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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

পরীক্ষা৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]তারিখতারিখ অনির্ধারিতসময়45 minutes
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Exam - 08 Topics: Real Analysis (a) Differentiation: Continuous function. Derivative of a function. Rolle’s Theorem, Meanvalue theorem, Toylor’s theorem. (b) Functions of Several Variables: Limit and Continuity. Partial differentiation. Schwarz’s theorem, Young’s theorem. [Source: Class - 06 and Relevant Books]
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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১] · তারিখ অনির্ধারিত · ৩৮ প্রশ্ন

১.
Which of the following is the correct definition of continuity at a point c?
  1. ঘ
    f(x) must be differentiable at cc
ব্যাখ্যা

 Continuity requires that the limit of the function at c exists and equals the function value at c. Differentiability is stronger than continuity, so D is wrong.

২.
Which of the following functions is continuous everywhere on R?
  1. ঘ
    All of these
ব্যাখ্যা

৩.
  1. ক
    Yes
  2. খ
    No
  3. গ
    Only from Left
  4. ঘ
    Only from right
ব্যাখ্যা

৪.
If a function is differentiable at x = c, then:
  1. ক
    It must be continuous at c.
  2. খ
    It may or may not be continuous at c.
  3. গ
    It must be discontinuous at c.
  4. ঘ
    None of the above.
ব্যাখ্যা

Differentiability implies continuity, but the converse is not always true (e.g. f(x) = ∣x∣ continuous at 0 but not differentiable).

৫.
The function f(x) = ∣x∣ at x = 0 is:
  1. ক
    Continuous and differentiable
  2. খ
    Continuous but not differentiable
  3. গ
    Discontinuous
  4. ঘ
    Differentiable but not continuous
ব্যাখ্যা

Continuous at 0, but left derivative = - 1, right derivative = 1. So, not differentiable.

৬.
Which of the following functions satisfies Rolle’s theorem on [0, π]?
  1. ক
    f(x) = sin⁡x
  2. খ
    f(x) = cosx 
  3. গ
    f(x) = x2
  4. ঘ
    f(x) = ex
ব্যাখ্যা

  • f(x) = sinx : f(0) = 0,f(π) = 0.  Conditions satisfied.

  • f(x) = cos⁡x : endpoints different.

  • f(x) = x2 : endpoints not equal. 

  • f(x) = ex : endpoints not equal

৭.
If f(x) = x2 on [1, 3], then by Mean Value Theorem, there exists c ∈ (1, 3) such that:
  1. ক
    f’(c) = 2
  2. খ
    f’(c) = 3
  3. গ
    f’(c) = 4
  4. ঘ
    f’(c) = 5
ব্যাখ্যা

৮.
  1. ক
    0
  2. খ
    1
  3. গ
    2
  4. ঘ
    None of these
ব্যাখ্যা

  • First check conditions:

    • Continuous? Yes (polynomial).

    • Differentiable? Yes.

    • f(0)=1,f(2)=1 → equal endpoints.

  • Apply Rolle’s theorem: f′(x)=2x−2

  • Solve f′(c)=0⟹2c−2=0⟹c=1.

৯.
Cauchy’s Mean Value Theorem generalizes the MVT by involving two functions f,g. Its result is:
  1. ক
    f‘(c) = 0
  2. খ
    g‘(c) = 0
  3. ঘ
    Both A and B
ব্যাখ্যা

১০.
If f(x) = ∣x∣ on [- 1, 1], does Rolle’s theorem apply?
  1. ক
    Yes, because f(-1) = f(1)
  2. খ
    No, because f(x) is not continuous on [- 1, 1]
  3. গ
    No, because f(x) is not differentiable at x = 0
  4. ঘ
    Yes, because derivative exists at all x ≠ 0
ব্যাখ্যা

Function: f(x)=∣x∣ on [−1,1].

Conditions for Rolle’s theorem:

f continuous on [a,b] → true.
f differentiable on (a,b) → fails at x=0.
f(a)=f(b)⟹f(−1)=f(1)=1 →  true.
Since differentiability fails at 0, Rolle’s theorem does not apply.

Correct: C

১১.
  1. ক
    1
  2. খ
    2
  3. গ
    3
  4. ঘ
    4
ব্যাখ্যা

১২.
For f(x) = sin⁡(2x) on [0, π], find all possible c given by Rolle’s theorem.
  1. ক
    c = π/4​ only
  2. খ
    c = π/4, 3π/4
  3. গ
    c = 0, π​
  4. ঘ
    None of these
ব্যাখ্যা

১৩.
  1. ক
    ec = 1
  2. খ
    ec = e
  3. গ
    ec = e - 1
  4. ঘ
    None of these
ব্যাখ্যা

১৪.
  1. ক
    0
  2. খ
    π/4
  3. গ
    π/2
  4. ঘ
    π
ব্যাখ্যা

  • Step 1: Check Rolle’s theorem:

    • Continuous on [0, π]? 

    • Differentiable on (0, π)? 

    • Endpoints equal? f(0)=f(π)=0

  • Step 2: Find derivative: f′(x)=cosx

  • Step 3: Solve f′(c)=0: cos⁡c=0  ⟹  c=π/2

১৫.
  1. ক
    c = - 1, 0, 1
  2. খ
    c = 1 only
  3. গ
    c = - 1 only
  4. ঘ
    None of these
ব্যাখ্যা







১৬.
Which of the following is necessary for Rolle’s theorem to hold?
  1. ক
    f(a) = f(b)
  2. খ
    f(x) continuous on [a, b]
  3. গ
    f(x) differentiable on (a, b)
  4. ঘ
    All of the above
১৭.
  1. ক
    c = 3/2
  2. খ
    c = 2/√3
  3. গ
    c = 0
  4. ঘ
    c = 1
ব্যাখ্যা



১৮.
Which of the following is true?
  1. ক
    Rolle’s theorem is a special case of MVT
  2. খ
    MVT is a special case of Rolle’s theorem
  3. গ
    Rolle’s theorem and MVT are not related
  4. ঘ
     None of the above
ব্যাখ্যা

Rolle’s Theorem is the case of MVT when f(a) = f(b).

১৯.
  1. ক
    0
  2. খ
    1
  3. গ
    2
  4. ঘ
    3
ব্যাখ্যা

২০.
  1. ক
    0
  2. খ
    1
  3. গ
    2
  4. ঘ
    Does not exist
ব্যাখ্যা


২১.
A function f(x, y) is continuous at (a, b) if:
  1. ঘ
    All of these
ব্যাখ্যা

Generalization of single - variable continuity to several variables.

২২.
    ব্যাখ্যা

    ২৩.
    1. ক
      1
    2. খ
      x
    3. গ
      y
    4. ঘ
      z
    ব্যাখ্যা

    ২৪.
    Schwarz’s theorem states that if f(x, y) has continuous second partial derivatives:
    1. ঘ
      None of the above
    ব্যাখ্যা

    ২৫.
    Which one is true?
    1. ক
      Young’s theorem is a generalization of Schwarz’s theorem
    2. খ
       Schwarz’s theoremis a generalization of Young’s theorem.
    3. গ
      Young’s theorem is a generalization of Mean Value Theorem.
    4. ঘ
      Schwarz’s theoremis a generalization of Taylor's Theorem.
    ব্যাখ্যা

    Young’s theorem provides conditions for equality of mixed partials even when continuity is weaker than Schwarz’s theorem.

    ২৬.
    1. ক
       6xy + 5
    ব্যাখ্যা

    ২৭.
    Young’s theorem is applicable when
    1. ক
      The function is differentiable but not continuous
    2. খ
      The mixed partial derivatives exist and are continuous
    3. গ
      The function is linear
    4. ঘ
      The limit exists along all paths but is path-dependent
    ব্যাখ্যা

    Young's theorem is essentially the same as Schwarz’s or Clairaut’s theorem for functions of two variables, stating that if f_xy and f_yx exist and are continuous in an open region, then they are equal at every point in that region. The key is the continuity of the second partials, which guarantees commutativity of differentiation.

    ২৮.
    Which of the following is continuous everywhere?
      ব্যাখ্যা

      • sin⁡x/x not defined at 0.

      • ln⁡x undefined at x≤0.

      • ex continuous everywhere

      • tan⁡x has poles at odd multiples of π/2

      ২৯.
      Which function is differentiable everywhere?
      1. ক
        ।x।
      2. খ
        x2sinx
      3. গ
        x/।x।
      4. ঘ
        All Of these
      ব্যাখ্যা

      • ∣x∣ not differentiable at 0.

      • x2sin⁡x smooth → differentiable everywhere

      • x/∣x∣ undefined at 0.

      ৩০.
      For f(x) = √x on [0, 1], why does Rolle’s theorem fail?
      1. ক
        Not continuous
      2. খ
        Not differentiable at 0
      3. গ
        Endpoints not equal
      4. ঘ
        None
      ব্যাখ্যা





      ৩১.
      For f(x) = x2 - 1 on [- 1, 1], c is:
      1. ক
        0
      2. খ
        - 1
      3. গ
        1
      4. ঘ
        None of these
      ব্যাখ্যা

      ৩২.
      Mean Value Theorem guarantees:
      1. ক
        A point where tangent is parallel to secant
      2. খ
        Function must be constant
      3. গ
        Function must be increasing
      4. ঘ
        Tangent = normal
      ব্যাখ্যা

      That’s the geometric meaning of MVT.

      ৩৩.
      Let f(x)=x​ on [0,4]. Which is true?
      1. ক
        Rolle’s theorem applies
      2. খ
        Mean Value Theorem applies
      3. গ
        Neither applies.
      4. ঘ
        Both apply.
      ব্যাখ্যা

      f is continuous on [0,4], differentiable on (0,4), so MVT applies. Rolle does not because f(0)≠f(4).

      ৩৪.
      The Cauchy Mean Value Theorem reduces to the ordinary Mean Value Theorem when:
      1. ক
        f(x) = g(x)
      2. খ
        g(x) = x
      3. গ
        f(x) = 0
      4. ঘ
        g(x) = f’(x)
      ব্যাখ্যা

      ৩৫.
      Which of the following is not a condition for Cauchy Mean Value Theorem?
      1. ক
        f, g continuous on [a, b]
      2. খ
        f, g differentiable on (a, b)
      3. গ
        g’(x) ≠ 0 for all x ∈ (a, b)
      4. ঘ
        f’(x) ≠ 0 for all x ∈ (a,b)
      ব্যাখ্যা


      ৩৬.
      If f(x) = arctan⁡x, g(x) = x on [0, 1], then CMVT ensures:
      1. ঘ
        None
      ব্যাখ্যা





      ৩৭.
      1. ক
        0
      2. খ
        1
      3. গ
        2
      4. ঘ
        Doesn't Exist
      ব্যাখ্যা

      ৩৮.
      1. ক
        Schwarz’s theorem
      2. খ
        Young’s theorem
      3. গ
        Mean Value theorem
      4. ঘ
        Taylor’s theorem
      ব্যাখ্যা

      Schwarz’s theorem is the standard result for equality of mixed partial derivatives.