Exercise 5.3 - Chapter 5 Numerical Methods 12th Maths Guide Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated On May 15, 2024

By SaraNextGen

Ex $5.3$
Choose the correct answer.
Question 1.
$
\Delta^2 \mathrm{y}_0=
$
(a) $\mathrm{y}_2-2 \mathrm{y}_1+\mathrm{y}_0$
(b) $\mathrm{y}_2+2 \mathrm{y}_1-\mathrm{y}_0$
(c) $\mathrm{y}_2+2 \mathrm{y}_1+\mathrm{y}_0$
(d) $y_2-y_1+2 y_0$
Answer:
(a) $y_2-2 y_1+y_0$
Hint:
$
\begin{aligned}
& \Delta^2 \mathrm{y}_0=\Delta\left(\Delta \mathrm{y}_0\right)=\Delta\left(\mathrm{y}_1-\mathrm{y}_0\right)=\Delta \mathrm{y}_1-\Delta \mathrm{y}_0 \\
& =\left(\mathrm{y}_2-\mathrm{y}_1\right)-\left(\mathrm{y}_1-\mathrm{y}_0\right) \\
& =\mathrm{y}_2-2 \mathrm{y}_1+\mathrm{y}_0
\end{aligned}
$
Question 2.
$\Delta \mathrm{f}(\mathrm{x})=$
(a) $f(x+h)$
(b) $f(x)-f(x+h)$
(c) $f(x+h)-f(x)$
(d) $f(x)-f(x-h)$
Answer:
(c) $f(x+h)-f(x)$
Hint:
$
\Delta \mathrm{f}(\mathrm{x})=\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}(\mathrm{x})
$
Question 3 .
$\mathrm{E}=$
(a) $1+\Delta$
(b) $1-\Delta$
(c) $1+\nabla$
(d) $1-\nabla$
Answer:
(a) $1+\Delta$

Hint:
$
\mathrm{E}=1+\Delta
$
Question 4.
If $\mathrm{h}=1$, then $\Delta\left(\mathrm{x}^2\right)=$
(a) $2 \mathrm{x}$
(b) $2 \mathrm{x}-1$
(c) $2 x+1$
(d) 1
Answer:
(c) $2 \mathrm{x}+1$
Hint:
$
\Delta\left(\mathrm{x}^2\right)=(\mathrm{x}+\mathrm{h})^2-\mathrm{x}^2=(\mathrm{x}+1)^2-\mathrm{x}^2=2 \mathrm{x}+1
$
Question 5 .
If $\mathrm{c}$ is a constant then $\Delta \mathrm{c}=$
(a) c
(b) $\Delta$
(c) $\Delta^2$
(d) 0
Answer:
(d) 0
Question 6.
If $\mathrm{m}$ and $\mathrm{n}$ are positive integers then $\Delta^{\mathrm{m}} \Delta^{\mathrm{n}} \mathrm{f}(\mathrm{x})=$
(a) $\Delta^{\mathrm{m}+\mathrm{n}} \mathrm{f}(\mathrm{x})$
(b) $\Delta^{\mathrm{m}} \mathrm{f}(\mathrm{x})$
(c) $\Delta^{\mathrm{n}} \mathrm{f}(\mathrm{x})$
(d) $\Delta^{\mathrm{m}-\mathrm{n}} \mathrm{f}(\mathrm{x})$
Answer:

(a) $\Delta^{\mathrm{m}+\mathrm{n}} \mathrm{f}(\mathrm{x})$
Question 7.
If ' $n$ ' is a positive integer $\Delta^{\mathrm{n}}\left[\Delta^{-\mathrm{n}} f(\mathrm{x})\right]$
(a) $f(2 x)$
(b) $f(x+h)$
(c) $f(x)$
(d) $\Delta \mathrm{f}(2 \mathrm{x})$
Answer:
(c) $f(x)$
Question 8.
$\mathrm{E} f(\mathrm{x})=$
(a) $f(x-h)$
(b) $f(x)$
(c) $f(x+h)$
(d) $f(x+2 h)$
Answer:
(c) $f(x+h)$
Question 9.
$\nabla=$
(a) $1+\mathrm{E}-1$
(b) $1-\mathrm{E}$
(c) $1-E^{-1}$
(d) $1+\mathrm{E}$
Answer:
(c) $1-\mathrm{E}^{-1}$

Question 10.
$\nabla \mathrm{f}(\mathrm{a})=$
(a) $f(a)+f(a-h)$
(b) $f(a)-f(a+h)$
(c) $f(a)-f(a-h)$
(d) $f(a)$
Answer:
(c) $f(a)-f(a-h)$
Question 11.
For the given points $\left(\mathrm{x}_0, \mathrm{y}_0\right)$ and $\left(\mathrm{x}_1, \mathrm{y}_1\right)$ the Lagrange's formula is
(a) $y(x)=\frac{x-x_1}{x_0-x_1} y_0+\frac{x-x_0}{x_1-x_0} y_1$
(b) $y(x)=\frac{x_1-x}{x_0-x_1} y_0+\frac{x-x_0}{x_1-x_0} y_1$
(c) $y(x)=\frac{x-x_1}{x_0-x_1} y_1+\frac{x-x_0}{x_1-x_0} y_0$
(d) $y(x)=\frac{x_1-x}{x_0-x_1} y_1+\frac{x-x_0}{x_1-x_0} y_0$
Answer:
(a) $y(x)=\frac{x-x_1}{x_0-x_1} y_0+\frac{x-x_0}{x_1-x_0} y_1$
Question 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

Question 13.
If $f(x)=x^2+2 x+2$ and the interval of differencing is unity then $\Delta f(x)$
(a) $2 x-3$
(b) $2 x+3$
(c) $x+3$
(d) $x-3$
Answer:
(b) $2 \mathrm{x}+3$
Hint:
$
\begin{aligned}
& \mathrm{f}(\mathrm{x})=2 \mathrm{x}^2+2 \mathrm{x}+2 \\
& \mathrm{~h}=1 \\
& \Delta \mathrm{f}(\mathrm{x})=(\mathrm{x}+1)^2+2(\mathrm{x}+1)+2-\mathrm{x}^2-2 \mathrm{x}-2 \\
& =\mathrm{x}^2+2 \mathrm{x}+1+2 \mathrm{x}+2+2-\mathrm{x}^2-2 \mathrm{x}-2 \\
& =2 \mathrm{x}+3
\end{aligned}
$
Question 14.
For the given data find the value of $\Delta^3 \mathrm{y}_0$ is

(a) 1
(b) 0
(c) 2
(d) $-1$
Answer:
(b) 0
Hint:
$
\begin{aligned}
\Delta^3 y_0 & =(\mathrm{E}-1)^3 y_0 \\
& =\left(\mathrm{E}^3-3 \mathrm{E}^2+3 \mathrm{E}-1\right) y_0 \\
& =\mathrm{E}^3 y_0-3 \mathrm{E}^2 y_0+3 \mathrm{E} y_0-y_0 \\
& =y_3-3 y_2+3 y_1-y_0 \\
& =18-3(15)+3(13)-12 \\
& =18-45+39-12=0
\end{aligned}
$