Calculus

Question 1

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)
Question 1 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 2

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
Question 2 Explanation: 
Note: Out of syllabus.
Question 3

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

A
Theory Explanation.
Question 4

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
Question 4 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 5

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
Question 5 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 6

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
Question 6 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 7

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
Question 7 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 8

(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.
Question 9

Which of the following statements is true?

A
S > T
B
S = T
C
S < T and 2S > T
D
2S ≤ T
Question 9 Explanation: 
S is continuously increasing function but T represent constant value so S>T.
Question 10

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

A
linear
B
non-linear
C
homogeneous
D
of degree two
Question 10 Explanation: 
Note: Out of syllabus.
d2y/dx2 + dy/dx + siny = 0
In this DE, degree is 1 then this represent linear equation.
Question 11

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
Question 11 Explanation: 
Note: Out of syllabus.
Question 12

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

A
B
C
D
Question 13
A
1
Question 13 Explanation: 
Since the given expression is in 0/0 form, so we can apply L-Hospital rule.
Question 14

The radius of convergence of the power series

A
Out of syllabus.
Question 15

The value of the double integral is

A
1/3
Question 15 Explanation: 
Question 16

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.
Question 17
Consider the following expression.

The value of the above expression (rounded to 2 decimal places) is _______
A
0.25
Question 17 Explanation: 
Question 18

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)
Question 18 Explanation: 
Note: Out of syllabus.
Question 19

What is the value of

A
-1
B
1
C
0
D
π
Question 19 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 20

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
Question 20 Explanation: 
Note: Out of syllabus.
Question 21

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
Question 21 Explanation: 
Note: Out of syllabus.
Question 22

The following definite integral evaluates to

A
1/2
B
π √10
C
√10
D
π
E
None of the above
Question 22 Explanation: 
Question 23

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)
Question 23 Explanation: 
If x = 3/2
f(x) = 25/8x = 25/8(3/2) = 25/12 = 2(1/12)
Question 24
The value of the following limit is _____________.
A
1/2
Question 24 Explanation: 

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


-1/2
Question 25

The value of is

A
0
B
1/2
C
1
D
Question 25 Explanation: 
Question 26

If for non-zero x, where a≠b then is

A
B
C
D
Question 26 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 27

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
Question 27 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 28

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
Question 28 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 29
A
-1
B
-2
C
-3
D
-4
Question 29 Explanation: 
Question 30
A
0.99
B
1.00
C
2.00
D
3.00
Question 30 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 31

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
Question 31 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 32
A
4
B
3
C
2
D
1
Question 32 Explanation: 
Question 33

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

A
B
C
D
Question 33 Explanation: 

There are 33 questions to complete.

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