Question 1 
A  
B  
C  
D 
Discuss it
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 
8.983  
9.003  
9.017  
9.045 
Discuss it
Question 2 Explanation:
Since the intervals are uniform, apply the uniform grid formula of trapezoidal rule.
This solution is contributed by Anil Saikrishna Devarasetty
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
One, at π/2  
One, at 3π/2  
Two, at π/2 and 3π/2  
Two, at π/4 and 3π/2 
Discuss it
Question 3 Explanation:
Question 4 
The bisection method is applied to compute a zero of the function f(x) = x^{4} – x^{3} – x^{2} – 4 in the
interval [1,9]. The method converges to a solution after ––––– iterations
1  
3  
5  
7 
Discuss it
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 6 
NewtonRaphson method is used to compute a root of the equation x^{2}13=0 with 3.5 as the initial value. The approximation after one iteration is
3.575  
3.676  
3.667  
3.607 
Discuss it
Question 6 Explanation:
In NewtonRaphson's method, We use the following formula to get the next value of f(x). f'(x) is derivative of f(x).
f(x) = x^{2}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 Lim_{n>∞}(11/n)^{2n} ?
0  
e^{2}  
e^{1/2}  
1 
Discuss it
Question 7 Explanation:
The value of e (mathematical constant) can be written as following
And the value of 1/e can be written as following.
Lim_{n> ∞}( 11/n)^{2n }= (Lim_{n> ∞}(11/n)^{n})^{2} = e^{2}
Question 8 
Two alternative packages A and B are available for processing a database having 10k records.Package A requires 0.0001n^{2} 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?
12  
10  
6  
5 
Discuss it
Question 8 Explanation:
B must be preferred on A when the time taken taken by B is more than A, i.e.,
0.0001 n^{2} < 10 n log_{10}n 10^{5}n < log_{10}n
Question 9 
0  
1  
ln 2  
1/2 ln 2 
Discuss it
Question 9 Explanation:
(1tanx)/(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 11 
0  
either 0 or 1  
one of 0, 1 or 1  
any real number  
any real number other than 5 
Discuss it
Question 11 Explanation:
The choice E was not there in GATE paper. We have added it as the given 4 choices don't seem correct.
Augment the given matrix as 1 1 2  1 1 2 3  2 1 4 a  4 Apply R2 < R2  R1 and R3 < R3  R1 1 1 2  1 0 1 1  1 0 3 a2  3 Apply R3 < R3  3R2 1 1 2  1 0 1 1  1 0 0 a5  0 So for the system of equations to have a unique solution, a  5 != 0 or a != 5 or a = R  {5}Thanks to Anubhav Gupta for providing above explanation. Readers can refer below MIT video lecture for linear algebra. https://www.youtube.com/watch?v=QVKj3LADCnA&index=2&list=PLE7DDD91010BC51F8
Question 12 
The minimum number of equal length subintervals needed to approximate to an accuracy of at least using the trapezoidal rule is
1000 l  
1000  
100 l  
100 
Discuss it
Question 12 Explanation:
Trapezoidal rule error :
Maximum error = 1/3 * 10^{6} (given)
Therefore, E_{n} < 1/3 * 10^{6}
a = 1 and b = 2 (given)
Therefore,
f''(x) = xe^{x} + 2e^{x}
f''(x) is maximum at x = 2.
Therefore, f''(x) = 4e^{2}
Thus, option (A) is correct.
Reference: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2008.html
Please comment below if you find anything wrong in the above post.
Question 13 
square of R  
reciprocal of R  
square root of R  
logarithm of R 
Discuss it
Question 13 Explanation:
According to NewtonRaphson method,
x_{n+1} = x_{n} − f(x_{n}) / f′(x_{n})So we try to bring given equation in above form. Given equation is :
x_{n+1} = x_{n}/2 + R/(2x_{n}) = x_{n} − x_{n}/2 + R/(2x_{n}) = x_{n} − (x_{n}^{2} − RSo clearly f(x) = x^{2} − R, so root of f(x) means x^{2} − R = 0 i.e. we are trying to find square root of R. So option (C) is correct. Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2008.html2)/(2x_{n})
Question 14 
P = Q  k  
P = Q + k  
P = Q  
P = Q +2 k 
Discuss it
Question 14 Explanation:
P is sum of odd integers from 1 to 2k Q is sum of even integers from 1 to 2k Let k = 5 P is sum of odd integers from 1 to 10 P = 1 + 3 + 5 + 7 + 9 Q is sum of even integers from 1 to 10 Q = 2 + 4 + 6 + 8 + 10 In general, Q can be written as Q = (1 + 3 + 5 + 9..... ) + (1 + 1 + .....) = P + k
Question 15 
A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x^{4}  16x^{3} + 24x^{2} + 37
0  
1  
2  
3 
Discuss it
Question 16 
Consider the following two statements about the function f(x)=x
P. f(x) is continuous for all real values of x Q. f(x) is differentiable for all real values of xWhich of the following is TRUE?
P is true and Q is false.  
P is false and Qis true.  
Both P and Q are true  
Both P and Q are false. 
Discuss it
Question 16 Explanation:
A function is continuous if for every value of 'x', we have a corresponding f(x). Here, for every x, we have f(x) which is actually the value of x itself, without the negative sign for x < 0.
But, the given function is not differentiable for x = 0 because for x < 0, the derivative is negative and for x > 0, the derivative is positive. So, the left hand derivative and right hand derivative do not match.
Hence, P is correct and Q is incorrect. Thus, A is the correct option.
Please comment below if you find anything wrong in the above post.
But, the given function is not differentiable for x = 0 because for x < 0, the derivative is negative and for x > 0, the derivative is positive. So, the left hand derivative and right hand derivative do not match.
Hence, P is correct and Q is incorrect. Thus, A is the correct option.
Please comment below if you find anything wrong in the above post.
Question 17 
Consider the series X_{n+1} = X_{n}/2 + 9/(8 X_{n}), X_{0} = 0.5 obtained from the NewtonRaphson method. The series converges to
1.5  
sqrt(2)  
1.6  
1.4 
Discuss it
Question 17 Explanation:
As per Newton Rapson's Method, X_{n+1} = X_{n} − f(X_{n})/f′(X_{n}) Here above equation is given in the below form X_{n+1} = X_{n}/2 + 9/(8 X_{n}) Let us try to convert in Newton Rapson's form by putting X_{n} as first part. X_{n+1} = X_{n}  X_{n}/2 + 9/(8 X_{n}) = X_{n}  (4*X_{n}^{2}  9)/(8*X_{n}) So f(X) = (4*X_{n}^{2}  9) and f'(X) = 8*X_{n}So clearly f(X) = 4X^{2} − 9. We know its roots are ±3/2 = ±1.5, but if we start from X^{0} = 0.5, according to equation, we cannot get negative value at any time, so answer is 1.5 i.e. option (A) is correct.
Question 18 
Consider the polynomial p(x) = a0 + a1x + a2x^{2} + a3x^{3} , where ai ≠ 0 ∀i. The minimum number of multiplications needed to evaluate p on an input x is:
3  
4  
6  
9 
Discuss it
Question 18 Explanation:
Background Explanation :
Horner's rule for polynomial division is an algorithm used to simplify the process of evaluating a polynomial f(x) at a certain value x = x0 by dividing the polynomial into monomials (polynomials of the 1st degree). Each monomial involves a maximum of one multiplication and one addition processes. The result obtained from one monomial is added to the result obtained from the next monomial and so forth in an accumulative addition fashion. To explain the above, let is rewrite the polynomial in its expanded form;
f(x0) = a0 + a1x0+ a2x0^2+ ... + anx0^n
This can, also, be written as:
f(x0) = a0 + x0(a1+ x0(a2+ x0(a3+ ... + (an1 + anx0)....)
The algorithm proposed by this rule is based on evaluating the monomials formed above starting
from the one in the innermost parenthesis and move out to evaluate the monomials in the outer
parenthesis.
Solution :
Using Horner's Rule, we can write the polynomial as following
a0 + (a1 + (a2 + a3x)x)x
In the above form, we need to do only 3 multiplications
p = a3 X x  (1) q = (a2 + p) X x (2) r = (a1 + q) X x (3) result = a0 + rReference : http://www.geeksforgeeks.org/hornersmethodpolynomialevaluation/ This solution is contributed by Nitika Bansal.
Question 19 
Consider the following system of equations:
3x + 2y = 1 4x + 7z = 1 x + y + z = 3 x – 2y + 7z = 0The number of solutions for this system is __________________
1  
0  
2  
3 
Discuss it
Question 19 Explanation:
rank(Augmented Matrix) = rank(Matrix) = no of unknowns.
Hence it has a unique solution.
Question 20 
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4by4 symmetric positive definite matrix is _____________________.
0  
1  
1  
2 
Discuss it
Question 20 Explanation:
The eigen vectors corresponding to different eigen values of a real symmetric matrix are orthogonal to each other. And dot product of orthogonal vectors is 0.
Question 21 
I only  
II only  
Both I and II  
Neither I nor II 
Discuss it
Question 22 
A nonzero polynomial f(x) of degree 3 has roots at x = 1, x = 2 and x = 3. Which one of the following must be TRUE?
f(0)f(4) < 0  
f(0)f(4) > 0  
f(0) + f(4) < 0  
f(0) + f(4) > 0 
Discuss it
Question 22 Explanation:
The graph of a degree 3 polynomial f(x) = a0 + a1x + a2(x^2) + a3(x^3), where a3 ≠ 0
is a cubic curve, as can be seen here
https://en.wikipedia.org/wiki/...
Now as given, the polynomial is zero at x = 1, x = 2 and x = 3, i.e. these are the only 3 real roots of this polynomial.
Hence we can write the polynomial as f(x) = K (x1)(x2)(x3) where K is some constant coefficient.
Now f(0) = 6K and f(4) = 6K ( by putting x = 0 and x = 4 in the above polynomial )
and f(0)*f(4) = 36(k^2), which is always negative. Hence option A.
We can also get the answer by just looking at the graph. At x < 1, the cubic graph (or say f(x) ) is at one side of xaxis, and at x > 3 it should be at other side of xaxis. Hence +ve and ve values, whose multiplication gives negative.
Question 23 
In the NewtonRaphson method, an initial guess of x0 = 2 is made and the sequence x0, x1, x2 … is obtained for the function
0.75x^{3} – 2x^{2} – 2x + 4 = 0 Consider the statements (I) x_{3} = 0. (II) 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 
Discuss it
Question 25 
I Only  
II Only  
Both I and II  
Neither I or II 
Discuss it
Question 27 
i  
i+1  
2i  
2^{i} 
Discuss it
Question 27 Explanation:
B is the correct option. Let us put values
S = 1 + 2x + 3x^{2} + 4x^{3} + .......... Sx = x + 2x^{2} + 3x^{3} + .......... S  Sx = 1 + x + x^{2} + x^{3} + .... S  Sx = 1/(1  x) [sum of infinite GP series with ratio < 1 is a/(1r)] S = 1/(1  x)^{2}
Question 28 
A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale).
If we use the NewtonRaphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be
1.3, 0.6, and 0.6 respectively  
0.6, 0.6, and 1.3 respectively  
1.3, 1.3, and 0.6 respectively  
1.3, 0.6, and 1.3 respectively 
Discuss it
Question 28 Explanation:
First of all, There is a mistake in coordinates of a given point. I have corrected that in red color.
Now in NewtonRaphson method, we draw a tangent from our guess point, and our new guess would be the point where this tangent cuts xaxis. Now we choose initial guess points one by one :
x0 : Tangent at this point is line AB itself, and that would cut xaxis at point (1.0,0.0) (found using equation of line AB). So our next guess would be 1.0. Point on the curve corresponding to this new guess 1.0 is shown as F. Now tangent at point F is line DE, which cuts xaxis at 1.3, and at this point, value of function is zero, so we found the root as 1.3. x1 : Tangent at this point is line BE, which cuts xaxis at 0.6, also function value is zero here, so we find root as 0.6. x2 : Tangent at this point is line CD, which cuts xaxis at 1.05 (again found by finding equation of line CD). Point on the curve corresponding to this new guess 1.05 is shown as G. Now tangent at point G is line DE, which cuts xaxis at 1.3, and at this point, value of function is zero, so we found the root as 1.3.Source: Question 60 of http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2003.html
Question 29 
The trapezoidal rule for integration give exact result when the integrand is a polynomial of degree:
0 but not 1  
1 but not 0  
0 or 1  
2 
Discuss it
Question 29 Explanation:
Question 30 
The NewtonRaphson iteration X_{n + 1} = (X_{n}/2) + 3/(2X_{n}) can be used to solve the equation
X^{2} = 3  
X^{3} = 3  
X^{2} = 2  
X^{3} = 2 
Discuss it
Question 30 Explanation:
In NewtonRaphson's method, We use the following formula to get the next value of f(x). f'(x) is derivative of f(x).
Option (A)
X^{2} = 3 f(x) = X^{2}  3 X_{n + 1} = X_{n}  (X_{n}^{2}  3) / (2*X_{n}) = (X_{n}/2) + 3/(2x_{n})
Question 32 
A polynomial p(x) satisfies the following:
p(1) = p(3) = p(5) = 1 p(2) = p(4) = 1The minimum degree of such a polynomial is
1  
2  
3  
4 
Discuss it
Question 32 Explanation:
p(1) = p(3) = p(5) = 1 p(2) = p(4) = 1 The polynomial touches 0 at least once from 1 to 2, so there is a root between 1 to 2 The polynomial touches 0 at least once from 2 to 3, so there is a root between 2 to 3 Similarly, there is at least one root from 3 to 4 and 4 to 5. So minimum degree is 4.
Question 33 
∞  
0  
1  
Not Defined 
Discuss it
Question 33 Explanation:
[Tex]\lim_{x\to\infty} 1/x[/Tex]
= 0. and
[Tex]\lim_{x\to\infty} x^0[/Tex]
= 1.
Alternate method : Using log lnm=lim x>infinity 1/x*lnx lim x>infinity lnx/x (numerator = finite value,denominator = infinity and finite/infinite=0 ) ln(m)=0 m=e^{0}=1
Alternate method : Using log lnm=lim x>infinity 1/x*lnx lim x>infinity lnx/x (numerator = finite value,denominator = infinity and finite/infinite=0 ) ln(m)=0 m=e^{0}=1
Question 34 
a  
b  
c  
d 
Discuss it
Question 34 Explanation:
g(h(x)) = g(x/(x1)) = 1  x/(x1) = 1/(x1) h(g(x)) = h(1x) = (1x)/((1x)  1) = (1x)/x g(h(x)) / h(g(x)) = [1/(x1)] / [(1x)/x] = x/(x1)^{2} x/(x1)^{2} is same as h(x) / g(x)
Question 35 
0.99  
1  
99  
0.9 
Discuss it
Question 35 Explanation:
S = 1/1*2 + 1/2*3 + 1/3*4 + ..... + 1/99*100 = (1  1/2) + (1/2  1/3) + (1/3  1/4) + .... + (1/99  1/100) = (1 + 1/2 + 1/3 .... 1/99)  (1/2 + 1/3 + 1/4 ... 1/100) = 1  1/100 = 99/100 = 0.99
Question 36 
0  
1  
1  
infinite 
Discuss it
Question 36 Explanation:
Let f(x) be the given function. We assume that \[\frac{1}{x} = z\]
Differentiating both sides, we get
[Tex]
\[\frac{1}{x^2} dx = dz\]
Now, accordingly, the lower limit of the integral is \[ z = \frac{1}{\frac{1}{\pi}} = \pi\] and the upper limit for the integral is \[ z = \frac{1}{\frac{2}{\pi}} = \frac{\pi}{2}\]
So, the given function now becomes
\[ f(x)=  \int_\pi^{\frac{\pi}{2}} cos(z) dz \]
\[ f(x)= \int_\frac{\pi}{2}^{\pi} cos(z) dz \]
\[f(x) = sin(z) ,\] and the upper limit is π and the lower limit is π/2
So, \[f(x) = sin(\pi)  sin(\frac{\pi}{2})\]
\[f(x) = 0  1\]
\[f(x) = 1\]
So, the required answer is 1.
[/Tex]
Question 37 
Consider a function f(x) = 1 – x on –1 ≤ x ≤ 1. The value of x at which the function attains a maximum and the maximum value of the function are:
0, –1  
–1, 0  
0, 1  
–1, 2 
Discuss it
Question 37 Explanation:
f(x) = 1 – x f(0) = 1 f(1) = f(1) = 0 f(0.5) = f(0.5) = 0.5The maximum is attained at x = 0, and the maximum value is 1.
Question 38 
Let f(x) = x ^{–(1/3)} and A denote the area of the region bounded by f(x) and the Xaxis, when x varies from –1 to 1. Which of the following statements is/are True?
1. f is continuous in [–1, 1] 2. f is not bounded in [–1, 1] 3. A is nonzero and finite
2 only  
3 only  
2 and 3 only  
1, 2 and 3 
Discuss it
Question 38 Explanation:
1 is false: function is not a Continuous function. As a change of 1 in x leads to ∞ change in f(x). For example when x is changed from 1 to 0. At x = 0, f(x) is ∞ and at x = 1, f(x) is finite.
2 is True: f(x) is not a bounded function as it becomes ∞ at x = 0.
3 is true: A denote the area of the region bounded by f(x) and the Xaxis. This area is bounded, we can calculate it by doing integrating the function [See this]
Question 39 
0  
1/2  
1  
∞ 
Discuss it
Question 39 Explanation:
This can be solved using L'Hôpital's rule that uses derivatives to help evaluate limits involving indeterminate forms.
Since [Tex]
\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=\infty, and
\lim_{x\to c}\frac{f'(x)}{g'(x)} exists[/Tex]
We get
[Tex] \lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}. [/Tex]
[Tex] \lim_{x\to \infty}\frac{1 + x^2}{e^x} = \lim_{x\to \infty}\frac{2x}{e^x} = \lim_{x\to \infty}\frac{2}{e^x} = 0 [/Tex]
Question 41 
The velocity v (in kilometer/minute) of a motorbike which starts from 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 0The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3rd rule is _________.
309.33  
105.33  
110.00  
405.6 
Discuss it
Question 41 Explanation:
Question 42 
Let A be an Let A be an n × n matrix of the following form.
What is the value of the determinant of A?
A  
B  
C  
D 
Discuss it
Question 42 Explanation:
The first thing you need to get by seeing these type of questions is: Go for substitution method. For n=2, the values will be A) 16 B) 26 C) 7 D) 8 As all the values are unique for a small value of n, it does not take much time. The given matrix will be A = [3 1] [1 3] So, det(A) = 3*31*1 = 8 Option (D) is the answer.
Alternative method: You can frame the relations in between det(An+1), det(An), det(An1) i.e. d(An+1) = 3*d(An)  d(An1) X = 3*X^0  X^1 X^2 = 3*x  1 Solution for this equation is (3+sqrt(5))/2, (3sqrt(5))/2 The only option which has roots of type (3+sqrt(5)) is D. From this, you can match the options easily.This explanation has been provided by Anil Saikrishna.
Question 43 
Let X and Y be two exponentially distributed and independent random variables with mean α and β, respectively. If Z = min(X,Y), then the mean of Z is given by
1/α+β  
min(α ,β)  
α β/α + β  
α + β 
Discuss it
Question 44 
If f(1) = 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 
Discuss it
Question 44 Explanation:
Using Lagrange’s interpolation formula :
f(x) = ((x  x2)(x  x4)/(x1  x2)(x1  x4)) * f1 + ((x  x1)(x  x4)/(x2  x1)(x2  x4)) * f2 + ((x  x1)(x  x2)/(x4  x1)(x4  x2)) * f4
f(3) = ((3  2)(3  4)/(1  2)(1  4)) * 2 + ((3  1)(3  4)/(2  1)(2  4)) * 4 + ((3  1)(3  2)/(4  1)(4  2)) * 16 f(3) = 2/3 + 4 + 16/3 f(3) = 8 (2/3)
Thus, option (C) is correct.
Please comment below if you find anything wrong in the above post.
Question 45 
Consider the following iterative root finding methods and convergence properties: Iterative root finding Convergence properties methods
(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 Approximation (IV) Order of convergence = 1 with no guarantee of convergence
QII RIV SlIlTI  
QIIIRII SI TIV  
QIIRI SIVTIII  
QI RIV SIl TIII 
Discuss it
Question 45 Explanation:
These type of questions are standard type. You can answer these if you have strong command on the subjects.
a) False Position  Order of convergence = 1 with guarantee of convergence
b) Newton Raphson  Order of convergence = 2
c) Secant  Order of convergence = 1.62
d) Successive Approximation  Order of convergence = 1 with no guarantee of convergence
This solution is contributed by Anil Saikrishna Devarasetty.
Question 46 
Let f(n), g(n) and h(n) be functions defined for positive inter such that
f(n) = O(g(n)), g(n) ≠ O(f(n)), g(n) = O(h(n)), and h(n) = O(g(n)).
Which one of the following statements is FALSE?
f(n) + g(n) = O(h(n)) + h(n))  
f(n) = O(h(n))  
fh(n) ≠ O(f(n))  
f(n)h(n) ≠ O(g(n)h(n)) 
Discuss it
Question 46 Explanation:
f(n), g(n), h(n) are three functions defined over n
Given f(n) = O(g(n)) but g(n) != O(f(n))
g(n) = O(h(n)) and h(n) = O(g(n))
So, f(n)*h(n) = O(g(n))*h(n) using above given relations
But it is stated that f(n)*h(n)!=O(g(n))*h(n) which is false
So, answer is option (D).
This solution is contributed by Anil Saikrishna Devarasetty .
Question 47 
If the trapezoidal method is used to evaluate the integral obtained _{0}∫^{1}x^{2}dx ,then the value obtained
is always > (1/3)  
is always < (1/3)  
is always = (1/3)  
may be greater or lesser than (1/3) 
Discuss it
Question 48 
1  
0  
1  
2 
Discuss it
Question 48 Explanation:
Matrices don’t have value associated with them, but determinant have value associated with them. Determinant of a matrix can be find out by taking any one row or one column, in this row or column, multiplying each element with its cofactor and summing the value up.
This solution is contributed by Sandeep Pandey.
Question 49 
0  
1  
2  
3 
Discuss it
Question 49 Explanation:
Put y = x  4.
So, the problem becomes lim_{y>0} (sin y) / y = 1. (Property of Limits on sin)
Thus, B is the correct choice.
Question 50 
The trapezoidal method is used to evaluate the numerical value of . Consider the following values for the step size h.
i. 10^{2}
ii. 10^{3}
iii. 10^{4}
iv. 10^{5}
For which of these values of the step size h, is the computed value guaranteed to be correct to seven decimal places. Assume that there are no roundoff errors in the computation.
i. 10^{2}
ii. 10^{3}
iii. 10^{4}
iv. 10^{5}
For which of these values of the step size h, is the computed value guaranteed to be correct to seven decimal places. 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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Question 51 
Find the Integral value of f(x) = x * sinx within the limits 0, π.
π  
2π
 
π/2
 
0

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Question 51 Explanation:
Let I = ∫x*sinx > 1
I = ∫( πx)*sin(πx) = ∫(( πx)*sin(x)  2
Adding both 1 and 2
=> I + I = ∫ π*sinx = π*[cosx] =  π*[cos π – cos0]
=> 2I =  π*2
=> I = π
The value of Integral is π
I = ∫( πx)*sin(πx) = ∫(( πx)*sin(x)  2
Adding both 1 and 2
=> I + I = ∫ π*sinx = π*[cosx] =  π*[cos π – cos0]
=> 2I =  π*2
=> I = π
The value of Integral is π
Question 52 
The value of the constant 'C' using Lagrange's mean value theorem for f(x) = 8x  x^{2} in [0,8] is:
4  
8  
0  
None of these 
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Question 52 Explanation:
f(x) = 8x  x^2 in [0,8]
C = (0 + 8) / 2 = 4
Since, the value of ‘C’ for px^2 + qx + r (irrespective of p,q,r) defined in [a,b] using mean value theorem is mid point of the interval, i.e. C = (a + b) / 2
C = (0 + 8) / 2 = 4
Since, the value of ‘C’ for px^2 + qx + r (irrespective of p,q,r) defined in [a,b] using mean value theorem is mid point of the interval, i.e. C = (a + b) / 2
There are 52 questions to complete.