Numerical Methods Exercise 5.3 Text Book Solutions

Dear Class 12 Samacheer Kalvi students, here are the Business Maths Chapter 5 Numerical Methods Exercise 5.3 Text Book Solutions for your study.

Exercise 5.3 – Choose the Correct Answer

$1. \; \Delta^{2}y_{0}=\\$
$(a) \; y_{2}-2 y_{1}+ y_{0}\\$
$(b)\; y_{2}+2 y_{1}- y_{0}\\$
$(c)\; y_{2}+2 y_{1}+ y_{0}\\$
$(d)\; y_{2}+ y_{1}+ 2y_{0}$

$Answer: \; (a) \; y_{2}-2 y_{1}+ y_{0}$

$2. \; \Delta f(x)=\\$
$(a) \; f(x+h)\\$
$(b)\; f(x)-f(x+h)\\$
$(c)\; f(x+h)- f(x)\\$
$(d)\; f(x)-f(x-h)$

$Answer: \;(c)\; f(x+h)- f(x)$

$3. E\equiv \\$
$(a) \; 1+\Delta\\$
$(b)\; 1-\Delta\\$
$(c)\; 1+\nabla\\$
$(d)\; 1-\nabla$

$Answer: \; (a) \; 1+\Delta$

$4.If\; h=1\; then\; \Delta (x^{2})=\\$
$(a) \; 2x\\$
$(b)\; 2x-1\\$
$(c)\; 2x+1\\$
$(d)\; 1$

$Answer: \; (c)\; 2x+1$

$5.If\; c \;is \;a \;constant \; then\; \Delta\; c=\\$
$(a) \; c\\$
$(b)\; \Delta\\$
$(c)\; \Delta^{2}\\$
$(d)\; 0$

$Answer: \; (d)\; 0$

$6.If\; m\;and \;n \;are\; positive\; integers \; then\; \Delta^{m}\; \Delta^{n} \; f(x)=\\$
$(a) \;\Delta^{m+n}\; f(x) \\$
$(b)\; \Delta^{m}\; f(x) \\$
$(c)\; \Delta^{n}\; f(x) \\$
$(d)\; \Delta^{m-n}\; f(x)$

$Answer: \; (a) \;\Delta^{m+n}\; f(x)$

$7.If\; n \;is \;a \;positive \;integer \; \Delta^{n} \; \left[ \Delta^{-n} f(x) \right]=\\$
$(a) \; f(2x) \\$
$(b)\; f(x+h) \\$
$(c)\; f(x) \\$
$(d)\; \Delta f(x)$

$Answer: \;(c)\; f(x)$

$8.E \;f(x) =\\$
$(a) \; f(x-h) \\$
$(b)\; f(x) \\$
$(c)\; f(x+h) \\$
$(d)\; f(x+2h)$

$Answer: \;(c)\; f(x+h)$

$9.\nabla\equiv \\$
$(a) \; 1+E \\$
$(b)\; 1-E \\$
$(c)\; 1-E^{-1} \\$
$(d)\; 1+E^{-1}$

$Answer: \;(c)\; 1-E^{-1}$

$10.\nabla\; f(a)=\\$
$(a) \;f(a)+f(a-h)\\$
$(b)\; f(a)-f(a+h)\\$
$(c)\; f(a)-f(a-h) \\$
$(d)\; f(a)$

$Answer: \;(c)\; f(a)-f(a-h)$

$11.\;For\; the\; given\; points\; (x_{0}, y_{0},) \; and \; (x_{1}, y_{1},)\; the\; Lagrange's \; formula; is\\$
$(a) \;y(x)=\dfrac{x-x_{1}}{x_{0}-x_{1}}y_{0}+\dfrac{x-x_{0}}{x_{1}-x_{0}}y_{1} \\$
$(b)\; y(x)=\dfrac{x_{1}-x}{x_{0}-x_{1}}y_{0}+\dfrac{x-x_{0}}{x_{1}-x_{0}}y_{1}\\$
$(c)\; y(x)=\dfrac{x-x_{1}}{x_{0}-x_{1}}y_{1}+\dfrac{x-x_{0}}{x_{1}-x_{0}}y_{0}\\$
$(d)\; y(x)=\dfrac{x_{1}-x}{x_{0}-x_{1}}y_{1}+\dfrac{x-x_{0}}{x_{1}-x_{0}}y_{0}$

$Answer: \; (a) \;y(x)=\dfrac{x-x_{1}}{x_{0}-x_{1}}y_{0}+\dfrac{x-x_{0}}{x_{1}-x_{0}}y_{1}$

$12.\;Lagrange's \;interpolation\; formula \;can \;be \;used \;for\\$
$(a) \;equal \;intervals \;only \\$
$(b)\; unequal \;intervals \;only\\$
$(c)\; both \;equal \;and\; unequal \;intervals\\$
$(d)\; none\; of \;these.$

$Answer: \;(c)\; both \;equal \;and\; unequal \;intervals$

$13.\;If \; f (x) \;= x^{2} + 2x + 2 \;and \;the\; interval\; of \;differencing\; is \;unity\; then \;\Delta f (x)\\$
$(a) \;2x-3 \\$
$(b)\; 2x+3\\$
$(c)\; x+3\\$
$(d)\; x-3$