Problems

(i) Evaluate $$\int_a^b \sqrt{[(x-a)(b-x)]} \, dx$$ where \(a\) and \(b\) (\(> a\)) are constants. (ii) If $$B(x) = \int_0^x e^{-t^4} dt,$$ prove that $$(2n+1)\int_0^x t^{2n}B(t) dt = x^{2n+1}B(x) - \frac{1}{4}n!\left[1-e^{-x^4}\sum_{r=0}^{\infty}\frac{x^{2r}}{r!}\right],$$ where \(n\) is a positive integer.

A family of plane curves has the property that if the tangent to \(f(x,y)\) of any one of the curves intersects the \(x\)-axis in \(N\), then the distance \(ON\) is equal to \(ky^2\), where \(O\) is the origin and \(k\) is a positive constant. Find the equation of the particular curve of the family that passes through the point \((0,1)\) and sketch it.

Suppose that \(a_j, b_j\) (\(1 \leq j \leq n\)) are given real numbers and that $$1 \leq a_j \leq A, \quad 1 \leq b_j \leq B \quad (1 \leq j \leq n)$$ for some \(A, B\). Show that $$a_j b_j \geq u_j B + v_j A \quad (1 \leq j \leq n),$$ where \(u_j, v_j\) are defined by the equations $$a_j^2 = u_j + v_j A^2, \quad b_j^2 = u_j B^2 + v_j.$$ Deduce that $$\frac{(\sum a_j^2)(\sum b_j^2)}{(\sum a_j b_j)^2} \leq \left(\frac{(AB)^{\frac{1}{2}} + (AB)^{-\frac{1}{2}}}{2}\right)^2.$$

The polynomial \(P(x)\) in the single variable \(x\) has real coefficients and is non-negative for every real value of \(x\). Show that there are polynomials \(Q(x), R(x)\) with real coefficients such that $$P(x) = \{Q(x)\}^2 + \{R(x)\}^2.$$

Given that the roots of the equation $$y^8 + 3y^2 + 2y - 1 = 0$$ are the fourth powers of the roots of an equation $$x^8 + ax^2 + bx + c = 0$$ with rational coefficients \(a, b, c\), find suitable values for \(a, b, c\).

Let \(N(k,l)\) be the number of sets of integers \(a_1, \ldots, a_k\) such that $$1 \leq a_{j+1} \leq 2a_j \quad (1 \leq j < k)$$ and $$a_1 = 1, \quad a_k = l.$$ Prove that $$N(k, 2s+2) - N(k, 2s) = N(k-1, s).$$ For \(k \geq 2\), \(0 \leq v < 2^{k-2}\), show that $$N(k, 2^{k-1} - 2v) = N(k, 2^{k-1} - 2v - 1) = c(v)$$ is independent of \(k\), and that $$\sum_{v=0}^{\infty} c(v)t^v = (1-t)^{-1}\prod_{r=0}^{\infty}(1-t^{2^r})^{-1}.$$

(i) \(A, B, C, D\) are the angles of a plane quadrilateral. Show that $$4\sin(A+B)\sin(B+D)\sin(D+A) = \sin 2A + \sin 2B + \sin 2C + \sin 2D.$$ (ii) If $$\arccos a + \arccos b = \frac{1}{4}\pi,$$ show that $$a^2 - 2\sqrt{2}ab + b^2 = \frac{1}{2}.$$

If, for \(n = 1, 2, 3, \ldots\), the polynomial $$\frac{1}{n!}x(x-1)(x-2)\cdots(x-n+1)$$ is denoted by \(P_n(x)\), show that $$P_{n+1}(x+1) - P_{n+1}(x) = P_n(x) \quad (n = 1, 2, 3, \ldots).$$ Hence, or otherwise, prove that each of the polynomials \(P_n(x)\) takes integral values for all integral values of \(x\).

If, for each real number \(x\), \(\{x\}\) denotes the distance of \(x\) from the nearest integer (so that, for example, \(\{\pi\} = \pi - 3\) and \(\{2\frac{3}{4}\} = \frac{1}{4}\)), show that, if \(n\) is any integer \(\geq 1\), $$\{(\sqrt{2} + 1)^n\} = (\sqrt{2} - 1)^n$$

Each of the following rules defines a map (or transformation) from the set \(Z\) of all integers (positive, negative, or zero) into the same set \(Z\):

1. [(a)] \(x \rightarrow \cos\pi x\);
2. [(b)] \(x \rightarrow 3x - 1\).

(For `\(x \rightarrow \cos\pi x\)' read `\(x\) is mapped into, or transformed into, \(\cos\pi x\)'.) If \(M\) is one of these maps and \(S\) is a subset of \(Z\) (which may be \(Z\) itself) then we denote by \(M(S)\) the subset of \(Z\) consisting of all integers expressible in the form \(M(s)\) for some integer \(s\) in \(S\). Thus in case (b) if \(S\) is the set of all positive integers divisible by 3 then \(M(S)\) is the set \(M(Z)\) in each of the cases (a) and (b). Furthermore, in case (a) find a non-empty subset \(S\) of \(Z\) such that \(M(S) = S\); and in case (b) show that there is no such non-empty subset.