Solve the equation: \[ 8x^6 - 54x^5 + 109x^4 - 108x^3 + 109x^2 - 54x + 8 = 0. \]
Solution: Solve the equation: \[ 8x^6 - 54x^5 + 109x^4 - 108x^3 + 109x^2 - 54x + 8 = 0. \] \begin{align*} && 0 &= 8x^6 - 54x^5 + 109x^4 - 108x^3 + 109x^2 - 54x + 8 \\ \Rightarrow &&0 &= 8(x^3+x^{-3}) -54(x^2+x^{-2}) +109(x+x^{-1}) - 108 \\ &&&= 8((x+x^{-1})^{3}-3(x+x^{-1})) - 54((x+x^{-1})^2-2)+109(x+x^{-1}) - 108 \\ &&&= 8z^3-24z-54z^2+108+109z -108 \\ &&&= 8z^3-54z^2+85z\\ &&&= z(8z^2-54z+85) \\ &&&= z(2z-5)(4z-17) \\ \Rightarrow && 0 &= x + \frac{1}{x} \tag{no soln} \\ && \frac52 &= x + \frac{1}{x} \\ \Rightarrow && 0 &= 2x^2-5x+2 \\ &&&= (2x-1)(x-2)\\ \Rightarrow && x &= 2, \frac12 \\ && \frac{17}{4} &= x + \frac{1}{x} \\ \Rightarrow && 0 &= 4x^2-17x + 4 \\ && &= (4x-1)(x-4) \\ && x &= 4, \frac14 \end{align*} Therefore \(x \in \left \{ 2, \frac12, 4, \frac14 \right \}\)
Prove that the arithmetic mean of a number of positive quantities is never less than their geometric mean. Prove that if \(u, v, w\) are positive quantities and \(u+v+w=1\), then \[ \frac{1}{u^2}+\frac{1}{v^2}+\frac{1}{w^2} \ge 27. \]
A recurring series whose \(n\)th term is \(u_n\) has the scale of relation: \[ u_{n+3}-6u_{n+2}+11u_{n+1}-6u_n=0. \] Show that \(u_n\) is of the general form \[ 3^n A + 2^n B + C, \] where \(A, B, C\) are independent of \(n\). Find the value of the \(n\)th term if \(u_1=1, u_2=6, u_3=14\).
Prove that the number of permutations of \(n\) things of which \(r\) are identical and the rest unlike is \(\dfrac{n!}{r!}\). A number \(n\) of identical cards are to be placed in \(p\) pigeon holes. Show that the number of ways in which the cards can be disposed is \[ \frac{(n+p-1)!}{(p-1)!n!}. \]
Evaluate the limits as \(x\) tends to infinity of the following expressions: \[ \sqrt{x^2+1}-x, \quad x^{-a}\log x \quad (a>0), \quad \text{and} \quad (x/\sqrt{x^2+1})^x. \]
State and prove Leibniz' theorem for the \(n\)th derivative of a product of two functions. If \[ y_n(x)=e^x \frac{d^n}{dx^n}(x^ne^{-x}), \] prove that \[ y_n = x \frac{dy_{n-1}}{dx} + (n-x)y_{n-1} \] and \[ \frac{1}{n} \frac{dy_n}{dx} = \frac{dy_{n-1}}{dx} - y_{n-1}. \] Hence show that the polynomial \(y_n\) satisfies a certain linear differential equation of the second order.
If \(m\) and \(n\) are positive integers greater than unity, prove that \[ I_{m,n} = \int_0^{\pi/2} \cos^m x \cos nx dx = \frac{m}{m-n}I_{m-1, n-1} = \frac{m}{m+n}I_{m-1, n-1}. \] Hence show, if \(p\) and \(q\) are positive integers, that \[ \int_0^{\pi/2} \cos^{p+q} x \cos(p-q)x dx = \pi(p+q)!/2^{p+q+1}p!q!. \]
A curve is such that its arc length \(s\) measured from a certain point and ordinate \(y\) are related by \[ y^2=s^2+c^2, \] where \(c\) is a constant. Show that referred to suitably chosen rectangular axes the curve has equation \[ y=c \cosh x/c. \] If \(C\) is the centre of curvature at a point \(P\) of the curve and \(G\) is the point in \(CP\) produced such that \(CP=PG\), prove that the locus of \(G\) is a straight line.
A point \(P\) is selected in the plane of a fixed triangle \(ABC\) and a function of the position of \(P\) is defined by \[ F(P) = PA^2.f(A) + PB^2.f(B) + PC^2.f(C), \] where \(f\) is a continuous function, and \(A, B, C\) denote the angles of the triangle. Find the form of the function \(f\) in order that, whatever the particular shape of the triangle, \(F(P)\) has its minimum value (i) at the centroid of the triangle, (ii) at the orthocentre, (iii) at the incentre, and (iv) at the circumcentre.
A circle of radius \(b\) is rotated about an axis in its own plane at perpendicular distance \(a\) (\(>b\)) from its centre to generate a solid ring. Prove that, if the ring is cut into two by a plane through the axis of rotation, the centre of gravity of either portion is at distance \((2a^2+\frac{1}{2}b^2)/\pi a\) from the axis. Show that the radius of gyration of either portion about the axis is \(\sqrt{(a^2+\frac{3}{4}b^2)}\), and hence find the radius of gyration of either portion about the line through its centre of mass parallel to the original axis.