A triangle \(ABC\) is inscribed in a circle, and chords \(Aa, Bb\) are drawn parallel to the sides \(BC, CA\). Prove that \(ab\) is parallel to the tangent at \(C\).
The lines which join the ends of any chord \(PQ\) of a given circle to a given point \(O\) cut the circle again in points \(p, q\). Prove that if the chords \(PQ\) all pass through a given point \(E\), then all the circles \(OPQ\) will pass through a point \(F\), all the circles \(Opq\) through a point \(G\), and all the chords \(pq\) through a point \(H\). State any properties you observe as to the positions of \(F, G, H\).
A uniform ladder weighing \(w\) lbs. rests against a vertical wall, the coefficient of friction between the ladder and the wall and between the ladder and the ground being \(0 \cdot 4\). The ladder makes an angle \(\theta\) with the horizontal, and a man weighing 120 lbs. can just ascend to the top. Draw a graph showing the relation between \(\tan\theta\) and \(w\).
Give an account of the method of Orthogonal Projection with illustrations of its use. Consider the following problems:
The angles of a parallelogram are bisected externally: prove that the bisectors form a rectangle whose diagonals are equal to the sum of two sides of the parallelogram.
Prove that the tangents from any point \(P\) to a central conic whose foci are \(S\) and \(S'\) are equally inclined to \(SP\) and \(S'P\). \(A, B, C, D\) are any four points in a plane. A conic with \(A\) as focus which touches the lines \(BC, CD, DB\) has \(A'\) as its second focus; and a conic with \(B\) as focus which touches the lines \(AC, CD, DA\) has \(B'\) as its second focus. Prove that \(CD\) cuts \(A'B'\) at right angles and bisects it.
Two forces act at the origin in directions making angles \(\tan^{-1}\frac{3}{4}\) and \(\tan^{-1}7\) with \(Ox\); also two forces act at the point \((a, a)\) in directions making angles \((\pi - \tan^{-1}\frac{3}{4})\) and \(-(\pi - \tan^{-1}7)\) with \(Ox\). If the forces are in equilibrium, indicate by means of a rough diagram how to find their relative magnitudes graphically, and prove that they are as 39, 60, 25, 52.
Discuss as systematically as you can the theory of the solutions of three linear equations of the type \(ax+by+cz=d\), paying special attention to the cases that are commonly regarded as exceptional.
Having given that \begin{align*} x + y + z &= 1, \\ x^2 + y^2 + z^2 &= 2, \\ x^3 + y^3 + z^3 &= 3, \end{align*} prove that \[ x^4 + y^4 + z^4 = 4\frac{1}{6}. \]
Find the function \(f(x) = ax + b\) for which \(f(1) = 1\), and for which \[ \int_0^1 [f(x)]^2 dx \] has its minimum value. Shew that \(f(x)\) vanishes when \(x=1-1/\sqrt{2}\).