Problems

Find a cubic polynomial in \(x\) which takes the values \[ \frac{1}{a-1}, \frac{1}{a}, \frac{1}{a+1}, \frac{1}{a+2} \] when \(x\) takes the values \(-1, 0, 1, 2\). Verify that \(x-a-1\) is one factor of the polynomial.

If \(n\) and \(s\) are given, show that the product of \(n\) positive integers whose sum is \(s\) is not greater than \(q^{n-r}(q+1)^r\), where \(q, r\) are respectively the quotient and remainder when \(s\) is divided by \(n\).

1. Prove that \(x^2+y^2+z^2-yz-zx-xy\) is a factor of \((y-z)^n+(z-x)^n+(x-y)^n\) provided \(n\) is a positive integer which is not an integral multiple of 3.
2. If \(x+y=a+b+c\) and \(x(x-a)(x-b)(x-c)=y(y-a)(y-b)(y-c)\) and \(x \ne y\), prove that \(x^2+y^2=a^2+b^2+c^2\). (Note: OCR says \(x^3+y^3 = a^3+b^3+c^3\). The image is blurry but looks more like squares.) Let's re-examine. It indeed looks like \(x^2+y^2=a^2+b^2+c^2\) on the scan.

A square of side 6 in. is divided into 36 inch squares. Find the number of paths 12 in. long which join a pair of opposite corners of the 6 in. square and which lie along the sides of the small squares.

Prove for positive values of \(x\), that if \(p>q>0\), then \[ q(x^p-1) \ge p(x^q-1). \] Hence, or otherwise, show that for, \(x \ge 1\), and \(p \ge q > 0\),

1. \(\frac{x^p-1}{p} \ge \frac{x^q-1}{q} - \frac{(p-q)}{pq}\log_e x\).
2. \(\frac{qn}{p-n}(x^p-1) \ge \frac{pn}{q-n}(x^q-1) - \frac{pq(p-q)}{(p-n)(q-n)}(x^n-1)\),

Prove that a necessary condition for the radius of curvature to be equal to the perpendicular from the origin on to the tangent for every point of a curve, is that the intrinsic equation is of the form \(s = \frac{1}{2}(a\psi^2+2b\psi+c)\). Show that with this form of intrinsic equation given, an origin can be found to satisfy the former property. Prove that in this case, the centre of curvature lies on a fixed circle of radius \(a\).

If \(y = \tan^{-1}\frac{x\sin\theta}{1+x\cos\theta}\), where \(\theta\) is constant, show that for \(n \ge 2\), \[ (1+2x\cos\theta+x^2)y_n + 2(n-1)(x+\cos\theta)y_{n-1} + (n-1)(n-2)y_{n-2}=0, \] where \(y_n\) denotes \(\frac{d^n y}{dx^n}\). Deduce that \(y_n(0) = (-1)^{n-1}(n-1)!\sin n\theta\).

If \(y = \frac{x^4+x^2-12}{x^4-4}\), determine the range of values possible for \(y\) when \(x\) is real. Sketch the graph of the function.

A solid in the form of a ring is generated by rotating a plane area possessing an axis of symmetry about an axis in its plane parallel to the axis of symmetry and not cutting the boundary of the area. Show that the radius of gyration \(\kappa\) of the ring about the axis of rotation is given by \(\kappa^2 = c^2 + \lambda^2\), where \(c\) is the distance of the centroid of the area from the axis of rotation, and \(\lambda\) is the radius of gyration of the area about its axis of symmetry. (Note: The question states \(\kappa^2=c^2+3\lambda^2\). The '3' is faint. Standard theorem is \(I = I_{cm} + Md^2\), so \(MK^2=M\lambda^2+Mc^2\), giving \(K^2=c^2+\lambda^2\). Let's stick with the scan.) Scan shows \(\kappa^2=c^2+3\lambda^2\). If the area is not symmetrical, show that for a uniform body having the same shape as the solid, the Moment of Inertia differs from \(MK^2\) by a term which is independent of \(c\).

1. Evaluate \[ \int_0^{2\pi} \frac{\cos(n-1)x - \cos nx}{1-\cos x} \, dx, \] where \(n\) is a positive integer.
2. Find the limit of \(\frac{1}{x^2}-\cot^2 x\), as \(x \to 0\).