10273 problems found
Prove that in any triangle \[ \tan \frac{1}{2} (B-C) = \frac{b-c}{b+c} \cot \frac{1}{2} A. \] X and Y are two stationary observers, of whom Y is one mile due east of X. An aeroplane, travelling with uniform horizontal velocity, is observed simultaneously by X and Y. Its bearings are found to be 50\(^\circ\) east of north, and 72\(^\circ\) west of north, respectively, by the two observers. Fifteen seconds later, the bearings of the plane are found to be 75\(^\circ\) east of south and 63\(^\circ\) west of south by X and Y, respectively. Find the speed of the plane to the nearest mile per hour.
A bag contains the ten numbers \(0, 1, 2, \dots, 9\). Three numbers are drawn from the bag simultaneously and at random. Verify that the probability of the three numbers so selected adding up to 13 is equal to 1/12. What is the probability that they add up to 14? \par Find the corresponding probabilities when the three numbers, instead of being drawn simultaneously, are drawn successively and each replaced in the bag before the next draw.
A smooth wire is bent into the form of a circle of radius \(a\) and is fixed with its plane vertical. A bead of mass \(M\) slides on the wire, and is attached to one end of a light string of length greater than \(2a\); the string passes over a smooth peg fixed at the highest point of the circle and carries a mass \(m\), hanging freely, at its other end. If \(m<2M\), show that positions of equilibrium exist in which the portion of the string between \(M\) and the peg is inclined to the vertical. \par Prove that in such positions the equilibrium is unstable.
If one triangle can be inscribed in a conic \(S_1\) and circumscribed about a conic \(S_2\), prove that there are infinitely many such triangles and that there is a conic \(S_3\) with respect to which they are all self-conjugate.
Prove that \[ (3 \cos \phi - \sec \phi)^2 \geq 12 \] for all real values of \(\phi\). \par Use this result to show that \[ \frac{\sin\theta - \cos\theta}{3 \cos \theta + 3 \cos \phi - \sec \phi} \] lies between \(1 - \sqrt{(5/3)}\) and \(1 + \sqrt{(5/3)}\) for all real values of \(\theta\) and \(\phi\).
As \(x\) tends to \(a\), the functions \(f(x), g(x), f'(x)\) and \(g'(x)\) tend to the limits \(0, 0, b\) and \(c\) respectively. Prove that, if \(c \neq 0\), \(f(x)/g(x)\) tends to \(b/c\). \par Evaluate the following limits:
Two trains \(A\) and \(B\) are travelling along the same straight line with velocities \(u\) and \(v\) respectively. Prove that, on certain assumptions, the velocity of \(B\) as measured by a man in \(A\) is \(v-u\). What are these assumptions?
If a variable chord of a parabola subtends a right angle at the focus, prove that the locus of its pole is a rectangular hyperbola. Generalize this result by projection.
Obtain the expansion of \(\sin x\) in ascending powers of \(x\). For what values of \(x\) is this series convergent? \par A small arc \(PQ\) of a circle of radius 1 is of length \(x\). The arc \(PQ\) is bisected at \(Q_1\) and the arc \(PQ_1\) is bisected at \(Q_2\). The chords \(PQ\), \(PQ_1\) and \(PQ_2\) are of lengths \(c\), \(c_1\) and \(c_2\), respectively. Prove that \(\frac{1}{45}(c - 20c_1 + 64c_2)\) differs from \(x\) by a quantity of order \(x^7\).
A heavy non-uniform rod, inclined at an angle \(\theta\) to the horizontal, is wedged between two rough planes inclined at angles \(\alpha_1, \alpha_2\) to the vertical, as shown in the figure. The planes intersect in a horizontal line, and the rod lies in a vertical plane at right angles to this line. The angles of friction between the rod and the two planes are \(\lambda_1, \lambda_2\) respectively. Prove geometrically, or otherwise, that equilibrium is possible if and only if \[ \lambda_1 - \alpha_1 > \theta > \alpha_2 - \lambda_2. \] (A diagram shows a rod wedged between two planes. The planes make angles \(\alpha_1\) and \(\alpha_2\) with the vertical. The rod makes an angle of \(90^\circ - \theta\) with the vertical.)