Find the conditions that \(ax^2+2bx+c\) may be positive for all real values of \(x\). Shew that the expression \(\frac{8x}{x+a} + \frac{18x}{x-a}\) cannot have values lying between 1 and 25.
Solution: We need \(a \geq 0\) so that it's positive when \(x \to \infty\). If \(a = 0\) then it's linear so \(b = 0, c > 0\) We need \(\Delta = (-2b)^2 - 4ac = 4(b^2-ac) <0\) therefore \(b^2 < ac\). Therefore the conditions are \(a > 0, b^2 < ac\) or \(a = b = 0, c > 0\) \begin{align*} && \frac{8x}{x+a} + \frac{18x}{x-a} &= \frac{8x^2-8ax+18x^2+18ax}{x^2-a^2} \\ &&&= \frac{26x^2+10ax}{x^2-a^2} \\ && k &= \frac{26x^2+10ax}{x^2-a^2} \\ && 0 &= (26-k)x^2+10ax+ka^2 \end{align*} this is always positive if \(k < 26\) and \begin{align*} && 25a^2 &< (26-k)ka^2 \\ && 25 &< (26-k)k \\ && 0 &< -k^2+26k-25 \\ &&&= -(k-25)(k-1) \end{align*} Therefore \(k \in (1,25)\) since all these values have \(k < 26\) too.
Find an expression giving all the angles which have the same sine as \(A\). Solve the equation \begin{align*} &\cos x \sin(x-a)\sin(x-b)\sin(x-c) \\ &+ \sin x \cos(x-a)\cos(x-b)\cos(x-c) = \sin a \sin b \sin c. \end{align*}
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 \[ \int_0^{10} \sqrt{6+5x-3x^2+x^3} \, dx. \]