Prove that the area of the curved surface and the volume of a segment of height \(h\) of a sphere of radius \(a\) are \(2\pi ah\) and \(\frac{1}{3}\pi h^2(3a-h)\). The whole area (curved and plane) of a segment of a sphere is given to be equal to \(\pi c^2\). Prove that when the volume is greatest the height of the segment is \(c\).
Trace the curve \(x^3+y^3-2ax^2=0\).
Find an expression for all the values of \(\theta\) satisfying the equation \(\sin\theta=\sin\alpha\). If \(\theta_1, \theta_2\) are the two values of \(\theta\) not differing by a multiple of \(\pi\) which satisfy the equation \[ \frac{\sin\theta\sin\phi}{\sin\alpha} + \frac{\cos\theta\cos\phi}{\cos\alpha} + 1 = 0, \] prove that \[ \cos(\theta_1+\theta_2) = \frac{\sin^2\alpha-\sin^2\phi}{\sin^2\alpha\cos^2\phi+\cos^2\alpha\sin^2\phi}. \]
In any triangle prove the formulae
Prove that \((\cos\theta+i\sin\theta)^{p/q}\), where \(p\) and \(q\) are integers, has \(q\) values. Find all the values of \((\sqrt{-1})^{\sqrt{-1}}\).
\(S\) is the area of a quadrilateral of which \(a,b,c,d\) are the sides, \(x,y\) the diagonals, and \(2\alpha\) the sum of two opposite angles, prove that \begin{align*} S^2 &= (s-a)(s-b)(s-c)(s-d)-abcd\cos^2\alpha, \\ x^2y^2 &= (ac+bd)^2 - 4abcd\cos^2\alpha, \end{align*} where \(2S = a+b+c+d\).
Prove that the sum of the moments of a system of two intersecting forces about any point in their plane is equal to the moment of their resultant. Forces \(P, 2P, 3P, 4P\) act along the sides \(AB, BC, CD, DA\) respectively of the square \(ABCD\). Find the magnitude of their resultant and the points in which it cuts the sides \(AB\) and \(BC\).
State the laws of statical friction. A heavy circular hoop is hung over a rough peg. A weight equal to that of the hoop is attached to it at a given point. Find the coefficient of friction between the peg and the hoop so that the system may hang in equilibrium whatever point of the hoop is placed in contact with the peg.
State the principle of Virtual Work, and prove it for a system of coplanar forces acting on a rigid body. A parallelogram \(ABCD\) is formed of uniform heavy rods freely jointed at their extremities. \(AB\) is held fixed in a horizontal position and the parallelogram is maintained in its form so that \(ADC\) is an acute angle \(\alpha\) by means of a string joining \(A\) to a point \(P\) in \(DC\). Prove that the tension of the string is \(W \cdot AP\cot\alpha/DP\), where \(W\) is half the weight of the parallelogram.
A particle is projected with velocity \(V\), from a point on an inclined plane, at an angle \(\beta\) to the line of greatest slope and in the vertical plane containing this line. Find the range on the inclined plane if its inclination to the horizontal is \(\alpha\). If \(\alpha\) is \(45^\circ\) and \(\beta\) is \(15^\circ\), shew that the direction of motion of the particle when it strikes the inclined plane makes an angle of \(30^\circ\) with the line of greatest slope.