Resolve into partial fractions \[ \frac{3x^2-6x+2}{(x^2+1)(x-3)^2}. \]
In how many ways can \(n\) things, of which \(p\) are exactly alike while the rest are all different, be arranged among themselves? Two players \(A\) and \(B\) have eight cards each. \(A\)'s cards are numbered 1 to 8. \(B\) has four 2's and four 5's. Each hand is shuffled and placed face downwards. Each player then draws a card from his own hand and the two cards thus drawn are paired off and removed from the two hands. A second pair of cards are then drawn and removed, the process being continued until the two hands have been completely paired off. What is the chance that there should be one and only one coincidence in numbers in the pairs thus selected at random?
Find the scale of relation, of the form \(u_{n+2}+pu_{n+1}+qu_n=0\), and the sum of the first \(n\) terms of the recurring series \[ 1+5+19+65+\dots. \]
A wireless signal from an aeroplane is intercepted at two direction-finding stations \(A\) and \(B\) which are five miles apart in a north and south line. From \(A\) the direction of the aeroplane is found to be \(66^\circ\) west of north and from \(B\) it is found to be \(20^\circ\) west of south. At the same time the altitude of the aeroplane is observed from \(A\) as \(40^\circ\). Find the height of the aeroplane above \(A\).
Prove that the radius of the inscribed circle of a triangle \(ABC\) is equal to \(a \sin\frac{1}{2}B \sin\frac{1}{2}C / \cos\frac{1}{2}A\). If \(P\) is a point on the side \(BC\) of the triangle such that the inscribed circles of the triangles \(APB, APC\) are equal, prove that \(a \cos APB = b-c\).
Find all the values of \(\theta\) lying between 0 and \(2\pi\) for the equation \[ a \cos\theta + b \sin\theta = c, \text{ where } c^2 < (a^2+b^2). \] Two men \(P\) and \(Q\) are at points on the sides of a square \(ABCD\) of side \(a\). \(P\) is at \(C\) and starts from \(C\) in a direction making an angle \(\theta\) with \(CB\), \(Q\) is at \(E\) on the side \(DA\), where \(DE=d\), \(EA=a-d\); \(Q\) starts from \(E\) at the same time as \(P\) starts from \(C\), and walks at twice \(P\)'s rate in a direction at right angles to the path of \(P\). Find the value of \(\theta\) necessary for \(P\) and \(Q\) to meet.
Obtain an expression for \(\sin x\) as a power series in \(x\), and give an expression for the remainder after \(n\) terms.
Draw the graph from \(x=0\) to \(x=\pi\) of \[ y = \sin x + \frac{1}{3}\sin 3x + \frac{1}{5}\sin 5x. \] Find the maximum and minimum points on the curve.
Evaluate the following: \[ \int_a^b \sqrt{(b-x)(x-a)}dx, \quad \int \frac{dx}{(a+b\sin x)\cos x}, \quad \int \frac{(1-x^2)dx}{(1+x^2)\sqrt{1+x^2+x^4}}. \]
Find an expression for the area of a closed curve in terms of polar coordinates. Show that the area enclosed by the curve \[ r=a \frac{\cos\theta+3\sin\theta}{(\cos\theta+2\sin\theta)^2} \] and the two radii from the origin for \(\theta=0\) and \(\theta=\frac{\pi}{4}\) is \(37a^2/162\).