Calculus MCQ Quiz - Objective Question with Answer for Calculus - Download Free PDF
Last updated on Apr 8, 2025
Latest Calculus MCQ Objective Questions
Calculus Question 1:
If f1 and f2 are differentiable scalar functions and v is differentiable vector function such that f1v = ∇f2, then v . curl v is
Answer (Detailed Solution Below)
Calculus Question 1 Detailed Solution
Explanation:
v . curl v = v . ∇ × v = [v ∇ v] = 0
Option (4) is true.
Calculus Question 2:
The value of the integral
Answer (Detailed Solution Below)
Calculus Question 2 Detailed Solution
Analysis:
Consider
Put sin θ = t
cos θ dθ = dt
If θ = 0 to
Now,
Calculus Question 3:
The value of integral
Answer (Detailed Solution Below)
Calculus Question 3 Detailed Solution
Concept:
Trigonometric Ratio Fundamental Identities
Integration by parts
when u and v are functions of x.
Integral of standard function
The derivative of standard function
Calculation:
Given:
we have
where u = x and
Integration by parts
when u and v are functions of x.
Integration by parts
when u and v are functions of x.
∫x × sec2 x dx = x tan x - ∫ tan x dx
∫x × sec2 x dx = x tan x - (-log (cos x))
∫ x × sec2 x dx = x tan x + log (cos x)
Calculus Question 4:
The length of the curve
Answer (Detailed Solution Below)
Calculus Question 4 Detailed Solution
Concept:
Arc length or curve length is the distance between two points along the section of the curve. Determining the length of an irregular section of the arc is termed as rectification of the curve.
The length of the curve y = f(x) from x = a to x = b is given as:
or,
If the curve is parametrized in the form x = f(t) and y = g(t) with the parameter t going from a to b then
Calculation:
Now, the arc length(l) is
Calculus Question 5:
The integral
Answer (Detailed Solution Below)
Calculus Question 5 Detailed Solution
Explanation:
The given integral is an improper integral of 1st kind.
I = log [log (∞)] – log [log (2)]
∴ I = ∞
Given integral is divergent and diverges to ∞
Additional Information
An improper integral of first kind is when integral limits have -∞ or +∞ or both.
An improper integral of second kind is when integral limits are finite but function is infinite at some value between those limits.
Top Calculus MCQ Objective Questions
The value of the definite integral
Answer (Detailed Solution Below)
Calculus Question 6 Detailed Solution
Download Solution PDFConcept:
We know that,
By Parts method
Where, u, v should follow the ILATE sequence.[I= Inverse, L= Logarithmic, A= Algebraic, T= Trigonometric, E= Exponential terms]
Calculation:
Given:
From the given Equation,
u = ln(x), v = √x
Now,
∴
The value of
Answer (Detailed Solution Below)
Calculus Question 7 Detailed Solution
Download Solution PDFConcept:
For limit evaluation of indeterminate forms, i.e. for
Calculation:
The given limit is,
By putting a limit we get
Again putting the value of limit we get
Now, by putting the limit, we get:
The value of
Answer (Detailed Solution Below)
Calculus Question 8 Detailed Solution
Download Solution PDFConcept:
To solve
Calculation:
Given:
Replacing x with ∞ in above expression, it is an indeterminate form
Take out common the highest degree term from numerator and denominator.
The volume determined from ∫∫∫v 8 xyz dv for V = [2, 3] × [1, 2] × [ 0,1 ] will be (in integer) ________.
Answer (Detailed Solution Below) 15
Calculus Question 9 Detailed Solution
Download Solution PDFExplanation
Given
Integral
∫∫∫v 8 xyz dv
Limits for x, y and z is given as
[2, 3] × [1, 2] × [0, 1]
Volume of the integral
V = ∫∫∫v 8 xyz dv
i.e. V = ∫ ∫ ∫V 8 xyz dxdydz
V = 5 × 3 × 1
V = 15
∴ Volume is 15
The values of x for which the function
is NOT continuous are
Answer (Detailed Solution Below)
Calculus Question 10 Detailed Solution
Download Solution PDFConcept:
A function f(x) is continuous at x = a, if the function is defined at x = a and,
Left limit = Right limit = Function value = Real and finite
A function is said to be differentiable at x =a if,
Left derivative = Right derivative = Well defined
Analysis:
This function is not defined for:
x2 + 3x – 4 = 0
⇒ (x + 4)(x - 1) = 0 i.e.
At x = 1 and x = - 4
∴ This function f(x) is not continuous at x = 1, -4
Answer (Detailed Solution Below)
Calculus Question 11 Detailed Solution
Download Solution PDFConcept:
L-Hospital Rule: Let f(x) and g(x) be two functions
Suppose that we have one of the following cases,
I.
II.
Then we can apply L-Hospital Rule as:
Calculation:
Given:
Applying L’ Hospital rule:
Once again by L’ Hospital rule,
The value (round off to one decimal place) of
Answer (Detailed Solution Below) 0
Calculus Question 12 Detailed Solution
Download Solution PDFExplanation
Given,
Function f(x) = x e|x|
Integral is -1 to 1.
If f(-x) = f(x) then the function is said to be even function
If f(-x) = - f(x) then the function is said to be odd function.
f(-x) = -x e|-x| = -x e|x| = - f(x)
∴ The given function is an odd function.
For an odd function:
For a even function
Now, as the function is odd
Find the area of triangle whose two sides are represented by the vectors 3i + 4j and 5i + 7j + k is
Answer (Detailed Solution Below)
Calculus Question 13 Detailed Solution
Download Solution PDFConcept:
If a triangle is formed by three vectors, then the sum of the vectors should be zero.
AB + BC + CA = 0
Cross product of the vectors:
For two vectors
The magnitude of the cross product is:
Area of a triangle:
If the vectors
Calculation:
Given:
Let, AB = 3i + 4j and CA = 5i +7j + k
If a triangle is formed by three vectors, then the sum of the vectors should be zero.
AB + BC + CA = 0 ⇒ 3i + 4j + BC + 5i +7j + k = 0
BC = - 8i - 11j - k
Let the adjacent vectors be AB (a), AC (b)
First, we will calculate the cross product as follow:
Therefore, the magnitude of the cross product is:
Using the formula for the area of the triangle, the area is given by:
Answer (Detailed Solution Below)
Calculus Question 14 Detailed Solution
Download Solution PDFConcept:
We know
⇒ 1 - cos x = 2 sin2(x/2)
⇒
Calculation:
Multiply and divide the denominator by 4
Function f(x) = (x + 1)cotx will be continuous at x = 0 if the value of f(0) is
Answer (Detailed Solution Below)
Calculus Question 15 Detailed Solution
Download Solution PDFConcept:
A function is said to be continuous if
Calculation:
Given:
f(x) = (x + 1)cotx
Taking log on both sides
log (f(x)) = cot x log (x + 1)
For checking continuity at x = 0
Using L-Hospital rule
⇒ log (f(0)) = 1
⇒ f(0) = e1 = e