By considering \((1-1)^n\), prove that \[\binom{n}{0}-\binom{n}{1}+\binom{n}{2}- \ldots + (-1)^n\binom{n}{n} = 0,\] for \(n = 1, 2, \ldots\). Hence or otherwise prove by induction that \[1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{n} = \binom{n}{1}\frac{1}{1} - \binom{n}{2}\frac{1}{2} + \ldots + (-1)^{n-1}\binom{n}{n}\frac{1}{n}\] for \(n = 1, 2, \ldots\). [You may assume without proof that \(\displaystyle \binom{n}{r} = \binom{n-1}r + \binom{n-1}{r-1}\)]
Decompose \[\frac{3x^2+2ax+2bx+ab}{x^3+(a+b)x^2+abx}\] into partial fractions. By considering the smallest denominator or otherwise, show that this expression takes the value 1 for only a finite number of positive integral values of \(x\), \(a\) and \(b\) (You are not required to find all such values.)
Whenever possible, solve the following simultaneous equations (in which \(\lambda\) is a real number). \begin{align*} \lambda x + y &= 1\\ x + (\lambda - 1)y &= 2\\ x + y + (\lambda - 2)z &= \lambda \end{align*} For what values of \(\lambda\) are there no solutions?
The equation \(x^4-8x^3+ax^2-28x+12\) has the property that the sum of a certain pair of roots is equal to the sum of the remaining two roots. Determine \(a\) and find all the roots.
Show that if \(n\) straight lines are drawn in a plane in such a way that no two are parallel and no three meet in a point, then the plane is divided into \(\frac{1}{2}(n^2+n+2)\) regions. How many of these regions stretch to infinity?
Show that angles subtended by a chord of a circle at the circumference and in the same segment are equal. A rod is bent so as to form an acute angle at \(X\). Another rod \(PQ\) slides with its ends \(P\) and \(Q\) on the two straight arms of the bent rod. At each position of \(P\) and \(Q\) lines \(PR\), \(QR\) are drawn perpendicular to the arms on which respectively \(P\) and \(Q\) move. Show that, when the bent rod is fixed and \(PQ\) moves, \(R\) moves on a circle. Show further that, when \(PQ\) is fixed and the bent rod is moved, \(R\) again moves on a circle, of radius half that of the former circle.
Sketch and describe the three curves given in polar coordinates by \begin{align*} (i)&~ r = \sin\theta \quad (0 < \theta < \pi);\\ (ii)&~ r^{-1} = \sin\theta \quad (0 < \theta < \pi);\\ (iii)&~ r^{-2} = \sin 2\theta \quad (0 < \theta < \pi/2). \end{align*} [No credit will be given for solutions obtained by numerical methods alone.]
A sequence of numbers \(u_1, u_2, u_3 \ldots\) is defined by the relations \begin{align*} u_1 &= a+b\\ u_n &= a+b-\frac{ab}{u_{n-1}}, \end{align*} where \(a+b \neq 0\). Show that if \(a \neq b\) then \[u_n = \frac{a^{n+1}-b^{n+1}}{a^n-b^n},\] and when \(a > b > 0\) determine the limit to which \(u_n\) tends as \(n\) tends to infinity. Find a formula for \(u_n\) when \(a = b\), and determine the limit to which \(u_n\) tends as \(n\) tends to infinity.
Express the function \[f(x) = \frac{x^3-x}{(x^2-4)^2}\] in partial fractions with constant numerators. Find the \(n\)th derivative of \(f(x)\) at \(x = 0\).
Show that if \[e^x\sin x = a_0 + \frac{a_1}{1!}x + \frac{a_2}{2!}x^2 + \ldots + \frac{a_n}{n!}x^n + \ldots\] then \(a_0 = 0\), \(a_{4n+1} = (-1)^n 4^n\); and determine \(a_{4n+2}\) and \(a_{4n+3}\).