topic
Calculus
Question 1 
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 2 
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 3 
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 4 
1/2 
When 0 is substituted, we get 0/0
Apply L Hospital rule
1/2
Question 5 
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 6 
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 7 
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 8 
(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 9 
Which of the following statements is true?
S > T  
S = T  
S < T and 2S > T  
2S ≤ T 
Question 10 
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 11 
(a) Determine the number of divisors of 600.
(b) Compute without using power series expansion
Theory Explanation. 
Question 12 
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 13 
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 14 
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 15 
Which of the following improper integrals is (are) convergent?
Question 16 
1 
Question 17 
The radius of convergence of the power series
Out of syllabus. 
Question 18 
The value of the double integral is
1/3 
Question 19 
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 20 
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 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 
The value of the above expression (rounded to 2 decimal places) is _______
0.25 
Question 23 
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 24 
The following definite integral evaluates to
1/2  
π √10  
√10  
π  
None of the above 
Question 25 
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 26 
1  
1  
∞  
∞ 
Question 27 
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 28 
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 29 
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 30 
4  
3  
2  
1 
Question 31 
If f(x) = Rsin(πx/2) + S, f'(1/2) = √2 and , then the constants R and S are, respectively.
Question 32 
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 33 
1  
Limit does not exist  
53/12  
108/7 