Problems

The series \(1+3x+7x^2+\dots+p_nx^n+\dots\) is such that \[ p_{n+1}=3p_n-2p_{n-1}; \] find the value of \(p_n\). Prove that the coefficient of \(x^n\) in the expansion, in ascending powers of \(x\), of \(\dfrac{(1+x)}{(1-x)^3}\) is \((n+2)2^{n-1}\).

If the sum of two positive numbers is given, prove that their product is greatest when they are equal. If \(a+b=4\), and \(a\) and \(b\) are positive, shew that \[ a^2+\frac{1}{a^2}+b^2+\frac{1}{b^2} \ge 8\frac{1}{2}. \]

The equation \(4x^5-57x^3+64x^2+108x-144=0\) has two roots which are equal in magnitude and opposite in sign. Solve it completely.

Deduce from the parallelogram of forces that the algebraic sum of the moments of two non-parallel forces about any point is equal to the moment of their resultant about that point. Forces \(P,Q,R\) act along the sides \(BC, AC, BA\) of a triangle respectively; find their ratios when their resultant is a force parallel to \(BC\) through the centroid of the triangle.

State the laws of friction, and explain the terms angle of friction, cone of friction. A rod rests in equilibrium against a hemispherical boss on a horizontal plane with one end on the plane. If the plane is smooth and the boss is rough, prove that the inclination of the rod to the horizontal cannot be greater than the angle of friction.

Two motor cars \(A, B\) are travelling along straight roads at right angles to one another, with uniform velocities of 21 miles an hour and 28 miles an hour respectively, towards \(C\) the point at which the roads cross. If \(AC\) is half a mile when \(BC\) is three-quarters of a mile, find the shortest distance between the cars during the subsequent motion.

A particle is projected at an angle \(\alpha+\theta\) to the horizontal from a point on an inclined plane of angle \(\alpha\) so that its path is in the same vertical plane as a line of greatest slope; if it strikes the plane at right angles, prove that \(\theta\) is given by \(2\tan\alpha\tan\theta=1\), and that the greatest distance from the plane is \[ \frac{u^2\cos^2\alpha}{2g(1+3\sin^2\alpha)}. \]

A mass \(M\) is moving with velocity \(V\). It encounters a constant resistance \(F\); write down equations to determine the time before it is brought to rest and the distance it has travelled, stating the principles on which these equations depend. Two moving masses are brought to rest by equal constant resistances. If the one mass moves for twice as long as the other but goes only half the distance, find the ratio of the masses and also that of their velocities.

If \[ (1+x)^n = c_0 + c_1 x + c_2 x^2 + \dots + c_n x^n, \] prove that

1. [(i)] \(c_0 - c_1 + c_2 - c_3 + \dots + (-1)^r c_r = \frac{(-1)^r(n-1)!}{r!(n-r-1)!} \quad (r
2. [(ii)] \(c_0c_1 + c_1c_2 + \dots + c_{n-1}c_n = \frac{2n!}{(n-1)!(n+1)!}\).

If \[ (1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \dots + a_{2n}x^{2n}, \] prove that \[ a_0 + a_3 + a_6 + \dots = a_1 + a_4 + a_7 + \dots = a_2 + a_5 + a_8 + \dots. \]

Shew that, if \(c^2=a^2d\), then the product of two of the roots of the equation \[ x^4 + ax^3 + bx^2 + cx + d = 0 \] is equal to the product of the other two. Hence, or otherwise, solve the equation \[ x^4 + x^3 + 2x^2 + 2x + 4 = 0. \]