\[For\,the\,given\,data\,{\text{points}}\,({x_{0,}}{y_0}),\,({x_1}{y_1}),\,({x_2}{y_2}),\,and\,({x_{3,}}{y_3})\,\,the\,zero - order\,divide\,difference\,will\,be\,given\,as\]

• A) \[y[{x_0}]\]
• B) \[y[{y_0}]\]
• C) \[y[{y_0},{y_1}]\]
• D) \[y[{x_0},{x_1}]\]

\begin{gathered} What\,will\,be\,the\,value\,of\,'a'\,in\,the\,given\,divide\,diffidence\,table? \hfill \\ \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D}&{3rdD.D} \\\ 1&{0.7}&{0.25}&{0.025}&{} \\\ 3&{1.2}&{0.35}&{ - 0.0625}&a \\\ 5&{1.9}&{0.1}&{}&{} \\\ 7&{2.1}&{}&{}&{} \end{array} \hfill \\\ \end{gathered}

• A) -0.0146
• B) -0.0387
• C) -0.0021
• D) -0.0245

If y(x) is approximated by a polynomial {P_n}(x) of degree n then the error is given by

• A) \varepsilon (x) = y(x)\,\, \div \,\,\,{P_n}(x)
• B) \varepsilon (x) = y(x) - {P_n}(x)
• C) \varepsilon (x) = y(x) + {P_n}(x)
• D) \varepsilon (x) = y(x)\,\, \times \,\,{P_n}(x)

\[For\,the\,given\,data\,{\text{points}}\,(1, - 3),\,(2,0),\,and\,(3,15),\,the\,first - order\,divide\,difference\,will\,be\,\]

• A) 3
• B) -2
• C) -3
• D) 2

If there are (n+2) values of y corresponding to (n+2) values of x, then we can represent the function f(x) by a polynomial of degree

• A) n+1
• B) n+2
• C) n-1
• D) n

In Romberg’s method, accuracy of Simpson and Trapezoidal rules is improved by ---------.

• A) extrapolation
• B) interpolation
• C)
• D)

\delta \,\, = \,\, - - - -

• A) {E^{\frac{1}{2}}}\,\, - \,\,{E^{ - \,\,\,\,\frac{1}{2}}}
• B) None
• C) \frac{{{E^{\frac{1}{2}}}\,\, + \,\,\,{E^{ - \,\,\,\,\frac{1}{2}}}}}{2}
• D) {E^{\frac{1}{2}}}\,\, + \,\,\,{E^{ - \,\,\,\,\frac{1}{2}}}

\Delta = - - -

• A) \frac{{E - 1}}{2}
• B) 1 - E
• C) None
• D) E\,\, - \,\,1

\begin{gathered} For\,the\,given\,four\,data\,{\text{point}}s,\,the\,{\text{degree}}\,of\,Lagrange's\,{\text{interpolation}}\,polynomial\,could\,be \hfill \\ \begin{array}{*{20}{c}} x&{0.3}&{0.7}&{0.9}&{1.0} \\\ y&{0.067}&{0.248}&{0.518}&{0.6812} \end{array} \hfill \\\ \end{gathered}

• A) three
• B) five
• C) six
• D) four

\[For\,the\,given\,data\,{\text{points}}\,(4,1.3),\,(8,1.5),\,and\,(12,1.9)\,\,the\,divide\,difference\,table\,will\,be\,given\,as\]

• A) \[\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 4&{1.3}&{0.0062}&{} \\\ 8&{1.5}&{0.1}&{0.05} \\\ {12}&{1.9}&{}&{} \end{array}\]
• B) \[\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 4&{1.3}&{0.05}&{} \\\ 8&{1.5}&{0.1}&{0.0062} \\\ {12}&{1.9}&{}&{} \end{array}\]
• C) \[\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 4&{1.3}&{0.1}&{} \\\ 8&{1.5}&{0.0062}&{0.05} \\\ {12}&{1.9}&{}&{} \end{array}\]
• D) \[\begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 4&{1.3}&{0.1}&{} \\\ 8&{1.5}&{0.35}&{0.0062} \\\ {12}&{1.9}&{}&{} \end{array}\]

\[\begin{gathered} What\,will\,be\,the\,value\,of\,'a'\,in\,the\,given\,divide\,diffidence\,table? \hfill \\ \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D}&{3rdD.D} \\\ 1&{0.7}&{0.25}&{0.025}&{} \\\ 3&{1.2}&{0.35}&{ - 0.0625}&a \\\ 5&{1.9}&{0.1}&{}&{} \\\ 7&{2.1}&{}&{}&{} \end{array} \hfill \\\ \end{gathered} \]

• A) -0.0146
• B) -0.0245
• C) -0.0387
• D) -0.0021

$\delta \,\, = \,\, - - - - $

• A) $\frac{{{E^{\frac{1}{2}}}\,\, + \,\,\,{E^{ - \,\,\,\,\frac{1}{2}}}}}{2}$
• B) None
• C) \[{E^{\frac{1}{2}}}\,\, - \,\,{E^{ - \,\,\,\,\frac{1}{2}}}\]
• D) ${E^{\frac{1}{2}}}\,\, + \,\,\,{E^{ - \,\,\,\,\frac{1}{2}}}$

\[If\,only\,two\,data\,{\text{point}}s\,are\,given,\,the\,formula\,for\,Lagrange's\,{\text{interpolation}}\,polynomial\,will\,be\]

• A) \[y = f(x) = \frac{{(x - {x_1})}}{{({x_0} - {x_1})}}{y_0} + \frac{{(x - {x_0})}}{{({x_1} - {x_0})}}{y_1}\]
• B) \[y = f(x) = \frac{{(x - {x_0})}}{{({x_1} - {x_0})}}{y_0} + \frac{{(x - {x_1})}}{{({x_0} - {x_1})}}{y_1}\]
• C) \[y = f(x) = \frac{{({x_1} - {x_0})}}{{(x - {x_0})}}{y_0} + \frac{{({x_0} - {x_1})}}{{(x - {x_1})}}{y_1}\]
• D) \[y = f(x) = \frac{{(x - {x_0})}}{{({x_0} - {x_1})}}{y_0} + \frac{{(x - {x_1})}}{{({x_1} - {x_0})}}{y_1}\]

\[For\,the\,given\,data\,{\text{points}}\,({x_{0,}}{y_0}),\,({x_1}{y_1}),\,({x_2}{y_2}),\,and\,({x_{3,}}{y_3})\,\,the\,{\text{second}} - order\,divide\,difference\,will\,be\,given\,as\]

• A) \[y[{x_0},{x_1},{x_2},{x_3}]\]
• B) \[y[{x_0},{x_1}]\]
• C) \[y[{x_0},{x_1},{x_2}]\]
• D) \[y[{x_0}]\]

\[\begin{gathered} For\,the\,giev\,two\,data\,{\text{point}}s\,,\,the\,{\text{degree}}\,of\,Lagrange's\,{\text{interpolation}}\,polynomial\,could\,be \hfill \\ \begin{array}{*{20}{c}} x&{0.3}&{0.7}&{} \\\ y&{0.067}&{0.248}&{} \end{array} \hfill \\\ \end{gathered} \]

• A) three
• B) two
• C) one
• D) four

For\,the\,given\,data\,{\text{points}}\,({x_{0,}}{y_0}),\,({x_1}{y_1}),\,({x_2}{y_2}),\,and\,({x_{3,}}{y_3})\,\,the\,{\text{second}} - order\,divide\,difference\,will\,be\,given\,as

• A) y[{x_0},{x_1}]
• B) y[{x_0}]
• C) y[{x_0},{x_1},{x_2},{x_3}]
• D) y[{x_0},{x_1},{x_2}]

\begin{gathered} What\,will\,be\,the\,value\,of\,'a'\,in\,the\,given\,divide\,difference\,table? \hfill \\ \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 3&{0.4}&{}&{} \\\ 6&{0.9}&{0.1667}&{} \\\ 9&{1.7}&{0.2667}&a \end{array} \hfill \\\ \end{gathered}

• A) 0.0211
• B) 0.0254
• C) 0.0167
• D) 0.0349

For\,the\,given\,data\,{\text{points}}\,(1,0.3),\,(3,1),\,and\,(5,1.2)\,\,the\,divide\,difference\,table\,will\,be\,given\,as

• A) \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.0625} \\\ 6&{1.2}&{}&{} \end{array}
• B) \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.525} \\\ 6&{1.2}&{}&{} \end{array}
• C) \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.125} \\\ 6&{1.2}&{}&{} \end{array}
• D) \begin{array}{*{20}{c}} x&y&{1stD.D}&{2ndD.D} \\\ 2&{0.3}&{0.35}&{} \\\ 4&1&{0.1}&{ - 0.225} \\\ 6&{1.2}&{}&{} \end{array}

\[If\,any\,three\,data\,{\text{point}}s\,are\,given,\,the\,formula\,for\,Lagrange's\,{\text{interpolation}}\,polynomial\,will\,be\]

• A) \[y = f(x) = \frac{{(x - {x_1})(x - {x_2})}}{{({x_1} - {x_0})({x_1} - {x_2})}}{y_0} + \frac{{(x - {x_0})(x - {x_2})}}{{({x_0} - {x_1})({x_0} - {x_2})}}{y_1} + \frac{{(x - {x_0})(x - {x_1})}}{{({x_2} - {x_0})({x_2} - {x_1})}}{y_2}\]
• B) \[y = f(x) = \frac{{(x - {x_1})(x - {x_2})}}{{({x_0} - {x_1})({x_0} - {x_2})}}{y_0} + \frac{{(x - {x_0})(x - {x_2})}}{{({x_1} - {x_0})({x_1} - {x_2})}}{y_1} + \frac{{(x - {x_0})(x - {x_1})}}{{({x_2} - {x_0})({x_2} - {x_1})}}{y_2}\]
• C) \[y = f(x) = \frac{{(x - {x_1})(x - {x_2})}}{{({x_0} - {x_1})({x_0} - {x_2})}}{y_2} + \frac{{(x - {x_0})(x - {x_2})}}{{({x_1} - {x_0})({x_1} - {x_2})}}{y_1} + \frac{{(x - {x_0})(x - {x_1})}}{{({x_2} - {x_0})({x_2} - {x_1})}}{y_0}\]
• D) \[y = f(x) = \frac{{({x_0} - {x_1})({x_0} - {x_2})}}{{(x - {x_1})(x - {x_2})}}{y_0} + \frac{{({x_1} - {x_0})({x_1} - {x_2})}}{{(x - {x_0})(x - {x_2})}}{y_1} + \frac{{({x_2} - {x_0})({x_2} - {x_1})}}{{(x - {x_0})(x - {x_1})}}{y_2}\]

\[For\,the\,given\,data\,{\text{points}}\,(4,45),\,(5,104),\,and\,(6,190),\,the\,{\text{zero}} - order\,divide\,difference\,will\,be\,\]

• A) 42
• B) none
• C) 35
• D) 46