Embibe Experts Solutions for Chapter: Numerical Methods, Exercise 1: CG PET 2019

One root of the equation $x^3-4x+1=0$ is between $1$ and $2$. The value of this root using Newton-Raphson method will be

One root of the equation, $x^3-3x-5=0$ lies between $2$ and $2.5$. Using Newton-Raphson method, the value of that root will be

For the solution of equation $f\left(x\right)=0$ by the Newton-Raphson method, the value of $x$ tends to root of the equation highly when $f^{\text{'}}\left(x_n\right)$ is

Dividing the interval $\left(1,2\right)$ into four equal parts and using Simpson's rule, the value of ${\int }_1^2\frac{dx}{x}$ will be

Taking four subintervals, the value of ${\int }_0^1\frac{dx}{1+x}$ by using trapezoidal rule will be

Taking two sub-intervals and using Simpson's $\frac{1}{3}$ rd rule, the value of ${\int }_0^1\frac{dx}{1+x}$ will be

Using trapezoidal rule and taking $n=4$, the approximate value of integral ${\int }_1^9{x}^{2}dx$ is $2\left[\frac{1}{2}\left(1+9^2\right)+{\alpha }^2+{\beta }^2+7^2\right]$, then

By Simpson's rule, the value of ${\int }_1^2\frac{dx}{x}$ dividing the interval $\left(1,2\right)$ into four parts is