Calculus
Question 1 |
The value of the above expression (rounded to 2 decimal places) is _______
0.25 |
Question 2 |
Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).
yn+1 = yn + hf(xn, yn) | |
yn+1 = yn + hf(xn+1, yn+1) | |
yn+1 = yn-1 + 2hf(xn, yn) | |
yn+1 = (1 + h) f(xn+1, yn+1) |
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 3 |
(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion 
Theory Explanation. |
Question 4 |
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
C1ex + C2e2x | |
C1e-x + C2e3x | |
C1e-x + C2e-2x | |
C1e-2x + C22-x |
Question 5 |
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]?
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. 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 6 |
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 7 |
The formula used to compute an approximation for the second derivative of a function f at a point X0 is
f(x0+h) + f(x0-h)/2 | |
f(x0+h) - f(x0-h)/2h | |
f(x0+h) + 2f(x0) + f(x0-h)/h2 | |
f(x0+h) - 2f(x0) + f(x0-h)/h2 |
f(x0+h) - 2f(x0) + f(x0-h)/h2
Question 8 |
What is the maximum value of the function f(x) = 2x2 - 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 9 |
(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 
Theory Explanation. |
Question 10 |
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 11 |
Which of the following statements is true?
S > T | |
S = T | |
S < T and 2S > T | |
2S ≤ T |
Question 12 |
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 __________
Out of syllabus. |
Question 13 |
The value of the double integral 
is
1/3 |
Question 14 |
The radius of convergence of the power series
Out of syllabus. |
Question 15 |
1 |
Question 16 |
Which of the following improper integrals is (are) convergent?
 | |
 | |
 | |
 |
Question 17 |
The differential equation
d2y/dx2 + dy/dx + siny = 0 is:
linear | |
non-linear | |
homogeneous | |
of degree two |
d2y/dx2 + dy/dx + siny = 0
In this DE, degree is 1 then this represent linear equation.
Question 18 |
Fourier series of the periodic function (period 2π) defined by
But putting x = π, we get the sum of the series.
π2/4 | |
π2/6 | |
π2/8 | |
π2/12 |
Question 19 |
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 20 |
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 Q-II, R-IV, S-II, T-I | |
Q-III, R-II, S-I, T-IV | |
Q-II, R-I, S-IV, T-III | |
Q-I, R-IV, S-II, T-III |
Question 21 |
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 22 |
If the trapezoidal method is used to evaluate the integral obtained 0∫1x2dx ,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 23 |
The following definite integral evaluates to 
1/2 | |
π √10 | |
√10 | |
π | |
None of the above |
Question 24 |
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 25 |
1/2 |
When 0 is substituted, we get 0/0
Apply L- Hospital rule
-1/2
Question 26 |
The value of the integral given below is
-2π | |
π | |
-π | |
2π |
Question 27 |
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 28 |
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 29 |
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
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 30 |
0.99 | |
1.00 | |
2.00 | |
3.00 |
= 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 31 |
-1 | |
-2 | |
-3 | |
-4 |
Question 32 |
If g(x) = 1 - x and h(x) 
, then 
is:
h(x)/g(x) | |
-1/x | |
g(x)/h(x) | |
x/(1-x)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 33 |
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.