Calculus

Question 1
Consider the following expression.

The value of the above expression (rounded to 2 decimal places) is _______
A
0.25
Question 1 Explanation: 
Question 2

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 3

The value of the double integral is

A
1/3
Question 3 Explanation: 
Question 4

The radius of convergence of the power series

A
Out of syllabus.
Question 5
A
1
Question 5 Explanation: 
Since the given expression is in 0/0 form, so we can apply L-Hospital rule.
Question 6

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

A
B
C
D
Question 7

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

A
linear
B
non-linear
C
homogeneous
D
of degree two
Question 7 Explanation: 
Note: Out of syllabus.
d2y/dx2 + dy/dx + siny = 0
In this DE, degree is 1 then this represent linear equation.
Question 8

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 8 Explanation: 
Note: Out of syllabus.
Question 9

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 9 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 10

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

A
Theory Explanation.
Question 11

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 11 Explanation: 
Note: Out of syllabus.
Question 12

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 12 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 13

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 13 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 14
 
A
1
B
Limit does not exist
C
53/12
D
108/7
Question 14 Explanation: 
Question 15

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 15 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 16

(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 17

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 17 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 18

Which of the following statements is true?

A
S > T
B
S = T
C
S < T and 2S > T
D
2S ≤ T
Question 18 Explanation: 
S is continuously increasing function but T represent constant value so S>T.
Question 19
A
0.289
B
0.298
C
0.28
D
0.29
Question 19 Explanation: 
Question 20
The value of the following limit is _____________.
A
1/2
Question 20 Explanation: 

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


-1/2
Question 21

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 21 Explanation: 
Note: Out of syllabus.
Question 22

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 22 Explanation: 
Note: Out of syllabus.
Question 23

What is the value of

A
-1
B
1
C
0
D
π
Question 23 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 24

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 24 Explanation: 
Note: Out of syllabus.
Question 25

The following definite integral evaluates to

A
1/2
B
π √10
C
√10
D
π
E
None of the above
Question 25 Explanation: 
Question 26

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 26 Explanation: 
If x = 3/2
f(x) = 25/8x = 25/8(3/2) = 25/12 = 2(1/12)
Question 27

If for non-zero x, where a≠b then is

A
B
C
D
Question 27 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 28

The value of is

A
0
B
1/2
C
1
D
Question 28 Explanation: 
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
A
II only
B
III only
C
II and III only
D
I, II and III
Question 29 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 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
A
-1
B
-2
C
-3
D
-4
Question 31 Explanation: 
Question 32

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 32 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 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 __________.

A
9
B
10
C
11
D
12
Question 33 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.
There are 33 questions to complete.

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