GATE-CH-2006-1-math-1mark

The ordinary differential equation \(dY/dt = f(Y)\) is solved using the approximation \(Y(t+\Delta t) = Y(t) + f[Y(t)]/\Delta t\). The numerical error introduced by the approximation at each step is

• proportional to \(\Delta t\)

• proportional to \((\Delta t)^2\)

• independent of \(\Delta t\)

• proportional to \((1/\Delta t)\)

GATE-CH-2006-2-math-1mark

The trepezoidal rule of integration when applied to \(\displaystyle \int _a^b f(x) dx\) will give the exact value of the integral

• if \(f(x)\) is a linear function of \(x\)

• if \(f(x)\) is a quadratic function of \(x\)

• for any \(f(x)\)

• for no \(f(x)\)

GATE-CH-2008-5-math-1mark

A nonlinear function \(f(x)\) is defined in the interval \(-1.2<x<4\) as illustrated in the figure below. The equation \(f(x)=0\) is solved for \(x\) within this interval by using the Newton-Raphson iterative scheme. Among the initial guesses (\(I_1,I_2,I_3\) and \(I_4\)), the guess that is likely to lead to the root most rapidly is

• \(I_1\)

• \(I_2\)

• \(I_3\)

• \(I_4\)

GATE-CH-2010-15-math-1mark

A root of the equation \(x^4-3x+1=0\) needs to be found using the Newton-Raphson method. If the initial guess, \(x_0\), is taken as 0, then the new estimate \(x_1\), after the first iteration is

• \(\dfrac {1}{3}\)

• \(-\dfrac {1}{3}\)

• \(3\)

• \(-3\)

GATE-CH-2016-1-math-1mark

Which one of the following is an iterative technique for solving a system of simultaneous linear algebraic equations?

• Gauss elimination

• Gauss-Jordan

• Gauss-Seidel

• LU decomposition

GATE-CH-2018-2-math-1mark

The fourth order Runge-Kutta (RK4) method to solve an ordinary differential equation \(\dfrac {dy}{dx}=f(x,y)\) is given as 

\( \begin {align*} y(x+h) &= y(x) + \frac {1}{6}(k_1+2k_2+2k_3+k_4) \\ k_1 &= hf(x,y) \\ k_2 &= h f\left (x+\frac {h}{2},y+\frac {k_1}{d}\right ) \\ k_3 &= h f\left (x+\frac {h}{2},y+\frac {k_2}{d}\right ) \\ k_4 &= hf(x+h, y+k_3) \end {align*} \)

For a special case when the function \(f\) depends solely on \(x\), the above RK4 method reduces to

• Euler’s explicit method

• Trapezoidal rule

• Euler’s implicit method

• Simpson’s 1/3 rule

GATE-CH-2017-3-math-1mark

The number of positive roots of the function \(f(x)\) shown below in the range \(0<x<6\) is

____________

• Answer

GATE-CH-2005-40-math-2mark

The function \(f(x)\) satisfies the equation \(f(x)=0\) at \(x=x_e\). The Newton-Raphson iterative method converges to the solution in one step, regardless of the initial guess, if

• \(f(x)\) is a linear function of \(x\)

• \(f(x)\) is a quadratic function of \(x\)

• \(f(x)\) is a cubic function of \(x\)

• \(f(x)\) is an exponential function of \(x\)

GATE-CH-2006-28-math-2mark

The Newton-Raphson method is used to solve the equation, \((x-1)^2+x-3=0\). The method will fail in the very first iteration if the initial guess is

• zero

• 0.5

• 1

• 3

GATE-CH-2008-28-math-2mark

Using Simpson's 1/3 rule and FOUR equally spaced intervals (\(n=4\)), estimate the value of the integral \(\displaystyle \int _0^{\pi /4}\frac {\sin x}{\cos ^3 x}dx\)

• 0.3887

• 0.4384

• 0.5016

• 0.5527

GATE-CH-2008-29-math-2mark

The following differential equation is to be solved numerically by the Euler’s explicit method. \[ \frac {dy}{dx} = x^2y-1.2y \quad \text { with } y(0)=1 \] A step size of 0.1 is used. The solution for \(y\) at \(x=0.1\) is

• 0.880

• 0.905

• 1.000

• 1.100

GATE-CH-2009-25-math-2mark

Using the trapezoidal rule and 4 equal intervals (\(n=4\)), the calculated value of the integral (rounded to the first place of decimal) \( \int _0^\pi \sin \theta \; d\theta \) is

• 1.7

• 1.9

• 2.0

• 2.1

GATE-CH-2011-30-math-2mark

In the fixed point iteration method for solving equations of the form \(x=g(x)\), the \((n+1)^{\text {th}}\) iteration value is \(x_{n+1}=g(x_n)\), where \(x_n\) represents the \(n^{\text {th}}\) iteration value. \(g(x)\) and corresponding initial guess value \(x_0\) in the domain of interest are shown in the following choices. Which ONE of these choices leads to a converged solution for \(x\) ?

• 
• 
• 
• 

GATE-CH-2012-28-math-2mark

The Newton-Raphson method is used to find the roots of the equation \[ f(x) = x-\cos \pi x \quad \quad \quad 0\le x\le 1 \] If the initial guess for the root is 0.5, then the value of \(x\) after the first iteration is

• 1.02

• 0.62

• 0.55

• 0.38

GATE-CH-2015-28-math-2mark

The solution of the non-linear equation \(x^3-x=0\) is to be obtained using Newton-Raphson method. If the initial guess is \(x=0.5\), the method converges to which one of the following values:

• \(-1\)

• 0

• 1

• 2

GATE-CH-2017-28-math-2mark

Match the problem type in Group-1 with the numerical method in Group-2.

Group-1	Group-2
P) System of linear algebraic equations	I) Newton-Raphson
Q) Non-linear algebraic equations	II) Gauss-Seidel
R) Ordinary differential equations	III) Simpson's rule
S) Numerical integration	IV) Runge-Kutta

• P-II, Q-I, R-III, S-IV

• P-I, Q-II, R-IV, S-III

• P-IV, Q-III, R-II, S-I

• P-II, Q-I, R-IV, S-III

GATE-CH-2013-29-math-2mark

The value of the integral \(\ \displaystyle \int _{0.1}^{0.5}e^{-x^{3}}dx \ \) evaluated by Simpson's rule using 4 subintervals (up to 3 digits after the decimal point) is ____________

• Answer

GATE-CH-2016-28-math-2mark

The model \(y=mx^2\) is to be fit to the data given below.

\(x\)	1	\(\sqrt {2}\)	\(\sqrt {3}\)
\(y\)	2	5	8

Using linear regression, the value (rounded off to the second decimal place) of \(m\) is ____________

• Answer

GATE-CH-2016-30-math-2mark

Values of \(f(x)\) in the interval \([0, 4]\) are given below.

\(x\)	0	1	2	3	4
\(f(x)\)	3	10	21	36	55

Using Simpson's 1/3 rule with a step size of 1, the numerical approximation (rounded off to the second decimal place) of \(\displaystyle \int _0^4 f(x) dx\) is ____________

• Answer