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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

পরীক্ষা৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]তারিখতারিখ অনির্ধারিতসময়45 minutes
মোট প্রশ্ন৪০
সিলেবাস
Exam- 12 Topics: (a) Numerical differentiation and integration. (b) Numerical solutions of ordinary differential equation. [Source: Class - 08 and Relevant Books]
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
উত্তর
উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১] · তারিখ অনির্ধারিত · ৪০ প্রশ্ন

১.
The forward difference formula for the first derivative is:
    ব্যাখ্যা

    ২.
    The central difference formula for the first derivative has an error of order:
    1. ক
      O(h)
    2. খ
      O(h2)
    3. গ
      O(h3)
    4. ঘ
      O(h4)
    ব্যাখ্যা

    Central difference is more accurate than forward/backward and has error of order h2, while forward/backward have error O(h).

    ৩.
    Which numerical integration rule requires that the number of subintervals nnn must be even?
    1. ক
      Trapezoidal Rule
    2. খ
      Simpson’s 1/3 Rule
    3. গ
      Simpson’s 3/8 Rule
    4. ঘ
      Midpoint Rule
    ব্যাখ্যা

    Simpson’s 1/3 rule works only if n is even, because it groups subintervals in pairs.

    ৪.
    If we apply Simpson’s 1/3 rule with n=2, it is equivalent to approximating the function with
    1. ক
      A straight line
    2. খ
      A quadratic polynomial
    3. গ
      A cubic polynomial
    4. ঘ
      A constant function
    ব্যাখ্যা

    .Simpson’s 1/3 rule is derived by fitting a parabola (quadratic polynomial) through 3 points: f(x0),f(x1),f(x2).
    That’s why it is more accurate than trapezoidal rule, which only fits a straight line.

    ৫.
    Which of the following gives the best accuracy for the same step size h?
    1. ক
      Trapezoidal Rule
    2. খ
      Simpson’s 1/3 Rule
    3. গ
      Simpson’s 3/8 Rule
    4. ঘ
      Midpoint Rule
    ব্যাখ্যা

    Trapezoidal rule error: O(h2).
    Midpoint rule error: O(h2).
    Simpson’s 1/3 rule error: O(h4) → much smaller error.
    Simpson’s 3/8 rule also has O(h4), but in practice 1/3 rule is often simpler and slightly better.

    ৬.
    Given f(x) = x2. Use forward difference with step size h = 0.1 to approximate f‘(2).
    1. ক
      4.1
    2. খ
      4
    3. গ
      4.41
    4. ঘ
      3.9
    ব্যাখ্যা

    ৭.
    Let f(x) = 2x2 + 1. Use central difference with step size h = 0.1 to approximate f‘(2)
    1. ক
      7.8
    2. খ
      7.9
    3. গ
      8
    4. ঘ
      8.1
    ব্যাখ্যা

    ৮.

    1. ক
      Trapezoidal Rule
    2. খ
      Simpson’s 1/3 Rule
    3. গ
      Both
    4. ঘ
      None
    ব্যাখ্যা

    Trapezoidal gave 0.375 (approx).
    Simpson’s 1/3 gave 0.333 (exact).
    Simpson’s rule is exact for polynomials up to degree 3.

    ৯.
    ‘For the same function f(x) = 2x2 + 1, use backward difference with h=0.1 to approximate f′(2).
    1. ক
      7.8
    2. খ
      8
    3. গ
      8.2
    4. ঘ
      None
    ব্যাখ্যা

    ১০.

    1. ক
      7.33
    2. খ
      7.5
    3. গ
      8
    4. ঘ
      8.5
    ব্যাখ্যা

    ১১.

    1. ক
      8.67
    2. খ
      8.5
    3. গ
      9
    4. ঘ
      8.33
    ব্যাখ্যা

    ১২.
    For f(x) = x2 + 3, approximate f‘(2) using central difference with h = 0.1.
    1. ক
      3.9
    2. খ
      4
    3. গ
      4.1
    4. ঘ
      7
    ব্যাখ্যা

    ১৩.

    1. ক
      14
    2. খ
      13.8
    3. গ
      14.2
    4. ঘ
      14.25
    ব্যাখ্যা

    ১৪.

    1. ক
      14
    2. খ
      14.2
    3. গ
      14.33
    4. ঘ
      13.67
    ১৫.

    1. ক
      7
    2. খ
      6.67
    3. গ
      6.5
    4. ঘ
      None
    ব্যাখ্যা

    ১৬.

    1. ক
      5.9
    2. খ
      6
    3. গ
      6.1
    4. ঘ
      6.2
    ব্যাখ্যা

    ১৭.

    1. ক
      11.8
    2. খ
      12
    3. গ
      12.2
    4. ঘ
      None
    ব্যাখ্যা

    ১৮.

    1. ক
      3.8
    2. খ
      4
    3. গ
      4.2
    4. ঘ
      4.5
    ব্যাখ্যা

    ১৯.
    The trapezoidal rule for integration is derived by approximating the curve by:
    1. ক
      Straight lines
    2. খ
      Parabolas
    3. গ
      Cubic polynomials
    4. ঘ
      None
    ব্যাখ্যা

    Trapezoidal rule assumes the area under the curve is made up of trapeziums formed by straight line joining successive points.

    ২০.
    Which of the following numerical integration methods is exact for all polynomials up to degree 3?
    1. ক
      Trapezoidal Rule
    2. খ
      Midpoint Rule
    3. গ
      Simpson’s 1/3 Rule
    4. ঘ
      Simpson’s 3/8 Rule
    ব্যাখ্যা

    Trapezoidal Rule → exact for degree ≤1
    Midpoint Rule → exact for degree ≤1
    Simpson’s 1/3 Rule → exact for degree ≤3
    Simpson’s 3/8 Rule → exact for degree ≤3

    Exactness:

    Simpson’s 3/8 Rule is exact for cubic polynomials (degree ≤3), if the number of intervals is a multiple of 3.
    That’s why in the MCQ about “exact for all polynomials up to degree 3,” 3/8 Rule was the correct choice.

    General Accuracy (for any smooth function, same step size h):
    Simpson’s 1/3 Rule is fourth-order accurate (error ∝ h4) and generally more efficient for most applications.
    Simpson’s 3/8 Rule is also fourth-order accurate, but requires more points per segment (4 points), so it’s slightly less convenient.
    Conclusion:

    If the question asks “best accuracy for the same step size”, Simpson’s 1/3 Rule  is usually preferred.
    If the question asks “exact for cubic polynomials”, Simpson’s 3/8 Rule is safer.

    ২১.

    1. ক
      1.01
    2. খ
      1.1
    3. গ
      1.05
    4. ঘ
      1.2
    ব্যাখ্যা

    ২২.
    Which of the following formulas corresponds to Heun’s method?
    1. ক
      yn + 1​ = yn ​+ hf(xn​, yn​)
    2. খ
      yn + 1​ = yn​ + (h​/2)[f(xn​, yn​) + f(xn + 1​, yn​ + hf(xn​, yn​))]
    3. গ
      yn + 1 ​= yn​ + h​(k1​ + 2k​2 + 2k3 ​+ k4​)/6
    4. ঘ
      yn + 1​ = yn​ + hf(xn+1​,yn​)
       
    ব্যাখ্যা

    Heun’s method improves Euler by averaging the slope at the current point and the predicted next point.
    This gives second-order accuracy (O(h2)), better than Euler.

    ২৩.
    What is the order of accuracy of Euler’s method?
    1. ক
      1
    2. খ
      2
    3. গ
      3
    4. ঘ
      4
    ব্যাখ্যা

    Euler is a first-order method, meaning global error is proportional to h (O(h)). Reducing h improves accuracy linearly.

    ২৪.
    Which step is not part of Runge-Kutta 4 (RK4) Method?
    1. ক
      k1​ = hf(xn​, yn​)
    2. খ
      k2​ = hf(xn​ + h/2, yn​ + k1​/2)
    3. গ
      k3​ = hf(xn​+ h/2, yn​ + k2​/2)
    4. ঘ
      All are the part of RK4 Method
    ব্যাখ্যা

    Steps A–C are part of RK4 slope calculations.
    RK4 has fourth-order accuracy (O(h4)) → highly accurate for moderate h.

    ২৫.
    Which method uses multiple previous points to compute the next value?
    1. ক
      Euler
    2. খ
      RK4
    3. গ
      Adams-Bashforth
    4. ঘ
      All of these
    ব্যাখ্যা

    Multistep methods like Adams-Bashforth use previous computed points to predict yn+1, increasing efficiency for long computations.
    Single-step methods (Euler, Heun, RK4) only use the current point.

    ২৬.
    Reducing the step size h in Euler’s method will:
    1. ক
      Increase error
    2. খ
      Decrease error
    3. গ
      Have no effect
    4. ঘ
      Make solution unstable
    ব্যাখ্যা

    Smaller h gives better slope approximation → lower truncation error.
    However, too small h may increase round-off error in computer calculations.

    ২৭.
    Which of the following is an initial value problem?
      ব্যাখ্যা

      An IVP specifies both the derivative and initial condition.
      Only option B provides y(x0​)=y0​, allowing step-by-step numerical solution.

      ২৮.

      1. ক
        1`
      2. খ
        1.1
      3. গ
        1.2
      4. ঘ
        1.3
      ব্যাখ্যা

      ২৯.
      Which is an initial value problem (IVP)?
        ব্যাখ্যা

        IVP requires first derivative and initial value. Option B specifies y(0)=0. Option D is second-order, but only one initial condition is given → incomplete for an IVP.

        ৩০.

        1. ক
          0.1
        2. খ
          0.105
        3. গ
          0.11
        4. ঘ
          0.115
        ব্যাখ্যা

        ৩১.
        Given:

        Compute y2 using Euler’s method.
        1. ক
          1.01
        2. খ
          1.02
        3. গ
          1.03
        4. ঘ
          1.04
        ব্যাখ্যা

        ৩২.
        Trapezoidal rule approximates an integral using:
        1. ক
          Midpoint values only
        2. খ
          Average of endpoints and interior points
        3. গ
          Only left endpoint
        4. ঘ
          Only right endpoint
        ব্যাখ্যা

        It uses linear interpolation between points (endpoints + 2*sum of interior points).

        ৩৩.
        Which numerical method is fourth-order accurate?
        1. ক
          Euler
        2. খ
          Heun
        3. গ
          Runge-Kutta 4
        4. ঘ
          Trapezoidal
        ব্যাখ্যা

        RK4 global error ∝ h4, very accurate even for moderate h.

        ৩৪.
        Approximate f‘(2) for f(x)=x2+x using central difference with h=0.1.
        1. ক
          4
        2. খ
          5
        3. গ
          4.1
        4. ঘ
          5.1
        ব্যাখ্যা

        ৩৫.
        Approximate

        using Simpson’s 1/3 Rule with n = 2.
        1. ক
          1.25
        2. খ
          1.5
        3. গ
          1.75
        4. ঘ
          2
        ব্যাখ্যা

        Exact value, since Simpson’s 1/3 rule is exact for cubic polynomials

        ৩৬.

        1. ক
          20
        2. খ
          22
        3. গ
          24
        4. ঘ
          26
        ব্যাখ্যা

        (exact value, since Simpson’s 3/8 rule is exact for cubic polynomials

        ৩৭.

        1. ক
          4
        2. খ
          4.5
        3. গ
          5
        4. ঘ
          3.5
        ব্যাখ্যা

        ৩৮.

        1. ক
          3
        2. খ
          3.5
        3. গ
          4
        4. ঘ
          4.5
        ব্যাখ্যা

        ৩৯.
        Approximate f′(1) for f(x) = 3x2 + 2x using forward difference with h = 0.1.
        1. ক
          7.8
        2. খ
          8
        3. গ
          8.3
        4. ঘ
          8.7
        ব্যাখ্যা

        ৪০.

        1. ক
          1.25
        2. খ
          1.33
        3. গ
          1.375
        4. ঘ
          1.67
        ব্যাখ্যা