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Recent questions and answers in Numerical Methods
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Number Theory
A prison houses 100 inmates, one in each of 100 cells, guarded by a total of 100 warders. One evening, all the cells are locked and the keys left in the locks. As the first warder leaves, she turns every key, unlocking all the doors. The second warder ... every third key and so on. Finally the last warder turns the key in just the last cell. Which doors are left unlocked and why?
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ISRO20173
Using NewtonRaphson method, a root correct to 3 decimal places of $x^3  3x 5 = 0$ 2.222 2.275 2.279 None of the above
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UGC NET NOV 2017 PAPER Q69
69. Consider the following LPP : Min Z=2x1+x2+3x3 Subject to : x1−2x2+x3 / 4 2x1+x2+x3 £ 8 x1−x3 / 0 x1 , x2 , x3 / 0 The solution of this LPP using Dual Simplex Method is : (1) x1=0, x2=0, x3=3 and Z=9 (2) x1=0, x2=6, x3=0 and Z=6 (3) x1=4, x2=0, x3=0 and Z=8 (4) x1=2, xx2=0, x3=2 andZ=10
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UGC NET NOV 2017 PAPER 3 Q68
68. Consider the following LPP : Max Z=15x1+10x2 Subject to the constraints 4x1+6x2 ≤ 360 3x1+0x2 ≤ 180 0x1+5x2 ≤ 200 x1 , x2> / 0 The solution of the LPP using Graphical solution technique is : (1) x1=60, x2=0 and Z=900 (2) x1=60, x2=20 and Z=1100 (3) x1=60, x2=30 and Z=1200 (4) x1=50, x2=40 and Z=1150
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UGCNETJune2014III60
The initial basic feasible solution of the following transportion problem: is given as 5 8 7 2 2 10 then the minimum cost is 76 78 80 82
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Nov 2, 2017
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GATE200821
The minimum number of equal length subintervals needed to approximate $\int_1^2 xe^x\,dx$ to an accuracy of at least $\frac{1}{3}\times10^{6}$ using the trapezoidal rule is 1000e 1000 100e 100
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Apr 25, 2017
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GATE2006IT28
The following definite integral evaluates to $$\int_{\infty}^{0} e^ {\left(\frac{x^2}{20} \right )}dx$$ $\frac{1}{2}$ $\pi \sqrt{10}$ $\sqrt{10}$ $\pi$
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Dec 30, 2016
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GATE19881i
Loosely speaking, we can say that a numerical method is unstable if errors introduced into the computation grow at _________ rate as the computation proceeds.
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GATE198711b
Use Simpson's rule with $h=0.25$ to evaluate $ V= \int_{0}^{1} \frac{1}{1+x} dx$ correct to three decimal places.
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Nov 15, 2016
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GATE198711a
Given $f(300)=2,4771; f(304) = 2.4829; f(305) = 2.4843$ and $f(307) = 2.4871$ find $f(301)$ using Lagrange's interpolation formula.
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Nov 15, 2016
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GATE19871xxv
Which of the following statements is true in respect of the convergence of the NewtonRephson procedure? It converges always under all circumstances. It does not converge to a tool where the second differential coefficient changes sign. It does not converge to a root where the second differential coefficient vanishes. None of the above.
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GATE19871xxiv
The simplex method is so named because It is simple. It is based on the theory of algebraic complexes. The simple pendulum works on this method. No one thought of a better name.
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Nov 9, 2016
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UGCNETDec2014III69
Five men are available to do five different jobs. From past records, the time (in hours) that each man takes to do each job is known and is given in the following table : Find out the minimum time required to complete all the jobs. $5$ $11$ $13$ $15$
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ISRO201348
The GuassSeidal iterative method can be used to solve which of the following sets? Linear algebraic equations Linear and nonlinear algebraic equations Linear differential equations Linear and nonlinear differential equations
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Jun 29, 2016
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ISRO200951
The formula $P_k = y_0 + k \triangledown y_0+ \frac{k(k+1)}{2} \triangledown ^2 y_0 + \dots + \frac{k \dots (k+n1)}{n!} \triangledown ^n y_0$ is Newton's backward formula Gauss forward formula Gauss backward formula Stirling's formula
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Jun 25, 2016
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isro2009
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ISRO201152
Given X: 0 10 16 Y: 6 16 28 The interpolated value X=4 using piecewise linear interpolation is 11 4 22 10
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Jun 24, 2016
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isro2011
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ISRO200947
The formula $\int\limits_{x0}^{xa} y(n) dx \simeq h/2 (y_0 + 2y_1 + \dots +2y_{n1} + y_n)  h/12 (\triangledown y_n  \triangle y_0)$ $ h/24 (\triangledown ^2 y_n + \triangle ^2 y_0) 19h/720 (\triangledown ^3 y_n  \triangle ^3 y_0) \dots $ is called Simpson rule Trapezoidal rule Romberg's rule Gregory's formula
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Jun 15, 2016
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ISRO200948
The cubic polynomial $y(x)$ which takes the following values: $y(0)=1, y(1)=0, y(2)=1$ and $y(3)=10$ is $x^3 +2x^2 +1$ $x^3 +3x^2 1$ $x^3 +1$ $x^3 2x^2 +1$
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Jun 15, 2016
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ISRO200946
The shift operator $E$ is defined as $E [f(x_i)] = f (x_i+h)$ and $E'[f(x_i)]=f (x_i h)$ then $\triangle$ (forward difference) in terms of $E$ is $E1$ $E$ $1E^{1}$ $1E$
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ISRO200944
A root $\alpha$ of equation $f(x)=0$ can be computed to any degree of accuracy if a 'good' initial approximation $x_0$ is chosen for which $f(x_0) > 0$ $f (x_0) f''(x_0) > 0$ $f(x_0) f'' (x_0) < 0$ $f''(x_0) >0$
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21
GATE20002.1
X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the following is true? k must be 1 k must be 0 k can take any value between 0 and 1 None of the above
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Jan 14, 2016
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gate2000
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22
GATE1999_1.23
The NewtonRaphson method is to be used to find the root of the equation $f(x)=0$ where $x_o$ is the initial approximation and $f’$ is the derivative of $f$. The method converges always only if $f$ is a polynomial only if $f(x_o) <0$ none of the above
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Jun 24, 2015
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gate1999
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GATE1997_4.10
The trapezoidal method to numerically obtain $\int_a^b f(x) dx$ has an error E bounded by $\frac{ba}{12} h^2 \max f’’(x), x \in [a, b]$ where $h$ is the width of the trapezoids. The minimum number of trapezoids guaranteed to ensure $E \leq 10^{4}$ in computing $\ln 7$ using $f=\frac{1}{x}$ is 60 100 600 10000
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Jun 6, 2015
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GATE1997_1.2
The NewtonRaphson method is used to find the root of the equation $X^22=0$. If the iterations are started from 1, the iterations will converge to 1 converge to $\sqrt{2}$ converge to $\sqrt{2}$ not converge
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Jun 5, 2015
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GATE1996_2.5
NewtonRaphson iteration formula for finding $\sqrt[3]{c}$, where $c > 0$ is $x_{n+1}=\frac{2x_n^3 + \sqrt[3]{c}}{3x_n^2}$ $x_{n+1}=\frac{2x_n^3  \sqrt[3]{c}}{3x_n^2}$ $x_{n+1}=\frac{2x_n^3 + c}{3x_n^2}$ $x_{n+1}=\frac{2x_n^3  c}{3x_n^2}$
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gate1996
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GATE1995_2.15
The iteration formula to find the square root of a positive real number $b$ using the Newton Raphson method is $x_{k+1} = 3(x_k+b)/2x_k$ $x_{k+1} = (x_{k}^2+b)/2x_k$ $x_{k+1} = x_k2x_k/\left(x^2_k+b\right)$ None of the above
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gate1995
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GATE1994_3.4
Match the following items (i) NewtonRaphson (a) Integration (ii) RungeKutta (b) Root finding (iii) GaussSeidel (c) Ordinary Differential Equations (iv) Simpson's Rule (d) Solution of Systems of Linear Equations
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Apr 28, 2015
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GATE1993_01.3
In questions 1.1 to 1.7 below, one or more of the alternatives are correct. Write the code letter(s) a, b, c, d corresponding to the correct alternative(s) in the answer book. Marks will be given only if all the correct alternatives have been selected and no ... is picked up. 1.3 Simpson's rule for integration gives exact result when $f(x)$ is a polynomial of degree 1 2 3 4
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GATE2007IT77
Consider the sequence $\left \langle x_n \right \rangle,\; n \geq 0$ defined by the recurrence relation $x_{n + 1} = c \cdot (x_n)^2  2$, where $c > 0$. For which of the following values of $c$, does there exist a nonempty open interval $(a, b)$ such that the sequence ... $a < x_0 < b$? $0.25$ $0.35$ $0.45$ $0.5$ i only i and ii only i, ii and iii only i, ii, iii and iv
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GATE2006IT27
Match the following iterative methods for solving algebraic equations and their orders of convergence. Method Order of Convergence 1. Bisection P. 2 or more 2. NewtonRaphson Q. 1.62 3. Secant R. 1 4. Regula falsi S. 1 bit per iteration IR, IIS, IIIP, IVQ IS, IIR, IIIQ, IVP IS, IIQ, IIIR, IVP IS, IIP, IIIQ, IVR
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GATE2004IT38
If f(l) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3) using Lagrange's interpolation formula? 8 8(1/3) 8(2/3) 9
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Apr 7, 2015
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GATE2004IT39
Consider the following iterative root finding methods and convergence properties: Iterative root finding methods Convergence properties Q. False Position I. Order of convergence = 1.62 R. Newton Raphson II. Order of convergence = 2 S. Secant III. Order of convergence = 1 with guarantee of convergence T. Successive ... , RII, SI, TIV QII, RI, SIV, TIII QI, RIV, SII, TIII
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GATE20153_50
The velocity $v$ (in kilometer/minute) of a motorbike which starts form rest, is given at fixed intervals of time $t$ (in minutes) as follows: t 2 4 6 8 10 12 14 16 18 20 v 10 18 25 29 32 20 11 5 2 0 The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson's $1/3^{rd}$ rule is ________.
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Feb 20, 2015
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GATE20152_39
The secant method is used to find the root of an equation $f(x)=0$. It is started from two distinct estimates $x_a$ and $x_b$ for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if $f(x_b)$ is very small and then $x_b$ is the solution. The ... / (x_bx_a)$ $x_b  (x_bx_a) f_b / (f_bf(x_a)) $ $x_a  (x_bx_a) f_a / (f_bf(x_a)) $
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calculus
The estimate of $\int_{0.5}^{1.5}\frac{dx}{x}$ obtained using Simpson’s rule with threepoint function evaluation exceeds the exact value by (A) 0.235 (B) 0.068 (C) 0.024 (D) 0.012
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2012 numerical methed
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GATE2013_23
Function $f$ is known at the following points: $x$ 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 $f(x)$ 0 0.09 0.36 0.81 1.44 2.25 3.24 4.41 5.76 7.29 9.00 The value of $\int_{0}^{3} f(x) \text{d}x$ computed using the trapezoidal rule is (A) 8.983 (B) 9.003 (C) 9.017 (D) 9.045
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GATE2005IT2
If the trapezoidal method is used to evaluate the integral obtained $\int_{0}^{1} x^2dx$, then the value obtained is always > (1/3) is always < (1/3) is always = (1/3) may be greater or lesser than (1/3)
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GATE2007IT22
The trapezoidal method is used to evaluate the numerical value of $\int_{0}^{1}e^x dx$. Consider the following values for the step size h. 102 103 104 105 For which of these values of the step size h, is the computed value guaranteed to be correct to ... Assume that there are no roundoff errors in the computation. iv only iii and iv only ii, iii and iv only i, ii, iii and iv
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GATE2014246
In the NewtonRaphson method, an initial guess of $x_0= 2 $ is made and the sequence $x_0,x_1,x_2\:\dots$ is obtained for the function $$0.75x^32x^22x+4=0$$ Consider the statements $x_3\:=\:0$ The method converges to a solution in a finite number of iterations. Which of the following is TRUE? Only I Only II Both I and II Neither I nor II
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