If \(a, b, c\) are the sides of a triangle and \(2s\) is their sum, prove that the area of the triangle is \[ \sqrt{s(s-a)(s-b)(s-c)}. \] If \(b+c=2a\) shew that the area is \(\frac{1}{2}a^2\tan\frac{A}{2}\) and find the maximum value of \(A\) subject to this condition.
If the diagonals of a quadrilateral intersect at right angles at \(O\), shew that the feet of the perpendiculars from \(O\) on the four sides lie on a circle, and that the points where these perpendiculars meet the opposite sides also lie on the same circle.
If the coordinates \((x,y)\) of a point are given by \[ x = at + \frac{b}{t}, \quad y = bt + \frac{a}{t}, \] shew that the point lies on a hyperbola and find the equation of the tangent at any point of the hyperbola in terms of its parameter \(t\).
In some experiments in hauling a truck along a level track, the following observations were made between the force \(P\) and the velocity \(V\):
Shew that \[ f(x+h) - f(x) = hf'(x+\theta h) \] for some value of \(\theta\) lying between 0 and 1, provided \(f(x)\) and its differential coefficient \(f'(x)\) satisfy certain conditions which are to be stated. Deduce that a function of \(x\) whose differential coefficient is positive, increases steadily as \(x\) increases. Hence shew that if \(0 < x < \frac{\pi}{2}\) \[ 1 > \cos x > 1 - \frac{x^2}{2} \quad \text{and} \quad x > \sin x > x - \frac{x^3}{6}. \]
Prove that if \((r, \theta)\) are the polar coordinates of a point on a curve and \(p\) is the length of the perpendicular from the origin on the tangent at the point, the radius of curvature is given by \[ \rho = r\frac{dr}{dp}. \] Shew that the radius of curvature at any point on a conic is \(2a \text{cosec}^3\phi\), where \(4a\) is the latus rectum and \(\phi\) the angle which the tangent at the point makes with a focal distance.
Perform the integrations \[ \int \frac{\sin 2x \, dx}{(a+b\cos x)^2}, \quad \int_0^{\frac{\pi}{2}} \sin^3x \cos^5x \, dx, \quad \int \frac{dx}{\sqrt{11x-5-2x^2}}. \]
Simpson's rule for finding areas by approximation is based on the property that, if \(y_1, y_2, y_3\) are three ordinates of the parabola \(y=a+bx+cx^2\) separated by equal intervals, the mean ordinate of the portion of the curve between the ordinates \(y_1\) and \(y_3\) is \(\frac{1}{3}(y_1+4y_2+y_3)\). Prove this and deduce the rule. Find an approximate value for \[ \int_0^{10} \sqrt{6+5x-3x^2+x^3} \, dx. \]
A circle cuts the sides of a triangle in \(P\) and \(P'\), \(Q\) and \(Q'\), \(R\) and \(R'\) respectively. Prove that, if the perpendiculars to the sides at \(P, Q\) and \(R\) are concurrent, then the perpendiculars at \(P', Q'\) and \(R'\) are also concurrent.
Prove that the inverse of a circle is in general another circle. Two circles cut orthogonally in \(A\) and \(B\). \(PQ\) is a fixed diameter of one and \(R\) is any point on the other. The circles \(APR, BQR\) intersect in \(X\). Prove that the locus of \(X\) is the line \(PQ\).