Problems

Prove that, if \(k\) is real and \(|k|<1\), the function \(\cot x + k \csc x\) takes all values as \(x\) varies through real values. Prove that, if \(|k|>1\), the function takes all values except those included in an interval of length \(2\sqrt{k^2-1}\). Give rough sketches of the graph of \[ y = \cot x + k\csc x \] for \(-\pi < x < \pi\), in the cases (i) \(01\).

Prove that, if \(X+Y+Z\) is equal to \(2n\) right angles, where \(n\) is an integer, then \[ \sin 2X + \sin 2Y + \sin 2Z \] is a numerical multiple of \(\sin X \sin Y \sin Z\). Prove that, if \(A, B, C\) are the angles of a triangle, then \begin{align*} \sin^3 A \cos A + \sin^3 B \cos B + \sin^3 C \cos C \\ = \frac{1}{4}(\sin 2A + \sin 2B + \sin 2C)(\cos 2A + \cos 2B + \cos 2C). \end{align*}

The coordinates of any point on a curve are given by \(x=\phi(t)/f(t)\), \(y=\psi(t)/f(t)\), where \(t\) is a parameter; prove that the equation of the tangent is \[ \begin{vmatrix} x & \phi(t) & \phi'(t) \\ y & \psi(t) & \psi'(t) \\ 1 & f(t) & f'(t) \end{vmatrix} = 0. \] Prove that the condition that the tangents at the points of the curve \[ x=at/(t^3+bt^2+ct+d), \quad y=a/(t^3+bt^2+ct+d), \] whose parameters are \(t_1, t_2, t_3\) may be concurrent is \[ 3(t_2t_3+t_3t_1+t_1t_2)+2b(t_1+t_2+t_3)+b^2=0. \]

Prove that, if \(f\) is a homogeneous polynomial in \(x\) and \(y\) of degree \(n\) and suffixes denote partial differentiations, then

1. \(xf_x+y f_y = nf\),
2. \(xf_{xx}+y f_{xy} = (n-1)f_x\),
3. \((n-1)\begin{vmatrix} f_{xx} & f_{xy} & f_x \\ f_{xy} & f_{yy} & f_y \\ f_x & f_y & 0 \end{vmatrix} = n f (f_{xy}^2-f_{xx}f_{yy})\).

Criticize the following arguments:

1. If \(y=(2x^2+3)/(x^2+4)\), then \(dy/dx=0\) if \(x=0\) or \(\pm 1\), and so the only maximum and minimum values are given by \(x=0\), \(y=\frac{3}{4}\) (minimum) and \(x=\pm 1\), \(y=1\) (maxima). Hence \(y\) lies between \(\frac{3}{4}\) and \(1\) for all values of \(x\).
2. If \(y^2=x^3-3x+1\), then \(dy/dx=0\) if \(x=\pm 1\). Also \(d^2y/dx^2\) does not vanish for these values of \(x\). Hence \(x=1\) and \(x=-1\) give points on the curve at which the tangents are parallel to the line \(y=0\).
3. \(\int_{-1}^3 \frac{dx}{1-x} = \left[-\log(1-x)\right]_{-1}^3 = \dots = -\log 2\).

Find the equation of the straight line which is asymptotic to the curve \[ x^2(x-y)+y^2=0. \] Prove also the following facts and give a sketch of the curve:

1. the origin is a cusp;
2. no part of the curve lies between \(x=0\) and \(x=4\);
3. the curve consists of two infinite branches, one lying in the first quadrant and the other in the second and third quadrants.

(i) Find \(\int \frac{1-\tan x}{1+\tan x}dx\). (ii) Prove that, if \(a>b>0\), \[ \int_0^\pi \frac{\sin^2 x dx}{a^2 - 2ab\cos x + b^2} = \frac{\pi}{2a^2}. \] What is the value of the integral, if \(b>a>0\)?

\(A'\) is a variable point on the circumcircle of a given triangle \(APQ\) such that \(A\) and \(A'\) lie on the same side of \(PQ\). \(A'P, A'Q\) are produced to points \(B', C'\) respectively such that the lengths \(A'B', A'C'\) are constant and in the ratio \(\frac{AQ}{AP}\) for all positions of \(A'\). Prove that \(B'C'\) touches a fixed circle centre \(A\).

\(P\) and \(Q\) are two points lying outside a circle \(C\). Establish a method of drawing a circle through \(P\) and \(Q\), (a) to touch \(C\), (b) to intersect \(C\) orthogonally. State the number of solutions, and discuss any exceptional cases.

\(D, E, F\) are respectively the feet of the perpendiculars drawn to the sides \(BC, CA, AB\) of a triangle \(ABC\) from a point \(O\) in its plane. \(P\) is the point of intersection of the straight line through \(EF\) with the line through \(O\) parallel to \(BC\); \(Q\) that of \(DF\) with the line through \(O\) parallel to \(CA\); \(R\) that of \(DE\) with the line through \(O\) parallel to \(AB\). Prove that the points \(P, Q, R\) are collinear.