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
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III only
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II only
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I and III only
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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 X-axis, 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 3 |
If f(x) is defined as follows, what is the minimum value of f(x) for x ∊ (0, 2] ?
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Question 3 Explanation:
Question 4 |
The following definite integral evaluates to
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None of the above |
Question 4 Explanation:
Question 5 |
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 5 Explanation:
Note: Out of syllabus.
Question 6 |
What is the value of
-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?
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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
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 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) = 2x2 - 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 X0 is
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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
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Question 13 Explanation:
Note: Out of syllabus.
Question 14 |
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Question 14 Explanation:
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 15 |
linear | |
non-linear | |
homogeneous | |
of degree two
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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 |
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Question 16 Explanation:
Note: Out of syllabus.
Question 17 |
Which of the following improper integrals is (are) convergent?
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Question 18 |
1 |
Question 18 Explanation:
Since the given expression is in 0/0 form, so we can apply L-Hospital rule.
There are 21 questions to complete.