If any of the following expressions are meaningless, explain why. Evaluate each of the integrals which has a meaning: \[ \int_0^1 \frac{dx}{e\sqrt[3]{x(1-x)^2}}; \quad \int_{-4}^{-2} \frac{dx}{2x+1}; \quad \int_0^{3\pi/4} \tan\theta\,d\theta. \]
Sketch the curve whose equation in polar coordinates is \[ r = \sin 3\theta - 2\sin\theta. \] Find any maximum or minimum values of \(r\). Prove that each of the smaller loops of the curve has area less than 0.005.
It is given that, for all \(x>0, y>0\), \[ \int_1^{xy} f(t)\,dt = \phi(y), \] where \(\phi(y)\) is independent of \(x\). Write down the results of differentiating this equation partially with respect to \(x\) and with respect to \(y\). Find the most general form of the function \(f(t)\).
N is the foot of the perpendicular from the origin, O, to the tangent at \((r, \theta)\) to the curve \[ r=a(1-e\cos\theta), \quad (0 < e < 1), \] and Q is the point on ON such that \(OQ \cdot ON = a^2\). Show that \[ ON^2 = a^2 \frac{(1-e \cos \theta)^4}{1-2e \cos \theta+e^2} \] and hence that, if \(e\) is so small that powers of \(e\) higher than the first can be neglected, the locus of Q is \[ r=a(1+e\cos\theta). \]
\(f(x), g(x)\) and \(h(x)\) are functions of \(x\) satisfying the equations \begin{align*} \frac{df}{dx} &= f+g+2h, \\ \frac{dg}{dx} &= 2g+2h, \\ \frac{dh}{dx} &= 7f+8g+24h. \end{align*} Show that \(f-g\) is of the form \(Ae^{\lambda x}\), where \(A\) is a constant. Find all the solutions of the form \[ f=f_0e^{\lambda x}, \quad g=g_0e^{\lambda x}, \quad h=h_0e^{\lambda x}, \] where \(\lambda\) is independent of \(x\), giving the possible values of \(\lambda\) and the corresponding ratios of the constants \(f_0, g_0, h_0\).
Two complex variables \(z=x+iy\), \(Z=X+iY\), are connected by the relation \[ Z = \sin(\tfrac{1}{2}\pi z). \] Show that to every point in the complex \(Z\)-plane there corresponds a point in the strip \(|x| \le \frac{1}{2}\) of the complex \(z\)-plane. Show also that the lines \(x=\text{constant}\), \(y=\text{constant}\) map into certain mutually orthogonal systems of ellipses and hyperbolae in the \(Z\)-plane.
If \[ X_n = e^{-x^2} \frac{d^n}{dx^n}(e^{x^2}), \quad (n=0, 1, 2, 3\dots) \] establish the relations
The coordinates of the points on a curve are given in terms of general homogeneous coordinates by the parametric relations \(x:y:z = \theta^3:\theta^2:1\). Prove that, if the points \(A, B, C\) with parameters \(a, b, c\) are collinear, then \(bc+ca+ab=0\); and that, if the points \(L, M, N\) with parameters \(l, m, n\) lie on a conic through the vertices \(X, Y, Z\) of the triangle of reference, then \(l+m+n=0\). State and prove the converse results. The points \(A, B, C\) of the curve are collinear, and the tangents at \(A, B, C\) meet the curve again in \(P, Q, R\) respectively. Prove that \(P, Q, R\) are collinear. The conic \(X, Y, Z, B, C\) meets the curve again in \(U\), and the tangent at \(U\) meets the curve again in \(V\). Prove that \(V\) lies on the conic through \(X, Y, Z, Q, R\).
Prove that, if two conics \(S\) and \(\Sigma\) are so related that there exists one triangle inscribed in \(S\) and circumscribed to \(\Sigma\), then such a triangle can be drawn with one of its vertices at any given point of \(S\). A circle \(S\) is drawn to pass through the focus \(F\) of a parabola \(\Sigma\). The tangents to \(\Sigma\) from any arbitrary point \(A\) of \(S\) cut \(S\) again in the points \(B, C\). Prove that \(BC\) is a tangent to \(\Sigma\). (It is assumed that the point \(A\) lies "outside" the parabola.)
A number \(n\) of equal uniform rectangular blocks are built into the form of a stairway, each block projecting the same distance \(a\) beyond the one below. The top block is supported from below at its outer edge. Show that the stairway can stand in equilibrium if, and only if, \(2l > a(n-1)\), where \(2l\) is the width of each block.