MTH213 TMA Questiona and Answers

Q1 One of these is not a process involved in the Gauss elimination method
 It is a direct method for solving large systems
 The coefficient matrix A is reduced using the elementary row operations
 The solution vector is obtained using the factorisation method
 For large system n, the total number of operations required in Gauss elimination method is approximately\[\frac{1}{3}n^3\]

Q2 Every non-singular matrix A can be written as
 the sum of the lower triangular matrix and the upper triangular matrix together with all principal minors of A
 the difference between the lower triangular matrix and and the upper triangular matrix are singular
 the product of a lower triangular matrix and an upper triangular matrix if all principal minors of A are singular
 the product of a lower riangular matrix and an upper triangular matrix if all principal minors of A are non-singular

Q3 The diagonal elements used as divisors are called
 diaments
 pivots
 eliminators
 nodes

Q4 Using the Gauss Eliminating method, the elementary row operations include the following except
 interchanging any two rows
 adding (or subtracting) any row by a non-zero constant
 multiplying (or dividing) any row by a non-zero constant
 adding (or subtracting) a constant multiple of one row to another row

Q5 Find the product of \[\begin{bmatrix}2&3&-1\end{bmatrix}\] and \[\begin{bmatrix}1\\4\\2\end{bmatrix}\]
 \[\begin{bmatrix}2&12&-2\end{bmatrix}\]
 \[\begin{bmatrix}2&3&-1\\1&4&2\end{bmatrix}\]
 12
 The product does not exist

Q6 The transpose of matrix \[\begin{bmatrix}1&0&-7\\0&-2&3\\4&5&6\end{bmatrix}\]
 \[\begin{bmatrix}-1&0&-4\\0&2&-5\\7&-3&-6\end{bmatrix}\]
 \[\begin{bmatrix}1&0&4\\0&-2&5\\-7&3&6\end{bmatrix}\]
 -83
 -59

Q7 The inverse matrix of \[\begin{bmatrix}2&3\\5&6\end{bmatrix}\]
 \[\begin{bmatrix}2&-1\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\]
 a 2x2 matrix
 \[\begin{bmatrix}-2&1\\\frac{5}{3}&-\frac{2}{3}\end{bmatrix}\]
 not defined

Q8 The determinant of \[\begin{bmatrix}2&3\\5&6\\1&-7\end{bmatrix}\]
 -3
 29
 infinity
 cannot be determined

Q9 The Lagrange form of interpolating polynomial can be determined for
 both equally and unequally spaced nodes
 equally spaced nodes only
 unequally spaced nodes only
 unique space

Q10 The interpolating polynomial for a given data is �?��?�.
 useful
 not very useful
 unique
 universal

Q11 Any problem involving inverse interpolation can always be solved
 if f(x) is continuous only
 if f(x) is continuous and strictly increasing or decreasing
 if f(x) is discontinuous and strictly increasing or decreasing
 if f(x) is continuous and strictly increasing only

Q12 �?��?��?�.. Theorem states that 'If f is a continuous function defined on [a, b] and differentiable on ]a, b[; and If f(a) = f(b), then there exists a number \[x_{0}\; in; ]a, b[ \;such\; that \;f '(x_{0}) = 0\]
 Pythagoras
 Chebyshev
 Rolle's
 Lagrange�??s Mean Value

Q13 One of the following describes Taylor's Theorem
 \[f(x)=f(x_{0})+\frac{x-x_0}{1!}f'(x_0)+\frac{x-x_0}^2{2!}f^(2)(x_0)+�?�+\frac{x-x_0}^n{n!}f^(n)(x_0)+\frac{x-x_0}^(n+1){(n+1)!}f^(n+1)(x_0)
 \[f(x)=f(x_{0})+\frac{x}{1!}f'(x_0)+\frac{x^2}{2!}f^(2)(x_0)+�?�+\frac{x-x_0}^n{n!}f^(n)(x_0)+\frac{x-x_0}^(n+1){(n+1)!}f^(n+1)(x_0)
 \[f(x)=f(x_{0})+\frac{x-x_0}{2!}f'(x_0)+\frac{x-x_0}^2{3!}f^(2)(x_0)+�?�+\frac{x-x_0}^n{n!}f^(n)(x_0)+\frac{x-x_0}^(n+1){(n+1)!}f^(n+1)(x_0)
 \[f(x)=f(x_{0})+\frac{x_0}{1!}f'(x_0)+\frac{x_0}^2{2!}f^(2)(x_0)+�?�+\frac{x_0}^n{n!}f^(n)(x_0)+\frac{x_0}^(n+1){(n+1)!}f^(n+1)(x_0)

Q14 \[If\; z_1, z_2, �?�, z_k\] are distinct zeros of the polynomial P(x), then
 \[P(x) = (x �?? z_1) (x �?? z_2)...(x �?? z_k)N(x) for some natural number N(x)
 \[P(x) = (x �?? z_1) (x �?? z_2)�?�.(x �?? z_k)R(x)\] for some natural number R(x)
 \[P(x) = (x �?? z) (x �?? z^2)...(x �?? z^k)R(x)\] for some polynomial R(x)
 \[P(x) = (x �?? z_1) (x �?? z_2)�?�.(x �?? z_k)R(x)\] for some polynomial R(x)

Q15 All but one are not true of divided difference
 The divided difference \[f[x^1, x^2, �?�., x^k]\] is not invariant under all permutations of the arguments \[x_1, x_2, �?�., x_k\]
 The divided difference \[f[x^1, x^2, �?�., x^k]\] is invariant under all permutations of the arguments \[x_1, x_2, �?�., x_k\]
 The divided difference \[f[x_1, x_2, �?�., x_k]\] is a variant under all permutations of the arguments \[x_1, x_2, �?�., x_k\]
 The divided difference \[f[x_1, x_2, �?�., x_k]\] is invariant under all permutations of the arguments \[x_1, x_2, �?�., x_k\]

Q16 This best describes the fundamental theorem of algebra
 If P(x) is a polynomial of degree n < 1 that is \[P(x) = a_{-n}x_{-n} + a_{n-1]x_{n-1} + �?� + a_1x + a_0, �?�, a_n\] real or complex numbers and \[a_n \neq 0,\] then P(x) has at least one zero, that is, there exists a real or complex number \[\gamma\] such that \[p(\gamma )= 0\]
 If P(x) is a polynomial of degree \[n \geq1\] that is \[P(x) = a_nx_n + a_{n-1]x_{n-1} + �?� + a_1x + a_0, �?�, a_n\] real or complex numbers and \[a_n \neq 0,\] then P(x) has at least one zero, that is, there exists a real or complex number \[\gamma\] such that \[p(\gamma )= 0\]
 If P(x) is a polynomial of degree \[n \leq 1\] that \[P(x) = a_nx_n + a_{n-1]x_{n-1} + �?� + a_1x + a_0, �?�, a_n\] real or complex numbers and \[a_n \neq 0,\] then P(x) has at least one zero, that is, there exists a real or complex number \[\gamma\] such that \[p(\gamma )= 0\]
 If P(x) is a polynomial of degree n >1, that is \[P(x) = a_{n-1]x_{n-1} + �?� + a_1x + a_0, �?�, a_n\] real or complex numbers and \[a_n \neq 0,\] then P(x) has at least one zero, that is, there exists a real or complex number \[\gamma\] such that \[p(\gamma )= 0\]

Q17 a polynomial P(x) of degree n which agrees with the values of f(x) at the given (n + 1) distinct points is known as �?��?�..
 notes
 nodes
 note
 nets

Q18 �?��?��?� is the process of determining the value of f(x) for a value of x lying outside the interval [a, b].
 interpolation
 intrapolation
 exterpolation
 extrapolation

Q19 The technique of determining an approximate value of f(x) for a non-tabular value of x which lies in the interval [a, b] is known as �?��?��?�.
 interpolation
 intrapolation
 exterpolation
 extrapolation

Q20 The eigenvalues of a diagonal matrix are �?��?��?�.
 0 & 1
 the diagonal elements themselves
 all zeroes
 all unity

Q21 The inverse of a matrix A is given by
 \[A^{-1} = \frac{1}{det(A)} adj(A)\]
 \[A = \frac{1}{det(A)}(A)\]
 \[A = det(A) adj(A)\]
 \[A = \frac{1}{det(A)} adj(A)\]

Q22 If AB = BA = I ,then A is the �?��?��?��?�.. of B
 index
 identity
 inverse
 inflection

Q23 One of the following is not true
 A square matrix is said to be lower triangular if all the elements below the main diagonal vanish
 Addition of matrices is defined only for matrices of same dimension
 A determinant can be expanded about any row or column
 The determinant of a diagonal matrix is same as the product of its diagonal elements

Q24 Iterative methods of the solutions of systems of equations are
 sequential
 non-sequential
 finite
 infinite

Q25 Solve the system of equations: x + y + z = 1; 4x +3y - z = 6; 3x + 5y + 3z = 4
 x = 0.5; y = 1; z = -0.5
 x = 0.5; y = 1; z = - 2
 x = 1.5; y = 1; z = - 2
 x = 1 ; y = 0.5 ; z = -0.5

Q26 A system of linear equations is said to be �?��?�..... if it has at least one solution
 homogeneous
 non-homogeneous
 consistent
 inconsistent

Q27 A system of linear equations is said to be �?��?��?�.. if all its right hand side is zero
 homogeneous
 non-homogeneous
 consistent
 inconsistent

Q28 A system of linear equations is said to be �?��?��?�.. if no solution exists
 homogeneous
 non-homogeneous
 consistent
 inconsistent

Q29 A system of linear equations is said to be �?��?�... if not all its right hand side is zero
 homogeneous
 non-homogeneous
 consistent
 inconsistent

Q30 The following statements are true except
 The divided difference is independent of the order of its arguments
 A square matrix is lower triangular if all the elements above the main diagonal vanish
 The divided difference is dependent of the order of its arguments
 The identity matrix I is a diagonal matrix in which each of the diagonal elements is unity

Q31 The zeroeth divided difference of the function f, with respect to \[ x_{i} \; denoted\; by\; f[x_{i}]\] can be written as
 \[f[x] = f(x_{i})\]
 \[f[x] = f(xi)\]
 \[ f[x_{i}] = f(x_{i})\]
 \[f[x_{0}] = f(x_{i})\]

Q32 The interpolating polynomial of degree \[\leq n\;with\; the\; nodes\; x_0, x_1, �?�., x_n\] can be written as
 \[ f[x_{i}] +f(x_{i}=0)\]
 \[ f[x_{i}] - f(x_{i}=0)\]
 \[ f[x_{i}] = f(x_{i})\]
 $$ f[x_{i}] + f(x^i)=1\]

Q33 Stirling's formula for interpolation is given by
 \[P_n (x) = f (x_0) + \frac{s}{2}[df_\frac{1}{2} + d_\frac{-1}{2}] + \frac{s^2}{2!}d^2f_0\]
 \[P_n (x) = f (x_0) + \frac{s - 1}{2}[df_1+ d_\frac{-1}{2}] + \frac{s^2}{2!}d^2f_0\]
 \[P_n (x) = f (x_0) + \frac{s}{2}[df_\frac{1}{2} + d_\frac{-1}{2}] + \frac{s^2}{2!}df_0\]
 \[P_n (x) = f (x_0) + \frac{s}{2}[df_\frac{1}{2} + d_\frac{-1}{2}] + \frac{s^2}{2}d^2f_0\]

Q34 When the number of rows is equal to the number of columns in a matrix, it is called �?��?�. matrix
 equal
 similar
 singular
 square

Q35 A matrix in which all the non-diagonal elements vanish is called �?�..... matrix.
 vanishing
 non-vanishing
 diagonal
 non-diagonal

Q36 The technique of determining an approximate value of f(x) for a non-tabular value of x which lies in the internal [a, b] is referred to as
 extrapolation
 exterpolation
 intrapolation
 interpolation

Q37 One of these describes the Lagrange�??s interpolating polynomial P(x)
 \[P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\]
 \[P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\]
 \[p(x) = L_{1}(x)f_{0} + L_{2}(x)f_{1} + L_{3}(x)f_{2} + L_{4}(x)f_{3}\]
 \[P(x) = L_{0}(x)f_{0} + L_{1}(x)f_{1} + L_{2}(x)f_{2} + L_{3}(x)f_{3}\]

Q38 Iterative methods of the solutions of systems of equations are �?��?��?��?��?�..
 sequential
 non-sequential
 finite
 infinite

Q39 One of these is a method of solving system of linear equations:
 Inverse method
 Lagrange Method
 Transformation Method
 Square Method

Q40 If all the elements above the main diagonal of a square matrix vanish, the matrix is known as
 lower triangular
 upper triangular
 half triangular
 side triangular