A light rod \(OA\) of length \(l\) rotates freely about a fixed point \(O\). A point particle of mass \(m\) attached to the rod at \(A\) is initially at rest vertically below \(O\). A projectile of mass \(m\) moving horizontally with speed \(v\) (\(v^2 < 16gl\)) embeds itself instantaneously in the target. Obtain the height \(h\) through which the target would rise before first coming to rest if undisturbed in the subsequent motion. However, after rising through a height \(3h/4\) another similar projectile embeds itself in the target. How much further will the target rise? If the total height through which the target rises is \(3h/4 + h'\), show that \(h'\) is greatest (for variable \(v\)) if \(v^2 = 16gl/3\).
If $$f(x) = \frac{(1+x)^{\frac{1}{2}} - 1}{1-(1-x)^{\frac{1}{2}}},$$ find (i) \(\lim_{x \to 0} f(x)\), and (ii) \(\lim_{x \to 0} \frac{df}{dx}\).
If \(y = (x^2-1)^n\), where \(n\) is a positive integer, prove that $$(1-x^2)\frac{dy}{dx} + 2nxy = 0.$$ By differentiating this equation \((n+1)\) times and using Leibniz' theorem, or otherwise, show that the function \(p_n(x)\), defined by $$p_n(x) = \frac{d^n}{dx^n}(x^2-1)^n,$$ satisfies the equation $$(1-x^2)\frac{d^2p_n}{dx^2} - 2x\frac{dp_n}{dx} + n(n+1)p_n = 0.$$ Show also that $$p_n(1) = (-1)^n p_n(-1) = 2^n n!$$
Find the minimum distance between the origin and the branch of the curve $$y = \frac{x}{x+c} \quad (c \neq 0)$$ that does not pass through the origin. Does your result remain true when \(c = 0\)?
(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 $$\sum_{v=0}^{\infty} c(v)t^v = (1-t)^{-1}\prod_{r=0}^{\infty}(1-t^{2^r})^{-1}.$$