Calculus
Question 1 
Consider the functions
Which of the above functions is/are increasing everywhere in [0,1]?
Which of the above functions is/are increasing everywhere in [0,1]?
II and III only
 
III only
 
II only
 
I and III only

Question 1 Explanation:
Question 2 
Let f(x) = x ^{1(1/3) } and A denote the area of the region bounded bu f(x) and rhe Xaxis, when x varies from 1 to1. Which of the followin 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 
Question 2 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.
∴ f is not bounced in [1, 1] and hence f is not continuous in [1, 1].
∴ Statement II & III are true.
Question 5 
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 5 Explanation:
Note: Out of syllabus.
Question 6 
1  
1  
0  
π 
Question 6 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 7 
If f(1) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3) using Lagrange’s interpolation formula?
Question 7 Explanation:
Note: Out of syllabus.
Question 8 
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 8 Explanation:
Note: Out of sylabus.
Question 9 
S > T  
S = T  
S < T and 2S > T  
2S ≤ T 
Question 9 Explanation:
S is continuously increasing function but T represent constant value so S>T.
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 
Question 10 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.
→ 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 
What is the maximum value of the function f(x) = 2x^{2}  2x + 6 in the interval
[0,2]?
6  
10  
12  
5.5 
Question 11 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.
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 12 
The formula used to compute an approximation for the second derivative of a function f at a point X_{0} is
Question 12 Explanation:
The formula which is used to compute the second derivation of a function f at point X is
Question 13 
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
Question 13 Explanation:
Note: Out of syllabus.
Question 14 
Question 14 Explanation:
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 15 
linear  
nonlinear  
homogeneous  
of degree two

Question 15 Explanation:
Note: Out of syllabus.
In this DE, degree is 1 then this represent linear equation.
In this DE, degree is 1 then this represent linear equation.
Question 16 
Question 16 Explanation:
Note: Out of syllabus.
Question 17 
Which of the following improper integrals is (are) convergent?
Question 18 
1 
Question 18 Explanation:
Since the given expression is in 0/0 form, so we can apply LHospital rule.
There are 21 questions to complete.