Numerical Methods and Calculus

  • Last Updated : 21 Jan, 2014


Question 1

Which one of the following functions is continuous at x = 3? 

 

gatecs20139


 

A

C
 

B

D
 

C

B
 

D

A
 

GATE CS 2013    Numerical Methods and Calculus    
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Question 1 Explanation: 

A function is continuous at some point c, 

Value of f(x) defined for x > c = Value of f(x) defined for x < c = Value of f(x) defined for x = c 

All values are 2 in option A
 

Question 2

Function f is known at the following points: 

 

gatecs201310



 

A

9.003 
 

B

9.017 
 

C

8.983 
 

D

9.045
 

GATE CS 2013    Numerical Methods and Calculus    
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Question 2 Explanation: 
"anil_m_4"

Since the intervals are uniform, apply the uniform grid formula of trapezoidal rule.
 

Question 3

Consider the function f(x) = sin(x) in the interval [π/4, 7π/4]. The number and location(s) of the local minima of this function are
 

A

One, at π/2

B

One, at 3π/2

C

Two, at π/2 and 3π/2

D

Two, at π/4 and 3π/2

GATE CS 2012    Numerical Methods and Calculus    
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Question 4

The bisection method is applied to compute a zero of the function f(x) = x4 – x3 – x2 – 4 in the 
interval [1,9]. The method converges to a solution after ––––– iterations
 

A

5
 

B

7
 

C

3
 

D

1
 

GATE CS 2012    Numerical Methods and Calculus    
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Question 4 Explanation: 

In bisection method, we calculate the values at extreme points of given interval, if signs of values are opposite, then we find the middle point. Whatever sign we get at middle point, we take the corner point of opposite sign and repeat the process till we get 0. 

f(1) < 0 and f(9) > 0 
mid = (1 + 9)/2 = 5 

f(5) > 0, so zero value lies in [1, 5] 
mid = (1+5)/2 = 3 

f(3) > 0, so zero value lies in [1, 3] 
mid = (1+3)/2 = 2 

f(2) = 0
 

Question 5

Given i=√-1, what will be the evaluation of the integral \int_{0}^{\pi/2} \frac{\cos x + i\sin x}{\cos x - i\sin x} dx ?
 

A

2
 

B

-i
 

C

0
 

D

i
 

GATE CS 2011    Numerical Methods and Calculus    
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Question 6

Newton-Raphson method is used to compute a root of the equation x2-13=0 with 3.5 as the initial value. The approximation after one iteration is
 

A

3.667
 

B

3.607
 

C

3.676
 

D

3.575
 

GATE CS 2010    Numerical Methods and Calculus    
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Question 6 Explanation: 

In Newton-Raphson's method, We use the following formula to get the next value of f(x). f'(x) is derivative of f(x). 
x_{n+1} = x_{n}- \frac{f(x_{n})}{f'(x_{n})}

 

f(x)  =  x2-13
f'(x) =  2x

Applying the above formula, we get
Next x = 3.5 - (3.5*3.5 - 13)/2*3.5
Next x = 3.607


 

Question 7

What is the value of Limn->∞(1-1/n)2n ?
 

A

1
 

B

e-2
 

C

e-1/2
 

D

0
 

GATE CS 2010    Numerical Methods and Calculus    
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Question 8
Two alternative packages A and B are available for processing a database having 10k records.Package A requires 0.0001n2 time units and package B requires 10nlog10n time units to process n records. What is the smallest value of k for which package B will be preferred over A?
A
12
B
10
C
6
D
5
GATE CS 2010    Numerical Methods and Calculus    
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Question 8 Explanation: 
Since, 10nlog10n ≤ 0.0001n2 Given n = 10k records. Therefore, ⟹10×(10k)log1010k ≤ 0.0001(10k)2 ⟹10k+1k ≤ 0.0001 × 102k ⟹k ≤ 102k−k−1−4 ⟹k ≤ 10k−5 Hence, value 5 does not satisfy but value 6 satisfies. 6 is the smallest value of k for which package B will be preferred over A. Option (C) is correct.
Question 9
CSE_2009_25 is equivalent to
A
0
B
1
C
ln 2
D
1/2 ln 2
GATE-CS-2009    Numerical Methods and Calculus    
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Question 9 Explanation: 
(1-tanx)/(1+tanx) = (cosx - sinx)/(cosx + sinx) Let cosx + sinx = t (-sinx + cosx)dx = dt (1/t)dt = ln t => ln(sinx + cosx) => ln(sin Π/4 + cos Π/4) => ln(1/√2 + 1/√2) => 1/2 ln 2
Question 10
1
A
1
B
-1
C
INF
D
-INF
GATE CS 2008    Numerical Methods and Calculus    
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Question 10 Explanation: 
 \lim_{x\leftarrow \infty } \frac{x - sin(x)}{x + cos(x)} = \lim_{x\leftarrow 0} \frac{1-\frac{sin(x)}{x}}{1+\frac{cos(x)}{x}} = \frac{1-\lim_{x\leftarrow \infty } \frac{Sin(x)}{x} }{1+\lim_{x\leftarrow \infty } \frac{Cos(x)}{x} }= \frac{1-0}{1+0} = 1
There are 93 questions to complete.
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