac{\cos^2 2x}{\sin^4 x \cos^2 x}, \quad x^m(\log x)^2. \]
(i) Evaluate \[ \int_0^\infty \frac{dx}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}, \] where \(a, b\) and \(c\) are positive. \item[(ii)] Find a reduction formula for the integral \[ \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^n x}, \] and evaluate the integral for the cases \(n=1, 2\).