Calculus

Question 1
 
A
1
B
Limit does not exist
C
53/12
D
108/7
       Engineering-Mathematics       Calculus       GATE 2019       Video-Explanation
Question 1 Explanation: 
Question 2
A
0.289
B
0.298
C
0.28
D
0.29
       Engineering-Mathematics       Calculus       GATE 2018       Video-Explanation
Question 2 Explanation: 
Question 3
The value of
A
is 0
B
is -1
C
is 1
D
does not exist
       Engineering-Mathematics       Calculus       GATE 2017 [Set-1]       Video-Explanation
Question 3 Explanation: 

If "x=1" is substituted we get 0/0 form, so apply L-Hospital rule

Substitute x=1
⇒ (7(1)6-10(1)4)/(3(1)2-6(1)) = (7-10)/(3-6) = (-3)/(-3) = 1
Question 4

If f(x) = Rsin(πx/2) + S, f'(1/2) = √2 and , then the constants R and S are, respectively.

A
B
C
D
       Engineering-Mathematics       Calculus       GATE 2017 [Set-2]       Video-Explanation
Question 4 Explanation: 

Question 5
A
4
B
3
C
2
D
1
       Engineering-Mathematics       Calculus       GATE 2016 [Set-1]       Video-Explanation
Question 5 Explanation: 
Question 6

Let f(x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f(x) + f(-x)) is 10, then the degree of (g(x) - g(-x)) is __________.

A
9
B
10
C
11
D
12
       Engineering-Mathematics       Calculus       GATE 2016 [Set-2]       Video-Explanation
Question 6 Explanation: 
If the degree of a polynomial is ‘n’ then the derivative of that function have (n – 1) degree.
It is given that f(x) + f(-x) degree is 10.
It means f(x) is a polynomial of degree 10.
Then obviously the degree of g(x) which is f’(x) will be 9.
Question 7

If g(x) = 1 - x and h(x) , then is:

A
h(x)/g(x)
B
-1/x
C
g(x)/h(x)
D
x/(1-x)2
       Engineering-Mathematics       Calculus       GATE 2015 [Set-1]
Question 7 Explanation: 
g(x)= 1 – x, h(x)=x/x-1 -------- (2)
Replace x by h(x) in (1), replacing x by g(x) in (2),
g(h(x))=1-h(x)=1-x/x-1=-1/x-1
h(g(x))=g(x)/g(x)-1=1-x/-x
⇒ g(h(x))/h(g(x))=x/(x-1)(1-x)=(x/x-1)/1-x=h(x)/g(x)
Question 8
A
-1
B
-2
C
-3
D
-4
       Engineering-Mathematics       Calculus       GATE 2015 [Set-1]
Question 8 Explanation: 
Question 9
A
0.99
B
1.00
C
2.00
D
3.00
       Engineering-Mathematics       Calculus       GATE 2015 [Set-1]
Question 9 Explanation: 

= 2-1/1(2)+3-2/2(3)+4-3/3(4)+…+100-99/99(100)
= 1/1-1/2+1/2-1/3+1/3…+1/98-1/99+1/99-1/100
= 1-1/100
= 99/100
= 0.99
Question 10

Let f(x) = x -(1/3) and A denote the area of the region bounded bu f(x) and the X-axis, when x varies from -1 to 1. Which of the following statements is/are TRUE?

    I) f is continuous in [-1,1]
    II) f is not bounded in [-1,1]
    III) A is nonzero and finite
A
II only
B
III only
C
II and III only
D
I, II and III
       Engineering-Mathematics       Calculus       GATE 2015 [Set-2]
Question 10 Explanation: 
Since f(0)→∞
∴ f is not bounced in [-1, 1] and hence f is not continuous in [-1, 1].

∴ Statement II & III are true.
Question 11

The value of is

A
0
B
1/2
C
1
D
       Database-Management-System       Calculus       GATE 2015 [Set-3]
Question 11 Explanation: 
Question 12

If for non-zero x, where a≠b then is

A
B
C
D
       Engineering-Mathematics       Calculus       GATE 2015 [Set-3]
Question 12 Explanation: 
Given,
af(x) + bf(1/x) = 1/x - 25 ------ (1)
Put x = 1/x,
af(1/x) + bf(x) = x - 25 ----- (2)
Multiply equation (1) with 'a' and Multiply equation (2) with 'b', then
abf(1/x) + a2 = a/x - 25a ----- (3)
abf(1/x) + b2 = bk - 25b ----- (4)
Subtract (3) - (4), we get
(a2 - b2) f(x) = a/x- 25a - bx + 25b
f(x) = 1/(a2 - b2) (a/x - 25a - bx +25b)
Now from equation,

Hence option (A) is the answer.
Question 13

Let the function

where and f(θ) denote the derivative of f with respect to θ. Which of the following is/are TRUE?

    (I) There exists such that such that f(θ)=0.
    (II) There exists such that such that f(θ)≠0.

A
I only
B
II only
C
Both I and II
D
Neither I nor II
       Engineering-Mathematics       Calculus       GATE 2014 [Set-1]
Question 13 Explanation: 
Rolle’s theorem:
Rolle’s theorem states that for any continuous, differentiable function that has two equal values at two distinct points, the function must have a point on the function where the first derivative is zero. The technical way to state this is: if f is continuous and differentiable on a closed interval [a,b] and if f(a) = f(b), then f has a minimum of one value c in the open interval [a, b] so that f'(c) = 0.
We can observe that, sin, cos are continuous, but, Tan is not continuous at π/2. As the mentioned interval does not contain π/2, we can conclude that it is continuous.
As per Rolls theorem both statement 1 and statement 2 are true.
Question 14

The function f(x) = x sinx satisfies the following equation: f''(x) + f(x) + tcosx = 0. The value of t is __________.

A
-2
B
-3
C
-4
D
-5
       Engineering-Mathematics       Calculus       GATE 2014 [Set-1]
Question 14 Explanation: 
f(x) = x Sinx
f ’(x) = x(Sinx)’ + Sin(x)(x’)
= xCosx + Sinx ---------①
f ’’(x) = x (Cosx)’ + Cos (x)’+ Cos x
= -x Sinx + 2Cosx -----------②
Given: f ’’(x) + f(x) + t Cosx = 0
Replace ① & ②,
-xSinx + 2Cosx + xSinx + tCosx = 0
2Cosx + tCosx = 0
t = -2
Question 15

A function f(x) is continuous in the interval [0,2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true?

A
There exists a y in the interval (0,1) such that f(y) = f(y+1)
B
For every y in the interval (0,1), f(y) = f(2-y)
C
The maximum value of the function in the interval (0,2) is 1
D
There exists a y in the interval (0, 1) such that f(y) = -f(2-y)
       Engineering-Mathematics       Calculus       GATE 2014 [Set-1]
Question 15 Explanation: 
Consider this function as sum of two functions as, g(y) = f(y) -f(y+1)
Since function f is continuous in [0, 2], therefore g would be continuous in [0, 1]
g(0) = -2, g(1) = 2
Since g is continuous and goes from negative to positive value in [0,1]. Therefore at some point g would be 0 in (0,1).
g=0 ⇒ f(y) = f(y+1) for some y in (0,1).
Apply similar logic to option D, Let g(y) = f(y) + f(2 - y)
Since function f is continuous in [0, 2], therefore g would be continuous in [0, 1] (sum of two continuous functions is continuous)
g(0) = -2, g(1) = 2
Since g is continuous and goes from negative to positive value in [0, 1]. Therefore at some point g would be 0 in (0, 1).
There exists y in the interval (0, 1) such that:
g=0 ⇒ f(y) = -f(2 – y)
Both A, D are answers.
Question 16

A non-zero polynomial f(x) of degree 3 has roots at x = 1, x = 2 and x = 3. Which one of the following must be TRUE?

A
f(0)f(4) < 0
B
f(0)f(4) > 0
C
f(0) + f(4) > 0
D
f(0) + f(4) < 0
       Engineering-Mathematics       Calculus       GATE 2014 [Set-2]
Question 16 Explanation: 
Roots of polynomial of degree ‘3’ are 1, 2, 3
Polynomial will be
f(x) = (x-1)(x-2)(x-3)
f(0) = -1 × -2 × -3 = -6
f(4) = 3 × 2 × 1 = 6
f(0) ∙ f(4) = - 36
f(0) + f(4) = 6 - 6 = 0
Option (A) is correct.
Question 17

If , then the value of k is equal to ________.

A
4
B
5
C
6
D
7
       Engineering-Mathematics       Calculus       GATE 2014 [Set-3]
Question 17 Explanation: 
The graph x.Sinx from 0 to 2π is

We have |xSinx|,

We can observe that it is positive from 0 to π and negative in π to 2π.
To get positive value from π to 2π we put ‘-‘ sign in the (π, 2π)
Question 18

With respect to the numerical evaluation of the definite integral ,  where a  and b are given, which of the following statements is/are TRUE?

    I) The value of obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
    II) The value of obtained using the Simpson’s rule is always equal to the exact value of the definite integral.

A
I only
B
II only
C
Both I and II
D
Neither I nor II
       Engineering-Mathematics       Calculus       GATE 2014 [Set-3]
Question 18 Explanation: 
Note: Numerical methods are out of syllabus for the GATE -CS.
Question 19

The value of the integral given below is

A
-2π
B
π
C
D
       Engineering-Mathematics       Calculus       GATE 2014 [Set-3]
Question 19 Explanation: 
Question 20

Consider the function f(x) = sin(x) in the interval x ∈ [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
       Engineering-Mathematics       Calculus       GATE 2012
Question 20 Explanation: 
f(x) = sin x
f’(x) = cos x

[just consider the given interval (π/4, 7π/4)]
f'(x) = 0 at π/2, 3π/2
To get local minima f ’’(x) > 0, f ’’(x) = - sin x
f ’’(x) at π/2, 3π/2
f ’’(x) = -1< 0 local maxima
f ’’ (3π/2) = 1 > 0 this is local minima
In the interval [π/4, π/2] the f(x) is increasing, so f(x) at π/4 is also a local minima.
So there are two local minima for f(x) at π/4, 3π/2.
Question 21

Given i=√-1, what will be the evaluation of the definite integral

A
0
B
2
C
-i
D
i
       Engineering-Mathematics       Calculus       GATE 2011
Question 21 Explanation: 
We know that,
Question 22

What is the value of

A
0
B
e-2
C
e-1/2
D
1
       Engineering-Mathematics       Calculus       GATE 2010
Question 22 Explanation: 
Question 23
is equivalent to
A
0
B
1
C
ln 2
D
1/2 ln 2
       Engineering-Mathematics       Calculus       GATE 2009
Question 23 Explanation: 
Question 24
equals
A
1
B
-1
C
D
-∞
       Engineering-Mathematics       Calculus       GATE 2008
Question 24 Explanation: 
Question 25

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 3x4 - 16x3 + 24x2 + 37 is

A
0
B
1
C
2
D
3
       Engineering-Mathematics       Calculus       GATE 2008
Question 25 Explanation: 
f(x) = 3x4 - 16x3 + 24x2 + 37
f’(x) = 12x3 + 48x2 + 48x = 0
12x(x2 - 4x + 4) = 0
x=0; (x-2)2 = 0
x=2
f’’(x) = 36x2 - 96x + 48
f ”(0) = 48
f ”(2) = 36(4) - 96(2) + 48
= 144 - 192 + 48
= 0
At x=2, we can’t apply the second derivative test.
f’(1) = 12; f’(3) = 36, on either side of 2 there is no sign change then this is neither minimum or maximum.
Finally, we have only one Extremum i.e., x=0.
Question 26

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 x 

Which of the following is TRUE?

A
P is true and Q is false.
B
P is false and Q is true.
C
Both P and Q are true.
D
Both P and Q are false.
       Engineering-Mathematics       Calculus       GATE 2007
Question 26 Explanation: 
f(x) = |x|
→ f(x) is continuous for all real values of x

For every value of x, there is corresponding value of f(x).
For x is positive, f(x) is also positive
x is negative, f(x) is positive.
So, f(x) is continuous for all real values of x.
→ f(x) is not differentiable for all real values of x. For x<0, derivative is negative
x>0, derivative is positive.
Here, left derivative and right derivatives are not equal.
Question 27

Which of the following statements is true?

A
S > T
B
S = T
C
S < T and 2S > T
D
2S ≤ T
       Engineering-Mathematics       Calculus       GATE 2000
Question 27 Explanation: 
S is continuously increasing function but T represent constant value so S>T.
Question 28

Consider the function y = |x| in the interval [-1,1]. In this interval, the function is

A
continuous and differentiable
B
continuous but not differentiable
C
differentiable but not continuous
D
neither continuous nor differentiable
       Engineering-Mathematics       Calculus       GATE 1998
Question 28 Explanation: 
The given function y = |x| be continuous but not differential at x= 0.
→ The left side values of x=0 be negative and right side values are positive.
→ If the function is said to be differentiable then left side and right side values are to be same.
Question 29

(a) Find the points of local maxima and minima, if any, of the following function defined in 0 ≤ x ≤ 6.

  x3 - 6x + 9x - 15 

(b) Integrate

A
Theory Explanation.
       Engineering-Mathematics       Calculus       GATE 1998
Question 30

What is the maximum value of the function f(x) = 2x2 - 2x + 6 in the interval [0,2]?

A
6
B
10
C
12
D
5.5
       Engineering-Mathematics       Calculus       GATE 1997
Question 30 Explanation: 
For f(x) to be maximum
f'(x) = 4x - 2 = 0
⇒ x = 1/2
So at x = 1/2, f(x) is an extremum (either maximum or minimum).
f(2) = 2(2)2 - 2(2) + 6 = 10
f(1/2) = 2 × (1/2)2 - 2 × 1/2 + 6 = 5.5
f(0) = 6
So, the maximum value is at x=2 which is 10 as there are no other extremum for the given function.
Question 31

The formula used to compute an approximation for the second derivative of a function f at a point X0 is

A
f(x0+h) + f(x0-h)/2
B
f(x0+h) - f(x0-h)/2h
C
f(x0+h) + 2f(x0) + f(x0-h)/h2
D
f(x0+h) - 2f(x0) + f(x0-h)/h2
       Engineering-Mathematics       Calculus       GATE 1996
Question 31 Explanation: 
The formula which is used to compute the second derivation of a function f at point X is
f(x0+h) - 2f(x0) + f(x0-h)/h2
Question 32

The solution of differential equation y'' + 3y' + 2y = 0 is of the form

A
C1ex + C2e2x
B
C1e-x + C2e3x
C
C1e-x + C2e-2x
D
C1e-2x + C22-x
       Engineering-Mathematics       Calculus       GATE 1995
Question 32 Explanation: 
Note: Out of syllabus.
Question 33

(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion

A
Theory Explanation.
       Engineering-Mathematics       Calculus       GATE 1995
Question 34

Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).

A
yn+1 = yn + hf(xn, yn)
B
yn+1 = yn + hf(xn+1, yn+1)
C
yn+1 = yn-1 + 2hf(xn, yn)
D
yn+1 = (1 + h) f(xn+1, yn+1)
       Engineering-Mathematics       Calculus       GATE 1994
Question 34 Explanation: 
dy/dx = f(x,y)
With initial value y(x0) = y0. Here the function f and the initial data x0 and y0 are known. The function y depends on the real variable x and is unknown. A numerical method produces a sequence y0, y1, y2, ....... such that yn approximates y(x0 + nh) where h is called the step size.
→ The backward Euler method is helpful to compute the approximations i.e.,
yn+1 = yn + hf(x n+1, yn+1)
Question 35

Consider the functions

    I. e-x
    II. x2-sin x
    III. √(x3+1)

Which of the above functions is/are increasing everywhere in [0,1]?

A
II and III only
B
III only
C
II only
D
I and III only
       Engineering-Mathematics       Calculus       GATE 2020
Question 35 Explanation: 
A function f(x) is said to be increasing if f'(x)>0 at each point in an interval.
I. e-x
II. f'(x) = -e-x
f'(x)<0 on the interval [0,1] so this is not an increasing function.
II. x2-sinx
f'(x) = 2x - cosx
at x=0, f'(0) = 2(0) - 1 = -1 < 0
f(x) = x2 - sinx is decreasing over some interval, increasing over some interval as cosx is periodic.
As the question is asked about increasing everywhere II is false.
III. √(x3+1) = (x3+1)1/2
f'(x) = 1/2(3x2/√(x3+1))>0
f(x) is increasing over [0,1].
Question 36

The differential equation
d2y/dx2 + dy/dx + siny = 0 is:

A
linear
B
non-linear
C
homogeneous
D
of degree two
       Engineering-Mathematics       Calculus       GATE 1993
Question 36 Explanation: 
Note: Out of syllabus.
d2y/dx2 + dy/dx + siny = 0
In this DE, degree is 1 then this represent linear equation.
Question 37

Fourier series of the periodic function (period 2π) defined by

But putting x = π, we get the sum of the series.

A
π2/4
B
π2/6
C
π2/8
D
π2/12
       Engineering-Mathematics       Calculus       GATE 1993
Question 37 Explanation: 
Note: Out of syllabus.
Question 38

Which of the following improper integrals is (are) convergent?

A
B
C
D
       Engineering-Mathematics       Calculus       GATE 1993
Question 39
A
1
       Engineering-Mathematics       Calculus       GATE 1993
Question 39 Explanation: 
Since the given expression is in 0/0 form, so we can apply L-Hospital rule.
Question 40

The radius of convergence of the power series

A
Out of syllabus.
       Engineering-Mathematics       Calculus       GATE 1993
Question 41

The value of the double integral is

A
1/3
       Engineering-Mathematics       Calculus       GATE 1993
Question 41 Explanation: 
Question 42

The differential equation yn + y = 0 is subjected to the boundary conditions.

    y (0) = 0        y(λ) = 0         

In order that the equation has non-trivial solution(s), the general value of λ is __________

A
Out of syllabus.
       Engineering-Mathematics       Calculus       GATE 1993
Question 43

If f(x) is defined as follows, what is the minimum value of f(x) for x∊(0,2] ?

A
2
B
2(1/12)
C
2(1/6)
D
2(1/2)
       Engineering-Mathematics       Calculus       GATE 2008-IT
Question 43 Explanation: 
If x = 3/2
f(x) = 25/8x = 25/8(3/2) = 25/12 = 2(1/12)
Question 44

The following definite integral evaluates to

A
1/2
B
π √10
C
√10
D
π
E
None of the above
       Engineering-Mathematics       Calculus       GATE 2006-IT
Question 44 Explanation: 
Question 45

If the trapezoidal method is used to evaluate the integral obtained 01x2dx ,then the value obtained

A
is always > (1/3)
B
is always < (1/3)
C
is always = (1/3)
D
may be greater or lesser than (1/3)
       Engineering-Mathematics       Calculus       GATE 2005-IT
Question 45 Explanation: 
Note: Out of syllabus.
Question 46

What is the value of

A
-1
B
1
C
0
D
π
       Engineering-Mathematics       Calculus       GATE 2005-IT
Question 46 Explanation: 

In the limits are be -π to π, one is odd and another is even product of even and odd is odd function and integrating function from the same negative value to positive value gives 0.
Question 47

If f(1) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3) using Lagrange’s interpolation formula?

A
8
B
8(1/3)
C
8(2/3)
D
9
       Engineering-Mathematics       Calculus       GATE 2004-IT
Question 47 Explanation: 
Note: Out of syllabus.
Question 48

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 
A
Q-II, R-IV, S-II, T-I
B
Q-III, R-II, S-I, T-IV
C
Q-II, R-I, S-IV, T-III
D
Q-I, R-IV, S-II, T-III
       Engineering-Mathematics       Calculus       GATE 2004-IT
Question 48 Explanation: 
Note: Out of syllabus.
Question 49
Consider the following expression.

The value of the above expression (rounded to 2 decimal places) is _______
A
0.25
       Engineering-Mathematics       Calculus       GATE 2021 CS-Set-1
Question 49 Explanation: 
Question 50
The value of the following limit is _____________.
A
1/2
       Engineering-Mathematics       Calculus       GATE 2022       Video-Explanation
Question 50 Explanation: 

When 0 is substituted, we get 0/0
Apply L- Hospital rule


-1/2
Question 51
The domain of the function log( log sin(x) ) is
A
0 < x < π
B
2nπ < x < (2n+1)π , for n in N
C
Empty set
D
None of the above
       Engineering-Mathematics       Calculus       ISRO-2018       Video-Explanation
Question 51 Explanation: 
→ The range of sinx value lies between -1 and 1 and whereas log(sinx) value will be the positive value.
→ So the domain of log(log sin(x)) is undefined which is empty Set.
Question 52
Consider the following C code segment   Of the following, which best describes the growth of f(x) as a function of x?
A
Linear
B
Exponential
C
Quadratic
D
Cubic
       Engineering-Mathematics       Calculus       ISRO-2018
Question 52 Explanation: 
Question 53
If x = -1 and x = 2 are extreme points of f(x) = α log |x| + βx2 + x then
A
α = -6, β = -1/2
B
α = 2, β = -1/2
C
α = 2, β = 1/2
D
α = -6, β =1/2
       Engineering-Mathematics       Calculus       ISRO-2017 December
Question 53 Explanation: 
Given data,
Step-1: x= -1 and x=2
f(x) = α log |x| + β x2 + x
f'(x)= α/x + 2βx + 1 = 0
Step-2: for extreme points f'(x)=0
α/x + 2βx + 1=0
Step-3: For x= -1 then we will get α+2β= 1 → (i)
For x= 2: then we will get α+8β= 2 → (ii)
from (i) and (ii) we can get the value of α=2 and β= -1/2
Question 54
If T(x) denotes x is a trigonometric function, P(x) denotes x is a periodic function and C(x) denotes x is a continuous function then the statement “It is not the case that some trigonometric functions are not periodic” can be logically represented as
A
¬∃(x) [ T(x) ⋀ ¬P(x) ]
B
¬∃(x) [ T(x) ⋁ ¬P(x) ]
C
¬∃(x) [ ¬T(x) ⋀ ¬P(x) ]
D
¬∃(x) [ T(x) ⋀ P(x) ]
       Engineering-Mathematics       Calculus       ISRO-2017 December
Question 54 Explanation: 
Option A implies "It is not the case that some trigonometric functions are not periodic”.
Hence it is correct .
Option B implies "It is not the case that some are trigonometric functions or they are not periodic”.
Option C implies "It is not the case that some of not trigonometric functions are not periodic”.
Option D implies "It is not the case that some trigonometric functions are periodic”.
Question 55
The function f: [0,3]→[1,29] defined by f(x) = 2x3 – 15x2 + 36x + 1 is
A
injective and surjective
B
surjective but not injective
C
injective but not surjective
D
neither injective nor surjective
       Engineering-Mathematics       Calculus       ISRO-2017 December
Question 55 Explanation: 
Question 56
x = a cos(t), y = b sin(t) is the parametric form of
A
Ellipse
B
Hyperbola
C
Circle
D
Parabola
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 56 Explanation: 
An ellipse can be defined as the locus of all points that satisfy the equations
x = a cos t
y = b sin t
where:
x,y are the coordinates of any point on the ellipse,
a, b are the radius on the x and y axes respectively,
t is the parameter, which ranges from 0 to 2π radians
Question 57
The value of x at which y is minimum for y = x2 − 3x + 1 is
A
-3/2
B
3/2
C
0
D
-5/4
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 57 Explanation: 
Given function is y = x2 – 3x + 1
To find the minimum value, calculate the derivative until at what point given function is minimum value.
Applying First order derivative to the function y is y’ = 2x – 3
Again applying derivative of y ’is y” = 2 (Since y” > 0, it has a minimum value)
So the minima value at that point is (2x – 3) = 0 and x = 3/2
Question 58
The formula:

is called
A
Simpson rule
B
Trapezoidal rule
C
Romberg’s rule
D
Gregory’s formula
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 58 Explanation: 
The above formula is for trapezoidal rule.
Question 59
The image

A
Newton’s backward formula
B
Gauss forward formula
C
Gauss backward formula
D
Stirling’s formula
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 59 Explanation: 
The above formula is NEWTON’S GREGORY BACKWARD INTERPOLATION FORMULA
Question 60
Which of the following statement is correct
A
△(UkVk) = Uk△Vk + Vk△Uk
B
△(UkVk) = Uk+1△Vk + Vk+1△Uk
C
△(UkVk) = Vk+1△Uk + Uk△Vk
D
△(UkVk) = Uk+1△Vk + Vk△Uk
       Engineering-Mathematics       Calculus       ISRO CS 2009
Question 61
n-th derivative of xn is
A
n xn-1
B
nn . n!
C
nxn !
D
n!
       Engineering-Mathematics       Calculus       ISRO CS 2011
Question 61 Explanation: 

Question 62

Which of the following statements is false about convex minimization problem ?

A
If a local minimum exists, then it is a global minimum
B
The set of all global minima is convex set
C
The set of all global minima is concave set
D
For each strictly convex function, if the function has a minimum, then the minimum is unique
       Engineering-Mathematics       Calculus       UGC-NET CS 2018 JUNE Paper-2
Question 62 Explanation: 
Properties of convex optimization problems:
1. Every local minimum is a global minimum
2. The optimal set is convex
3. If the objective function is strictly convex, then the problem has at most one optimal point.
These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma.
Question 63

The following LPP

    Maximize z = 100x1 + 2x2 + 5x3
    Subject to 14x1 + x2 − 6x3 + 3x4 = 7
32x1 + x2 − 12x3 ≤ 10 
3x1 − x2 − x3 ≤ 0 
x, x2 , x3 , x4 ≥ 0 has 
A
Solution : x1 = 100, x2 = 0, x3 = 0
B
Unbounded solution
C
No solution
D
Solution : x1 = 50, x2 = 70, x3 = 60
       Engineering-Mathematics       Calculus       UGC-NET CS 2018 JUNE Paper-2
Question 63 Explanation: 
An unbounded solution of a linear programming problem is a situation where objective function is infinite.
A linear programming problem is said to have unbounded solution if its solution can be made infinitely large without violating any of its constraints in the problem.
Question 64
A polynomial p(x) is such that p(0)=5, p(1)=4, p(2)=9 and p(3)=20 The minimum degree it can have is
A
1
B
2
C
3
D
4
       Engineering-Mathematics       Calculus       Nielit Scientist-B CS 2016 march
Question 64 Explanation: 
Let's take p(x) = ax + b
p(0) = 5 ⇒ b = 5
p(1) = 4 ⇒ a+b = 4 ⇒ a = -1
p(2) = 9 ⇒ 40+b = 9 ⇒ -4+5 = 9, which is false.
So degree 1 is not possible.
Let's take p(x) = ax​ 2​ + bx +c
p(0) = 5 ⇒ c = 5
p(1) = 4 ⇒ a+b+c = 4 ⇒ a+b = -1 -----(1)
p(2) = 9 ⇒ 4a+2b+c = 9 ⇒ 2a+b = 2 -----(2)
(2) - (1)
⇒ a = 3, b = -1-1 = -4
p(3) = 20 ⇒ 9a+3b+c = 20
⇒ 27-12+5 = 20
⇒ 20 = 20, True
Hence, minimum degree it can have.
Question 65
Which of the following is the derivative of f(x) = xx when x > 0?
A
xx
B
xx In x
C
xx + xx In x
D
(xx)(xxIn x)
E
None of the above; function is not differentiable for x > 0
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2018
Question 66
Consider the function f(x)=sin(x) in the interval [​ π ​ /4, 7​ π ​ /4]. The number and location(s) of the minima of this function are:
A
One, at π/2
B
One, at 3​ π ​ /2
C
Two, at ​ π ​ /2 and 3​ π ​ /2
D
Two, at ​ π ​ /4 and 3​ π ​ /2
       Engineering-Mathematics       Calculus       Nielit Scientist-B CS 4-12-2016
Question 66 Explanation: 
The local minima is at x =3π/2
This is very obvious from the graph of f (x) = sin x
On a second look at the graph below, I believe x =π/4
is also a local minimum. This is because it is lesser than all other values within its locality.
Thus we have two local minima: x = π /4 , 3π
Question 67

Which of the following statements about the Newton-Raphson method is/are correct?

    (i) It is quadratic convergent
    (ii) If f'(x) is zero, it fails
    (iii) It is also used to obtain complex root
A
(i), (ii) and (iii)
B
only (i) and (iii)
C
only (i) and (ii)
D
only (i)
       Engineering-Mathematics       Calculus       JT(IT) 2018 PART-B Computer Science
Question 67 Explanation: 
Above all statements are true for Newton-Raphson method.
Note: Newton-Raphson method is mainly used for Interpolation.
Question 68
What is the maximum value of the function f(x)=2x​ 2​ -2x +6 in the interval [0,2]?
A
6
B
10
C
12
D
5,5
       Engineering-Mathematics       Calculus       Nielit Scientific Assistance CS 15-10-2017
Question 68 Explanation: 
An open interval does not include its endpoints, and is indicated with parentheses. For example, (0,1) means greater than 0 and less than 1.
A closed interval is an interval which includes all its limit points, and is denoted with square brackets. For example, [0,1] means greater than or equal to 0 and less than or equal to 1.
A half-open interval includes only one of its endpoints, and is denoted by mixing the notations for open and closed intervals. (0,1] means greater than 0 and less than or equal to 1, while [0,1) means greater than or equal to 0 and less than 1.
Given function f(x)=2x​ 2​ -2x +6 and the given interval is [0,2]
According to the given interval , we need to check the function at the values 0,1,2.
f(0)=2x0-2x0+6=6
f(1)=2x1​ 2​ -2x1+6=2-2+6=6
f(2)=2x2​ 2​ -2x2+6=8-4+6=10
Question 69
The value of the integral
A
(x+2)/2
B
2/(π-2)
C
π-2
D
π+2
       Engineering-Mathematics       Calculus       Nielit Scientific Assistance CS 15-10-2017
Question 69 Explanation: 
The integral value of x​ 2​ sin(x)dx is [−x​ 2​ cos(x)+2xsin(x)+2cos(x)+C]​ with the interval [π/2,0]
=[−(​ π/2)​ 2​ cos(​ π/2​ )+2​ π/2​ sin(​ π/2​ )+2cos(​ π/2​ )+C]​ - ​ [−(​ 0)​ 2​ cos(​ π/2​ )+2(0)sin(​ π/2​ )+2cos(​ 0 ​ )+C]
=[0+​ π+0+C-2-C]
=π-2
Question 70

The number of strips required in simpson’s 3/8th rule is a multiple of:

A
1
B
2
C
3
D
6
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 70 Explanation: 
Simpson’s 3/8th rule is also known as Simpson's 2nd rule:
Area = 3h/ 8 [( a + 3 b + 3 c + d )]
Simpson's Second Rule:
Multipliers:
Question 71

The points at which the function attains extreme values are called:

A
Turning points
B
End points
C
Higher points
D
Extreme points
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 71 Explanation: 
The value of the function, the value of y, at either a maximum or a minimum is called an extreme value. The points at which the function attains extreme values are called Turning points.
Question 72

If f(x) = ax2 + bx + c the f(x-(b/2a)) is:

A
An even function for all a except a=0
B
An even function for all a
C
Neither even nor odd
D
An odd function for all a except a=0
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 72 Explanation: 
• A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.
• A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.
Question 73
The partial differential equation ∂2z/∂t2 = c2(∂2z/∂x2) represents:
A
Harmonic function
B
Laplace equation
C
Wave equation
D
Homogeneous
       Engineering-Mathematics       Calculus       JT(IT) 2016 PART-B Computer Science
Question 73 Explanation: 
Let us derive the d’Alembert’s formula in an alternate way.
Note that the wave equation can be factored as
Question 74
The functions mapping R into R are defined as : f(x) = x​ 3​ – 4x, g(x) = 1/(x​ 2​ + 1) and h(x) = x​ 4​ . Then find the value of the following composite functions : hog(x) and hogof(x)
A
(x​ 2​ + 1)4 and [(x​ 3​ – 4x)​ 2​ + 1]​ 4
B
(x​ 2​ + 1)4 and [(x​ 3​ – 4x)​ 2​ + 1]​ -4
C
(x​ 2​ + 1)-4 and [(x​ 3​ – 4x)​ 2​ + 1]​ 4
D
(x​ 2​ + 1)-4 and [(x​ 3​ – 4x)​ 2​ + 1]​ -4
       Engineering-Mathematics       Calculus       UGC NET CS 2017 Jan -paper-2
Question 74 Explanation: 
Step-1: Given data,
f(x) = x​ 3​ – 4x, g(x) = 1/(x​ 2​ + 1) and h(x) = x​ 4
hog(x)=h(1/(x​ 2​ + 1))
=h(1/(x​ 2​ )+1)​ 4
= 1/(x​ 2​ +1)​ 4
= (x​ 2​ +1)​ -4
hogof(x)= hog(x​ 3​ -4x)
= h(1/(x​ 3​ -4x)​ 2​ +1)
= h(1/(x​ 3​ -4x)​ 2​ +1)​ 4
= h((x​ 3​ -4x)​ 2​ +1)​ -4
So, option D id is correct answer.
Question 75
Let f and g be the functions from the set of integers to the set integers defined by f(x) = 2x + 3 and g(x) = 3x + 2 Then the composition of f and g and g and f is given as
A
6x + 7, 6x + 11
B
6x + 11, 6x + 7
C
5x + 5, 5x + 5
D
None of the above
       Engineering-Mathematics       Calculus       UGC NET CS 2013 Dec-paper-2
Question 75 Explanation: 
Given data,
f(x) = 2x+3
g(x) = 3x+2
fo(g(x)) = f(3x+2)
= 2(3x+2)+3
= 6x+7
go(f(x)) = g(2x+3)
= 3(2x + 3)+2
= 6x +11
So, option-A is correct answer.
Question 76
Domain and Range of the function Y = -√(-2x + 3) is
A
x ≥3/2, y ≥ 0
B
x >3/2, y ≤ 0
C
x ≥3/2, y ≤ 0
D
x ≤3/2, y ≤ 0
       Engineering-Mathematics       Calculus       UGC NET CS 2011 Dec-Paper-2
Question 77
Given y = arctan(tanh x) dy/dx = ?
A
1/cosh 2x
B
1/sinh 2x
C
1/(1+tanh2 x)
D
None of the above
       Engineering-Mathematics       Calculus       NIELIT Junior Teachnical Assistant_2016_march
Question 78
For the function f (x) = x 2 + x − 1 , there exists a value x = c in the interval [ − 1 , 1 ] where f′(c) = 0 , then c is equal to
A
(√5 − 1 )/2
B
(√5 + 1 )/2
C
( − √5 − 1 )/2
D
( − √5 + 1 )/2
       Engineering-Mathematics       Calculus       NIELIT Junior Teachnical Assistant_2016_march
Question 79
∫(xex /(1+x)2) equals to
A
(xex /(1+x))
B
x/(1+x)
C
ex/(1+x)2
D
ex/(1+x)3
       Engineering-Mathematics       Calculus       NIELIT Junior Teachnical Assistant_2016_march
Question 80
A
1
B
-1
C
0
D
       Engineering-Mathematics       Calculus       NIELIT Junior Teachnical Assistant_2016_march
Question 81
A
2/3
B
-2/3
C
3/2
D
-3/2
E
None of the Above
       Engineering-Mathematics       Calculus       NIELIT Junior Teachnical Assistant_2016_march
Question 82
Solution of the differential equation x (dy/dx) + y = y 2 is given by
A
y − 1 = c xy
B
y = c xy − 1
C
x y = 1 + c x
D
y + 1 = c xy
       Engineering-Mathematics       Calculus       NIELIT Junior Teachnical Assistant_2016_march
Question 83
Given y = ln(ln n mx),dy/dx = ?
A
nm /x ln mx
B
n /nx ln mx
C
n / x ln mx
D
m / nx ln mx
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 84
Given 2 x + 2 y = 2 x+y , dy/dx = ?
A
-2y-x
B
2 x (2 y +1)/2 y (2 x -1)
C
2 x (2 y -2)/2 y (2 x -1)
D
2 x (2 y -1)/2 y (2 x -1)
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 85
A
B
C
D
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 86
A
1
B
1/2
C
-1
D
2
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 87
A
e 10
B
e -10
C
e 2
D
e 1/2
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 88
A
8/9
B
-8/9
C
-9/8
D
9/8
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 89
A
-π / 2
B
2 / π
C
π / 2
D
-2 / π
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 90
The solution of the differential equation (dy / dx) + (y + x) = x 2 with y = 1 when x = 1 is
A
x y = x 4 + 2
B
2 x y = x 4 + 4
C
3 x y = x 4 + 4
D
4 x y = x 4 + 3
       Engineering-Mathematics       Calculus       NIELIT Technical Assistant_2016_march
Question 91
The value of the derivative of the sigmoid function given by
f(x)= 1 / (1+e(-2x)) at x=0 is
A
0
B
1/2
C
1/4
D
       Engineering-Mathematics       Calculus       UGC NET June-2019 CS Paper-2
Question 92
Let limn→∞ f(n) = ∞ and limn→∞ g(n) = ∞. Then which of the following is necessarily TRUE.
A
limn→∞ |f(n) − g(n)| = ∞
B
limn→∞ |f(n) − g(n)| = 0
C
limn→∞ |f(n)/g(n)| = ∞
D
limn→∞ |f(n)/g(n)| = 1
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2018
Question 93
Consider

What can we say about limx→∞ f(x)?
A
The function f(x) does not have a limit as x→∞
B
limx→∞ f(x) = e2
C
limx→∞ f(x) = e1/2
D
limx→∞ f(x) = 0
E
limx→∞ f(x) = ∞
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2018
Question 94
In order for a function to be invertible, it should be _______
A
one-one
B
onto
C
both one-one and onto
D
into
       Engineering-Mathematics       Calculus       APPSC-2016-DL-CS
Question 94 Explanation: 
A function is invertible if and only if it is both one-one and onto.
Question 95
What is the maximum value of the function f(x)=x2-3x+5 in the internal [0,5]?
A
15
B
5
C
3
D
9
       Engineering-Mathematics       Calculus       APPSC-2016-DL-CS
Question 95 Explanation: 
Question 96
A
0.00
B
0.02
C
0.10
D
0.33
E
1.00
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2019
Question 97
Consider a function f : R --> R such that f (x) = 1 if x is rational, and f (x) = 1-∈, where 0 < ∈ < 1, if x is irrational. Which of the following is TRUE?
A
limx→∞ f (x) = 1
B
limx→∞ f (x) = 1 − ∈
C
limx→∞ f (x) exists, but is neither 1 nor 1 − ∈
D
maxx≥1 f (x) = 1
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2019
Question 98
Let f (x) = √(x2-4x + 4), for x ∈ (-∞, ∞) Here, √y denotes the non-negative square root of y when y is non-negative. Then, which of the following is TRUE?
A
f(x) is not continuous but differentiable
B
f(x) is continuous and differentiable
C
f(x) is continuous but not differentiable
D
f(x) is neither continuous nor differentiable
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2019
Question 99
Consider the function f(x) = ex2 - 8x2 for all x on the real line. For how many distinct values of x do we have f(x)=0?
A
1
B
4
C
2
D
3
E
5
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2019
Question 100
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2019
Question 101
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2017
Question 102
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2017
Question 103
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2017
Question 104
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2016
Question 105
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2016
Question 106

A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2016
Question 107
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2015
Question 108
Suppose that f(x) is a continuous function such that 0.4 ≤ f(x) ≤ 0.6 for 0 ≤ x ≤ 1. Which of the following is always true?
A
f(0.5) = 0.5
B
There exists x between 0 and 1 such that f(x) = 0.8x
C
There exists x between 0 and 0.5 such that f(x) = x.
D
f(0.5) > 0.5.
E
None of the above statements are always true.
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2015
Question 109
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2015
Question 110
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 111
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 112
Which of these functions grows fastest with n?
A
en/n
B
en−0.9 log n
C
2n
D
(log n)n−1
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 113
A
e
B
1
C
21/3
D
0
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 114
A
g must be identically zero
B
g(π/2) = 1.
C
g need not be identically zero
D
g(π)=0.
E
None of the above.
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 115
Consider a square pulse g(t) of height 1 and width 1 centred at 1/2. Define fn(t) = 1/n (g(t) ∗n g(t)), where ∗n stands for n-fold convolution. Let f(t) = limn→∞ fn(t). Then, which of the following is TRUE?
A
The area under the curve of f(t) is zero
B
The area under the curve of f(t) is ∞
C
f(t) has width ∞ and height 1
D
f(t) has width 0 and height ∞.
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 116
Let function f : R → R be convex, i.e., for x, y ∈ R, α ∈ [0, 1], f(αx + (1 − α)y) ≤ αf(x) + (1 − α)f(y). Then which of the following is TRUE?
A
f(x) ≤ f(y) whenever x ≤ y.
B
For a random variable X, E(f(X)) ≤ f(E(X))
C
The second derivative of f can be negative
D
If two functions f and g are both convex, then min{f,g} is also convex
E
For a random variable X, E(f(X)) ≥ f(E(X))
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 117
A
0
B
π/2
C
1/√2
D
2/π
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2014
Question 118
The maximum value of the function
f(x, y, z) = (x − 1/3)2 + (y − 1/3)2 + (z − 1/3)2
subject to the constraints
x + y + z = 1, x ≥ 0, y ≥ 0, z ≥ 0
A
1/3
B
2/3
C
1
D
4/3
E
4/9
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2012
Question 119
A
B
1
C
1/2
D
0
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2012
Question 120
Consider the differential equation dx/dt = (1−x)(2−x)(3−x). Which of its equilibria is unstable?
A
x = 0
B
x = 1
C
x = 2
D
x = 3
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2012
Question 121
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2012
Question 121 Explanation: 
Given question and answer is wrong.
Question 122
Let α1, α2, · · · , αk be complex numbers. Then

is
A
0
B
C
αk
D
α1
E
maxjj |
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2012
Question 123
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2012
Question 124
A
a
B
b
C
c
D
d
E
e
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2012
Question 125
Consider a function f : [0, 1] --> [0, 1] which is twice differentiable in (0, 1). Suppose it has exactly one global maximum and exactly one global minimum inside (0, 1). What can you say about the behaviour of the first derivative f' and and second derivative f'' on (0, 1) (give the most precise answer)?
A
f' is zero at exactly two points, f'' need not be zero anywhere
B
f' is zero at exactly two points, f'' is zero at exactly one point
C
f' is zero at at least two points, f'' is zero at exactly one point
D
f' is zero at at least two points, f'' is zero at at least one point
E
f' is zero at at least two points, f'' is zero at at least two points
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2020
Question 126
A
h(2t)
B
h(t)
C
h(t − 1)
D
h(t + 1)
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2020
Question 127
Let f, g : R --> R be two functions that are continuous and differentiable. Consider the following statements:
1. min{f, g} is continuous
2. max{f, g} is continuous
3. max{f, g} is differentiable
Which of the following is TRUE?
A
Only statement 1 is correct
B
Only statement 2 is correct
C
Only statement 3 is correct
D
Only statements 1 and 2 are correct
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2020
Question 128
A
g(x) is aperiodic
B
g(x) is periodic with period 1
C
g(x) is periodic with period 1
D
The value of h determines whether or not g(x) is periodic
E
None of the above
       Engineering-Mathematics       Calculus       TIFR PHD CS & SS 2020
Question 129
Let f(x)=x2 sin 1/x2 for x > 0. Which of the following is a correct statement
A
f is unbounded
B
f is bounded by limz→ ∞ f(x) does not exist
C
limz→ ∞ f(x) = 0
D
limz→ ∞ f(x) = 1
       Engineering-Mathematics       Calculus       HCU PHD CS 2018 December
Question 129 Explanation: 
Question 130
The minimum value of x2 + y2 subject to x+y=1 is
A
½
B
1/√2
C
¼
D
1
       Engineering-Mathematics       Calculus       HCU PHD CS 2018 December
Question 130 Explanation: 


=1/2
There are 130 questions to complete.