Mathematics Essay Example | Topics and Well Written Essays - 2500 words

Mathematics - Essay Example (c) Let us take values u1 = 1 u2 = 2 u3 = 3 u4 = 4 p = q = 1 Putting these values in the MS Office Excel gives the following output. Input Parameters u1 = 1 u2 = 2 u3 = 3 u4 = 4 p = 1 q = 1 Output Values a = 1 b = 1 c = 2 d = 0 The excel file is also attached an if the input values for u1, u2, u3, u4, p and q are changed the solution values a, b, c an d will also get suitably changed. Q1. (d) A set of ‘m’ linear equations in ‘n’ variables is expressed by the following equation in terms of matrix notation: Ax = b Where A is ‘mxn’ matrix of the coefficients of the system x is ‘nx1’ column vector and b is ‘mx1’ column vector If the ‘b’ vector is a zero vector i.e. all the elements of this vector are zero, then the system of equations is called a ‘Homogeneous System’ If the ‘b’ vector is non-zero vector i.e. if even one of the elements is non-zero then the system of equations is termed as ‘Nonhomogenous System’. Q1. (e) A homogeneous system always has a trivial solution i.e. a solution vector with all the elements being zero. However, for a homogeneous system to have a non-trivial solution the Determinant of Matrix A must be equal to zero. i.e. for non – trivial solution of Ax = b (b = 0) Determinant A = 0 Q2. (a) The profile of the boiler shell is made by revolution of a parabola. Let us assume that equation of the parabola is y = a + bx2 Let us place centre of the co-ordinate system at the middle of the shell. Then, at x = 0 y = 2 i.e. 2 = a + b*02; or, a = 2 And at x = 4 y = 1.5 i.e. 1.5 = 2 + b*(42) or, b = - (1/32) Hence equation of the parabola is Y = 2 – (1/32)x2 where, -4 < x < 4 Plot of the parabola is shown below. If this parabolic profile is rotated about x-axis, it will produce the shell of the boiler and that will enclose a volume, which will be the volume of water that can be contained and hence boiled in this boiler. Esse ntially, what is required is to calculate the volume enclosed by this rotation. The volume of such a boiler will be Therefore, V = 85 m3 Hence, 85 m3 water can be boiled in the boiler. Q2. (b) Integration is essentially summation and therefore, it is important to realize as what is it that is integrated or summed up. It is essentially the product of the dependent variable (y) and infinitesimally small increment in the independent variable or ?x which is continuously summed up. If we know from which point to which point this summation is to be done, then we get a definite answer and this integral is known as definite integral. Mathematically it is expressed by indicating the limits or boundaries of integration as shown below. This is a definite integral with integration being carried out between ‘a’ and ‘b’ (a < b) for y, which is a function of x. This definite integral gives many useful parameters like area under curve, area of a curved surface, volume of a container etc. as shown in Q2. (a), where volume of the boiler was calculated using a definite integral. However, many times we many not know the limits of integration, this is where we are not solving any particular physical or engineering problem but just interested in