Calculus
Question 1 
1  
Limit does not exist  
53/12  
108/7 
Question 2 
0.289  
0.298  
0.28  
0.29 
Question 3 
is 0  
is 1  
is 1  
does not exist 
If "x=1" is substituted we get 0/0 form, so apply LHospital rule
Substitute x=1
⇒ (7(1)^{6}10(1)^{4})/(3(1)^{2}6(1)) = (710)/(36) = (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.
Question 5 
4  
3  
2  
1 
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 __________.
9  
10  
11  
12 
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:
h(x)/g(x)  
1/x  
g(x)/h(x)  
x/(1x)^{2} 
Replace x by h(x) in (1), replacing x by g(x) in (2),
g(h(x))=1h(x)=1x/x1=1/x1
h(g(x))=g(x)/g(x)1=1x/x
⇒ g(h(x))/h(g(x))=x/(x1)(1x)=(x/x1)/1x=h(x)/g(x)
Question 9 
0.99  
1.00  
2.00  
3.00 
= 21/1(2)+32/2(3)+43/3(4)+…+10099/99(100)
= 1/11/2+1/21/3+1/3…+1/981/99+1/991/100
= 11/100
= 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 Xaxis, 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
II only  
III only  
II and III only  
I, II and III 
∴ 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
0  
1/2  
1  
∞ 
Question 12 
If for nonzero x, where a≠b then is
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) + a^{2} = a/x  25a  (3)
abf(1/x) + b^{2} = bk  25b  (4)
Subtract (3)  (4), we get
(a^{2}  b^{2}) f(x) = a/x 25a  bx + 25b
f(x) = 1/(a^{2}  b^{2}) (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.
I only  
II only  
Both I and II  
Neither I nor II 
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 __________.
2  
3  
4  
5 
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?
There exists a y in the interval (0,1) such that f(y) = f(y+1)  
For every y in the interval (0,1), f(y) = f(2y)  
The maximum value of the function in the interval (0,2) is 1  
There exists a y in the interval (0, 1) such that f(y) = f(2y) 
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 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 
Polynomial will be
f(x) = (x1)(x2)(x3)
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 ________.
4  
5  
6  
7 
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.
I only  
II only  
Both I and II  
Neither I nor II 
Question 19 
The value of the integral given below is
2π  
π  
π  
2π 
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
One, at π/2  
One, at 3π/2  
Two, at π/2 and 3π/2  
Two, at π/4 and 3π/2 
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
0  
2  
i  
i 
Question 22 
What is the value of
0  
e^{2}  
e^{1/2}  
1 
Question 23 
0  
1  
ln 2  
1/2 ln 2 
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 3x^{4}  16x^{3} + 24x^{2} + 37 is
0  
1  
2  
3 
f’(x) = 12x^{3} + 48x^{2} + 48x = 0
12x(x^{2}  4x + 4) = 0
x=0; (x2)^{2} = 0
x=2
f’’(x) = 36x^{2}  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?
P is true and Q is false.  
P is false and Q is true.  
Both P and Q are true.  
Both P and Q are false. 
→ 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?
S > T  
S = T  
S < T and 2S > T  
2S ≤ T 
Question 28 
Consider the function y = x in the interval [1,1]. In this interval, the function is
continuous and differentiable  
continuous but not differentiable  
differentiable but not continuous  
neither continuous nor differentiable 
→ 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.
x^{3}  6x + 9x  15
(b) Integrate
Theory Explanation. 
Question 30 
What is the maximum value of the function f(x) = 2x^{2}  2x + 6 in the interval [0,2]?
6  
10  
12  
5.5 
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 X_{0} is
f(x_{0}+h) + f(x_{0}h)/2  
f(x_{0}+h)  f(x_{0}h)/2h  
f(x_{0}+h) + 2f(x_{0}) + f(x_{0}h)/h^{2}  
f(x_{0}+h)  2f(x_{0}) + f(x_{0}h)/h^{2} 
f(x_{0}+h)  2f(x_{0}) + f(x_{0}h)/h^{2}
Question 32 
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
C_{1}e^{x} + C_{2}e^{2x}  
C_{1}e^{x} + C_{2}e^{3x}  
C_{1}e^{x} + C_{2}e^{2x}  
C_{1}e^{2x} + C_{2}2^{x} 
Question 33 
(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion
Theory Explanation. 
Question 34 
Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).
y_{n+1} = y_{n} + hf(x_{n}, y_{n})  
y_{n+1} = y_{n} + hf(x_{n+1}, y_{n+1})  
y_{n+1} = y_{n1} + 2hf(x_{n}, y_{n})  
y_{n+1} = (1 + h) f(x_{n+1}, y_{n+1}) 
With initial value y(x_{0}) = y_{0}. Here the function f and the initial data x_{0} and y_{0} are known. The function y depends on the real variable x and is unknown. A numerical method produces a sequence y_{0}, y_{1}, y_{2}, ....... such that y_{n} approximates y(x_{0} + nh) where h is called the step size.
→ The backward Euler method is helpful to compute the approximations i.e.,
y_{n+1} = y_{n} + hf(x _{n+1}, y_{n+1})
Question 35 
Consider the functions
 I. e^{x}
II. x^{2}sin x
III. √(x^{3}+1)
Which of the above functions is/are increasing everywhere in [0,1]?
II and III only  
III only  
II only  
I and III only 
I. e^{x}
II. f'(x) = e^{x}
f'(x)<0 on the interval [0,1] so this is not an increasing function.
II. x^{2}sinx
f'(x) = 2x  cosx
at x=0, f'(0) = 2(0)  1 = 1 < 0
f(x) = x^{2}  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. √(x^{3}+1) = (x^{3}+1)^{1/2}
f'(x) = 1/2(3x^{2}/√(x^{3}+1))>0
f(x) is increasing over [0,1].
Question 36 
The differential equation
d^{2}y/dx^{2} + dy/dx + siny = 0 is:
linear  
nonlinear  
homogeneous  
of degree two 
d^{2}y/dx^{2} + 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.
π^{2}/4  
π^{2}/6  
π^{2}/8  
π^{2}/12 
Question 38 
Which of the following improper integrals is (are) convergent?
Question 39 
1 
Question 40 
The radius of convergence of the power series
Out of syllabus. 
Question 41 
The value of the double integral is
1/3 
Question 42 
The differential equation y^{n} + y = 0 is subjected to the boundary conditions.
y (0) = 0 y(λ) = 0
In order that the equation has nontrivial solution(s), the general value of λ is __________
Out of syllabus. 
Question 43 
If f(x) is defined as follows, what is the minimum value of f(x) for x∊(0,2] ?
2  
2(1/12)  
2(1/6)  
2(1/2) 
f(x) = 25/8x = 25/8(3/2) = 25/12 = 2(1/12)
Question 44 
The following definite integral evaluates to
1/2  
π √10  
√10  
π  
None of the above 
Question 45 
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) 
Question 46 
What is the value of
1  
1  
0  
π 
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?
8  
8(1/3)  
8(2/3)  
9 
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
QII, RIV, SII, TI  
QIII, RII, SI, TIV  
QII, RI, SIV, TIII  
QI, RIV, SII, TIII 
Question 49 
The value of the above expression (rounded to 2 decimal places) is _______
0.25 
Question 50 
1/2 
When 0 is substituted, we get 0/0
Apply L Hospital rule
1/2
Question 51 
0 < x < π  
2nπ < x < (2n+1)π , for n in N  
Empty set  
None of the above 
→ So the domain of log(log sin(x)) is undefined which is empty Set.
Question 52 
Linear  
Exponential  
Quadratic  
Cubic 
Question 53 
α = 6, β = 1/2  
α = 2, β = 1/2  
α = 2, β = 1/2  
α = 6, β =1/2 
Step1: x= 1 and x=2
f(x) = α log x + β x2 + x
f'(x)= α/x + 2βx + 1 = 0
Step2: for extreme points f'(x)=0
α/x + 2βx + 1=0
Step3: 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 
¬∃(x) [ T(x) ⋀ ¬P(x) ]  
¬∃(x) [ T(x) ⋁ ¬P(x) ]  
¬∃(x) [ ¬T(x) ⋀ ¬P(x) ]  
¬∃(x) [ T(x) ⋀ P(x) ] 
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 
injective and surjective  
surjective but not injective  
injective but not surjective  
neither injective nor surjective 
Question 56 
Ellipse  
Hyperbola  
Circle  
Parabola 
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 
3/2  
3/2  
0  
5/4 
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 
is called
Simpson rule  
Trapezoidal rule  
Romberg’s rule  
Gregory’s formula 
Question 59 
Newton’s backward formula  
Gauss forward formula  
Gauss backward formula  
Stirling’s formula 
Question 60 
△(U_{k}V_{k}) = U_{k}△V_{k} + V_{k}△U_{k}  
△(U_{k}V_{k}) = U_{k+1}△V_{k} + V_{k+1}△U_{k}  
△(U_{k}V_{k}) = V_{k+1}△U_{k} + U_{k}△V_{k}  
△(U_{k}V_{k}) = U_{k+1}△V_{k} + V_{k}△U_{k} 
Question 61 
n x^{n1}  
n^{n} . n!  
nx^{n} !  
n! 
Question 62 
Which of the following statements is false about convex minimization problem ?
If a local minimum exists, then it is a global minimum
 
The set of all global minima is convex set
 
The set of all global minima is concave set  
For each strictly convex function, if the function has a minimum, then the minimum is unique 
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 = 100x_{1} + 2x_{2} + 5x_{3}
Subject to 14x_{1} + x_{2} − 6x_{3} + 3x_{4} = 7
32x_{1} + x_{2} − 12x_{3} ≤ 10 3x_{1} − x_{2} − x_{3} ≤ 0 x_{1 }, x_{2} , x_{3} , x_{4} ≥ 0 has
Solution : x_{1} = 100, x_{2} = 0, x_{3} = 0
 
Unbounded solution  
No solution
 
Solution : x_{1} = 50, x_{2} = 70, x_{3} = 60

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 
1  
2  
3  
4 
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 = 11 = 4
p(3) = 20 ⇒ 9a+3b+c = 20
⇒ 2712+5 = 20
⇒ 20 = 20, True
Hence, minimum degree it can have.
Question 65 
x^{x}  
x^{x} In x  
x^{x} + x^{x} In x  
(x^{x})(x^{x}In x)  
None of the above; function is not differentiable for x > 0 
Question 66 
One, at π/2  
One, at 3 π /2  
Two, at π /2 and 3 π /2  
Two, at π /4 and 3 π /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 NewtonRaphson 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
(i), (ii) and (iii)
 
only (i) and (iii)
 
only (i) and (ii)  
only (i)

Note: NewtonRaphson method is mainly used for Interpolation.
Question 68 
6  
10  
12  
5,5 
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 halfopen 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)=2x02x0+6=6
f(1)=2x1 ^{2} 2x1+6=22+6=6
f(2)=2x2^{ 2} 2x2+6=84+6=10
Question 69 
(x+2)/2  
2/(π2)  
π2  
π+2 
=[−( 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+C2C]
=π2
Question 70 
The number of strips required in simpson’s 3/8^{th} rule is a multiple of:
1  
2  
3  
6 
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:
Turning points  
End points
 
Higher points  
Extreme points

Question 72 
If f(x) = ax^{2} + bx + c the f(x(b/2a)) is:
An even function for all a except a=0
 
An even function for all a
 
Neither even nor odd
 
An odd function for all a except a=0 
• 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 
Harmonic function  
Laplace equation
 
Wave equation
 
Homogeneous

Note that the wave equation can be factored as
Question 74 
(x ^{2} + 1)4 and [(x^{ 3} – 4x)^{ 2} + 1]^{ 4}  
(x ^{2} + 1)4 and [(x^{ 3} – 4x)^{ 2} + 1]^{ 4}  
(x ^{2} + 1)4 and [(x^{ 3} – 4x)^{ 2} + 1]^{ 4}  
(x ^{2} + 1)4 and [(x^{ 3} – 4x)^{ 2} + 1]^{ 4} 
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 
6x + 7, 6x + 11  
6x + 11, 6x + 7  
5x + 5, 5x + 5  
None of the above 
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, optionA is correct answer.
Question 76 
x ≥3/2, y ≥ 0  
x >3/2, y ≤ 0  
x ≥3/2, y ≤ 0  
x ≤3/2, y ≤ 0 
Question 77 
1/cosh 2x  
1/sinh 2x  
1/(1+tanh^{2} x)  
None of the above 
Question 78 
(√5 − 1 )/2  
(√5 + 1 )/2  
( − √5 − 1 )/2  
( − √5 + 1 )/2 
Question 79 
(xe^{x} /(1+x))  
x/(1+x)  
e^{x}/(1+x)^{2}  
e^{x}/(1+x)^{3} 
Question 81 
2/3  
2/3  
3/2  
3/2  
None of the Above 
Question 82 
y − 1 = c xy  
y = c xy − 1  
x y = 1 + c x  
y + 1 = c xy 
Question 83 
nm /x ln mx  
n /nx ln mx  
n / x ln mx  
m / nx ln mx 
Question 84 
2^{yx}  
2 ^{x} (2 ^{y} +1)/2 ^{y} (2 ^{x} 1)  
2 ^{x} (2 ^{y} 2)/2 ^{y} (2 ^{x} 1)  
2 ^{x} (2 ^{y} 1)/2 ^{y} (2 ^{x} 1) 
Question 87 
e ^{10}  
e ^{10}  
e ^{2}  
e ^{1/2} 
Question 89 
π / 2  
2 / π  
π / 2  
2 / π 
Question 90 
x y = x ^{4} + 2  
2 x y = x ^{4} + 4  
3 x y = x ^{4} + 4  
4 x y = x ^{4} + 3 
Question 91 
f(x)= 1 / (1+e^{(2x)}) at x=0 is
0  
1/2  
1/4  
∞ 
Question 92 
lim_{n→∞} f(n) − g(n) = ∞  
lim_{n→∞} f(n) − g(n) = 0  
lim_{n→∞} f(n)/g(n) = ∞  
lim_{n→∞} f(n)/g(n) = 1  
None of the above 
Question 93 
What can we say about lim_{x→∞} f(x)?
The function f(x) does not have a limit as x→∞  
lim_{x→∞} f(x) = e^{2}  
lim_{x→∞} f(x) = e^{1/2}  
lim_{x→∞} f(x) = 0  
lim_{x→∞} f(x) = ∞ 
Question 94 
oneone  
onto  
both oneone and onto  
into 
Question 95 
15  
5  
3  
9 
Question 97 
lim_{x→∞} f (x) = 1  
lim_{x→∞} f (x) = 1 − ∈  
lim_{x→∞} f (x) exists, but is neither 1 nor 1 − ∈  
max_{x≥1} f (x) = 1  
None of the above 
Question 98 
f(x) is not continuous but differentiable  
f(x) is continuous and differentiable  
f(x) is continuous but not differentiable  
f(x) is neither continuous nor differentiable  
None of the above 
Question 99 
1  
4  
2  
3  
5 
Question 108 
f(0.5) = 0.5  
There exists x between 0 and 1 such that f(x) = 0.8x  
There exists x between 0 and 0.5 such that f(x) = x.  
f(0.5) > 0.5.  
None of the above statements are always true. 
Question 112 
e^{n}/n  
e^{n−0.9 log n}  
2^{n}  
(log n)^{n−1}  
None of the above 
Question 114 
g must be identically zero  
g(π/2) = 1.  
g need not be identically zero  
g(π)=0.  
None of the above. 
Question 115 
The area under the curve of f(t) is zero  
The area under the curve of f(t) is ∞  
f(t) has width ∞ and height 1  
f(t) has width 0 and height ∞.  
None of the above 
Question 116 
f(x) ≤ f(y) whenever x ≤ y.  
For a random variable X, E(f(X)) ≤ f(E(X))  
The second derivative of f can be negative  
If two functions f and g are both convex, then min{f,g} is also convex  
For a random variable X, E(f(X)) ≥ f(E(X)) 
Question 118 
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
1/3  
2/3  
1  
4/3  
4/9 
Question 120 
x = 0  
x = 1  
x = 2  
x = 3  
None of the above 
Question 121 
a  
b  
c  
d  
e 
Question 122 
is
0  
∞  
α_{k}
 
α1  
max_{j} α_{j}  
Question 125 
f' is zero at exactly two points, f'' need not be zero anywhere  
f' is zero at exactly two points, f'' is zero at exactly one point  
f' is zero at at least two points, f'' is zero at exactly one point  
f' is zero at at least two points, f'' is zero at at least one point  
f' is zero at at least two points, f'' is zero at at least two points 
Question 126 
h(2t)  
h(t)  
h(t − 1)  
h(t + 1)  
None of the above 
Question 127 
1. min{f, g} is continuous
2. max{f, g} is continuous
3. max{f, g} is differentiable
Which of the following is TRUE?
Only statement 1 is correct  
Only statement 2 is correct  
Only statement 3 is correct  
Only statements 1 and 2 are correct  
None of the above 
Question 128 
g(x) is aperiodic  
g(x) is periodic with period 1  
g(x) is periodic with period 1  
The value of h determines whether or not g(x) is periodic  
None of the above 
Question 129 
f is unbounded  
f is bounded by lim_{z→ ∞} f(x) does not exist  
lim_{z→ ∞} f(x) = 0
 
lim_{z→ ∞ }f(x) = 1 
Question 130 
½  
1/√2  
¼  
1 
=1/2