Mathematics-III (Probability and Statistics) 2025 — MAKAUT · BSM-301 · Mid Sem PYQ with solutions

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Maulana Abul Kalam Azad University of Technology

Solutions

25 answers across 3 groups

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Group AVery Short Answer Type Question · answer any 10 · 1 mark each

  1. i

    Find fxf_x and fyf_y where f(x,y)=x2+xyf(x,y) = x^2 + xy.

    Free sample

    Answer: fx=2x+yf_x = 2x + y, fy=xf_y = x

    Explanation

    Given f(x,y)=x2+xyf(x,y) = x^2 + xy, Differentiating partially with respect to xx (treating yy as constant): fx=∂f∂x=∂∂x(x2+xy)=2x+yf_x = \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2 + xy) = 2x + y Differentiating partially with respect to yy (treating xx as constant): fy=∂f∂y=∂∂y(x2+xy)=xf_y = \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2 + xy) = x

  2. ii

    The value of ∫10∫01(x+y)dxdy\int_1^0 \int_0^1 (x + y) dx dy is..................

  3. iii

    The general solution of px+appx + \frac{a}{p}, where p=dydxp = \frac{dy}{dx} is..........

  4. iv

    The value of 1D(x2)\frac{1}{D}(x^2) is...................

  5. v

    If a path is considered as a sub graph, then the degree of the intermediate vertices is...............

  6. vi

    The series ∑n=1∞1np\sum_{n=1}^\infty \frac{1}{n^p} is convergent if.............

  7. vii

    If u+v=x,uv=yu + v = x, uv = y, then ∂(x,y)∂(u,v)=\frac{\partial(x,y)}{\partial(u,v)} =

  8. viii

    The value of ∫0π/2∫01rsin⁡θdrdθ\int_0^{\pi/2} \int_0^1 r \sin\theta dr d\theta is...................

  9. ix

    The singular solution of the equation y=px+a(1+p2)y = px + a\sqrt{(1 + p^2)} is ...................

  10. x

    The value of D(sin⁡3x)D(\sin^3 x) is..............

  11. xi

    A binary tree has exactly ..............

  12. xii

    The co-efficient of x3x^3 in the infinite series expansion of f(x)=sin⁡xf(x) = \sin x about x=π2x = \frac{\pi}{2} is

Group BShort Answer Type Question · answer any 3 · 5 marks each

  1. 2

    Show that (∂∂x+∂∂y+∂∂z)2u=−9(x+y+z)2\left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right)^2 u = \frac{-9}{(x+y+z)^2} if u=log⁡(x3+y3+z3−3xyz)u = \log(x^3 + y^3 + z^3 - 3xyz)

  2. 3

    Evaluate ∬Rxy(x+y)dxdy\iint_R xy(x+y) dx dy over the region RR bounded by y=x2y = x^2 and y=xy = x.

  3. 4

    Solve the ODE: xp2+(y−x)p−y=0xp^2 + (y - x)p - y = 0, where p=ddxp = \frac{d}{dx}

  4. 5

    Find the Taylor's series expansion of sin⁡x\sin x.

  5. 6

    Solve by variation of parameters, d2ydx2+a2y=cos⁡ax\frac{d^2 y}{dx^2} + a^2 y = \cos ax.

Group CLong Answer Type Question · answer any 3 · 15 marks each

  1. 7(a)

    Find the maxima and minima of the function f(x,y)=x3+y3−3x−12y+20f(x,y) = x^3 + y^3 - 3x - 12y + 20. Also find the saddle points.

    10 marks
  2. 7(b)

    Find div F⃗\text{div } \vec{F} and curl F⃗\text{curl } \vec{F} where F⃗=grad(x3+y3+z3−3xyz)\vec{F} = \text{grad}(x^3 + y^3 + z^3 - 3xyz)

    5 marks
  3. 8(a)

    Evaluate ∬ydxdy\iint y dx dy over part of the plane bounded by y=xy = x and the parabola y=4x−x2y = 4x - x^2.

    8 marks
  4. 8(b)

    Change the order of integration and hence evaluate the integral ∫01∫eyedxdyylog⁡y\int_0^1 \int_{e^y}^e \frac{dx dy}{y \log y}

    7 marks
  5. 9(a)

    Solve the ODE: (2x+3y+7)dx+(3x−5y+2)dy=0(2x + 3y + 7)dx + (3x - 5y + 2)dy = 0

    8 marks
  6. 9(b)

    Find the orthogonal trajectory of the family of curve x2+y2=r2x^2 + y^2 = r^2, where r being the variable of parameter.

    7 marks
  7. 10

    Solve the ODE: (x2D2+3xD+2)y=cos⁡(log⁡x)(x^2D^2 + 3xD + 2)y = \cos(\log x), where D=ddxD = \frac{d}{dx}

  8. 11

    Describe KrusKal and Prim's algorithm for finding the minimal spanning tree with examples.

About this paper

The MAKAUT Mathematics-III (Probability and Statistics) (BSM-301) 2025 Mid Sem previous year question paper, set in semester 3. Download the verified PDF or read it online, then check your answers against the worked solutions above.

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What is the subject code for Mathematics-III (Probability and Statistics)?

BSM-301 is the MAKAUT subject code for Mathematics-III (Probability and Statistics).

Which exam is this Mathematics-III (Probability and Statistics) question paper from?

The MAKAUT 2025 Mid Sem exam, semester 3. A student shared it, and it was checked before publishing.

Which branches is this Mathematics-III (Probability and Statistics) paper for?

It's a common paper for 3 B.Tech branches: Electrical Engineering, Electronics and Communication Engineering and Mechanical Engineering. Every branch gets the same questions.

Where can I find more BSM-301 question papers?

On the Mathematics-III (Probability and Statistics) subject page, which collects every year's papers along with notes and the syllabus.

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