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  • a)
    ∞    
  • b)
    2    
  • c)
    0
  • d)
    1
Correct answer is option 'C'. Can you explain this answer?

Yash Patel answered
the squeeze theorem for this. Recall that sinx is only defined on −1≤sinx≤1. Therefore

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 f oll owi ng is TRU E? 
  • a)
    P is true and Q is false  
  • b)
    P is false and Q is true  
  • c)
    Both P and Q are true  
  • d)
    Both P and Q are false  
Correct answer is option 'A'. Can you explain this answer?

Avinash Sharma answered
The graph of f(x) is  
f(x) is continuous for all real values of x   Lim |x| = Lim |x| = 0 
as can be seen from graph of |x|. 
and  Lim f(x) = +1 as can be seen from graph of |x| 
 x → 0+ 
Left deriva tive ≠ Rig ht derivative 
So |x| is continuous but not differentiable at x = 0. 

The minimum point of the function f(x) = (x2/3) – x is at 
  • a)
     x = 1 
  • b)
    x = -1
  • c)
     x = 0 
  • d)
    x = 1/√3
Correct answer is option 'A'. Can you explain this answer?

Avinash Sharma answered
Correct Answer :- a
Explanation : f(x) = (x^2/3) - x
f'(x) = 2/3(x-1/2) - 1
f"(x) = -1/3(x-3/2)
For critical points. f′(x)=0
=> 2/3(x-1/2) - 1 = 0 
f has minimum value of x = 1

Suppose C is the closed curve defined as the circle x2 + y2 = 1 with C oriented anti-clockwise. The value of ∮(xy2dx + x2ydy) over the curve C equals ________
    Correct answer is between '-0.03,0.03'. Can you explain this answer?

    Engineers Adda answered
    Concept:
    Green’s theorem:
    Let R be a closed bounded region in the xy plane whose boundary C consists of finitely many
    smooth curves.
    Let F1(x, y) & F(x, y) be functions that are continuous and have continuous partial
    derivatives 
     
    ∂F1 / ∂y and ∂F2 / ∂x. Then
    Analysis:
    Given curve C: x2 + y2 = 1
    = 0

    If x is real, find the maximum value of (-x2 + 3x + 7)
    • a)
      36/5
    • b)
      37/7
    • c)
      37/4
    • d)
      36/7
    Correct answer is option 'C'. Can you explain this answer?

    Bijoy Mehra answered
    Given Equation:
    - x is real, find the maximum value of (-x^2 + 3x + 7)

    Step 1: Find the vertex of the parabola
    - The given equation is in the form of a quadratic equation, -x^2 + 3x + 7.
    - To find the maximum value, we need to find the vertex of the parabola represented by this equation.
    - The x-coordinate of the vertex is given by the formula: x = -b/2a, where a=-1 and b=3 in this case.
    - Substituting the values of a and b, we get x = -3/(2*(-1)) = 3/2.
    - Now, substitute x = 3/2 back into the equation to find the maximum value.

    Step 2: Calculate the maximum value
    - Substitute x = 3/2 into the equation: (-3/2)^2 + 3*(3/2) + 7
    - Simplify the expression to find the maximum value: -9/4 + 9/2 + 7 = 37/4
    Therefore, the maximum value of the given equation (-x^2 + 3x + 7) when x is real is 37/4. Hence, the correct answer is option 'C'.

    The value of the integral 
    • a)
       − π
    • b)
      − π/2
    • c)
      π/2
    • d)
      π
    Correct answer is option 'D'. Can you explain this answer?

    Invento Id answered
    That integral is tan^-1 (x) and applying limits from -infinity to +infinity . so it will become (π/2 - (-π/2))= π. tan^-1(infinity) = π/2

    The distance between the origin and the point nearest to it on the surface z2 = 1 + xy is 
    • a)
      1
    • b)
      √3/2
    • c)
      √3
    • d)
      -2
    Correct answer is option 'A'. Can you explain this answer?

    Baishali Bajaj answered

    or pr – q^2 = 4 – 1 = 3 > 0 and r = +ve

    so f(xy) is minimum at (0,0)

    Hence, minimum value of d^2 at (0,0)

    d2 = x^2 + y^2 + xy + 1 = (0)^2 + (0)^2 + (0)(0) + 1 = 1

    Then the nearest point is

    z^2 = 1 + xy = 1+ (0)(0) = 1

    or z = 1

    Mathematically, the functions in Green’s theorem will be
    • a)
      Continuous derivatives
    • b)
      Discrete derivatives
    • c)
      Continuous partial derivatives
    • d)
      Discrete partial derivatives
    Correct answer is option 'C'. Can you explain this answer?

    Engineers Adda answered
    The Green’s theorem states that if L and M are functions of (x,y) in an open region containing D and having continuous partial derivatives then,
    ∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path taken anticlockwise.

    The divergence theorem value for the function x2 + y2 + z2 at a distance of one unit from the origin is
    • a)
      0
    • b)
      1
    • c)
      2
    • d)
      3
    Correct answer is option 'D'. Can you explain this answer?

    Sanya Agarwal answered
    Div (F) = 2x + 2y + 2z.
    The triple integral of the divergence of the function is ∫∫∫(2x + 2y + 2z)dx dy dz, where x = 0->1, y = 0->1 and z = 0->1. On integrating,
    we get 3 units.

    The voltage of a capacitor 12F with a rating of 2J energy is
    • a)
      0.57
    • b)
      5.7
    • c)
      57
    • d)
      570
    Correct answer is option 'A'. Can you explain this answer?

    Sanvi Kapoor answered
    We can compute the energy stored in a capacitor from Stoke’s theorem as 0.5Cv2. Thus given energy is 0.5 X 12 X v2.
    We get v = 0.57 volts.

    Gauss theorem uses which of the following operations?
    • a)
      Gradient
    • b)
      Curl
    • c)
      Divergence
    • d)
      Laplacian
    Correct answer is option 'C'. Can you explain this answer?

    Bibek Mukherjee answered
    Gauss's theorem, also known as Gauss's divergence theorem or Gauss's flux theorem, is a fundamental concept in vector calculus. It relates the flux of a vector field through a closed surface to the divergence of the vector field in the region enclosed by the surface. The theorem uses the operation of divergence to establish this relationship.

    The divergence of a vector field is a scalar quantity that measures the rate at which the vector field "spreads out" or "converges" at a given point. It is represented by the operator ∇ · F, where ∇ is the del operator and · denotes the dot product.

    The Gauss theorem states that the flux of a vector field F through a closed surface S is equal to the volume integral of the divergence of F over the region V enclosed by S. Mathematically, it can be expressed as:

    ∫∫S F · dA = ∫∫∫V ∇ · F dV

    where ∫∫S denotes the surface integral over S, F · dA represents the dot product of F and the infinitesimal area vector dA, and ∫∫∫V is the volume integral over V.

    In other words, the flux of F through S is equal to the sum of the divergences of F at each point within V, integrated over the entire volume.

    The divergence theorem is a powerful tool in various fields, including fluid mechanics, electromagnetism, and heat transfer. It allows for the conversion of a surface integral, which may be difficult to evaluate, into a volume integral, which is often easier to handle mathematically.

    Overall, the Gauss theorem utilizes the operation of divergence (∇ · F) to establish the relationship between the flux of a vector field through a closed surface and the divergence of the vector field within the enclosed region.

    The directional derivative of 1/r in the direction of is
    • a)
      1/r2
    • b)
      -1/r2
    • c)
      1/r3
    • d)
      -/r3
    Correct answer is option 'B'. Can you explain this answer?

    Pioneer Academy answered
    Concept:
    Let f(r) be a function then directional derivative of the function f(r) is given by: 
    Calculation:
    Given:
    f(r) = 1/r
    As we know that, if f(r) is a function then directional derivative of the function f(r)is given by: 
    f(r) = 1/r
    ∵ 
    Here, we have to find the directional derivative of f(r) in the direction of . It will be given by:

    Stokes theorem is used to convert __________ into _________.
    • a)
      Surface integral, Volume integral
    • b)
      Line integral, Volume integral
    • c)
      Line integral, ​Surface integral
    • d)
      None of the above
    Correct answer is option 'C'. Can you explain this answer?

    Sanya Agarwal answered
    Stokes theorem:
    (i) Stoke's theorem enables us to transform the surface integral of the curl of the vector field A into the line integral of that vector field A over the boundary C of that surface and vice-versa. The theorem states.
    (ii) The flux of the curl of a vector function A over any surface S of any shape is equal to the line integral of the vector field A over the boundary C of that surface i.e.
    Stokes Theorem is given as:
    It converts a line integral to a surface integral and uses the curl operation.

    The area between the parabolas y2 = 4ax and x2 = 4ay is
    • a)
    • b)
    • c)
    • d)
    Correct answer is option 'C'. Can you explain this answer?

    Sanvi Kapoor answered
    y2 = 4ax & x2 = 4ay
    We have to find shaded region area.
    So area drawn by y= x2 / 4a on x-axis = A1 (say)
    Then shaded area = |A1 – A2|
    So, 
    ⇒ 
    ⇒ Shaded area = 

    Using Green’s theorem, the value of the integral  , where C is the square, cut from the first quadrant by the lines x = 1 and y = 1, will be.
    • a)
      1
    • b)
      1/2
    • c)
      5/3
    • d)
      3/2
    Correct answer is option 'D'. Can you explain this answer?

    Engineers Adda answered
    Concept:
    If M(x,y), N(x,y), ∂N/∂y and ∂M/∂x be continuous functions over region R bounded by a simple closed curve c in x-y plane, then according to this theorem:
    It is used to simplify the vector integration.
    It gives the relation between the closed line and open surface integration.
    Calculation:
    Given:
    Comparing with the standard equation Mdx + Ndy; M = -y2 and N = xy.
    ∴ 
    = 3/2

    Stokes theorem connects  
    • a)
      A line integral and a surface integral  
    • b)
      A surface integral and a volume integral
    • c)
      A line integral and a volume integral  
    • d)
      Gradient of a function and its surface integral 
    Correct answer is option 'A'. Can you explain this answer?

    Ayush Chawla answered
    Stokes' theorem connects a line integral and a surface integral.

    Introduction:
    Stokes' theorem is a fundamental theorem in vector calculus that relates a line integral around a closed curve to a surface integral over the region bounded by that curve. It provides a powerful tool for calculating circulations and fluxes in vector fields.

    Statement of Stokes' theorem:
    Stokes' theorem states that the line integral of a vector field F around a closed curve C is equal to the surface integral of the curl of F over the surface S bounded by C. Mathematically, it can be written as:

    ∮C F · dr = ∬S (curl F) · dS

    where ∮C represents the line integral around the closed curve C, F is the vector field, dr is an infinitesimal vector along the curve C, ∬S represents the surface integral over the surface S, curl F is the curl of the vector field F, and dS is an infinitesimal vector normal to the surface S.

    Explanation:
    Stokes' theorem connects a line integral and a surface integral by relating the circulation of a vector field around a closed curve to the flux of its curl through the surface bounded by that curve. Here's a step-by-step explanation of how it works:

    1. Consider a closed curve C in a vector field F.
    2. Divide the surface bounded by the curve C into infinitesimal surface elements.
    3. Calculate the curl of the vector field F at each point on the surface.
    4. Take the dot product of the curl of F with the infinitesimal vector normal to the surface at each point.
    5. Sum up all the dot products over the entire surface to obtain the surface integral of the curl of F.
    6. Calculate the line integral of the vector field F along the closed curve C.
    7. According to Stokes' theorem, the line integral is equal to the surface integral.

    Significance:
    Stokes' theorem provides a powerful tool for calculating the circulation of a vector field around a closed curve or the flux of its curl through a surface. It allows us to relate these two seemingly different concepts and provides a bridge between line integrals and surface integrals. This theorem has applications in various fields, including fluid dynamics, electromagnetism, and differential geometry.

    Chapter doubts & questions for Calculus - Engineering Mathematics 2025 is part of Civil Engineering (CE) exam preparation. The chapters have been prepared according to the Civil Engineering (CE) exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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