Mathematics-III (Differential Calculus) 2024 — MAKAUT · BSC-301 · Mid Sem PYQ with solutions

  • Solved
  • Verified

Maulana Abul Kalam Azad University of Technology

Solutions

32 answers across 3 groups

Try each question first, then open it to check your answer.

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

  1. I

    Find the number of vertices in a graph with 15 edges if each vertex has degree 2.

    Free sample

    Answer: 15

    Explanation

    By the Handshaking Lemma, the sum of the degrees of all vertices is equal to twice the number of edges. Let nn be the number of vertices. Since each vertex has degree 2, the sum of degrees is 2n2n. Given the number of edges is 15: 2n=2×152n = 2 \times 15 2n=302n = 30 n=15n = 15

  2. II

    Determine whether the sequence {xn}\{x_n\}, where xn=11⋅2+12⋅3+13⋅4+⋯+1n⋅(n+1)x_n = \frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \cdots + \frac{1}{n\cdot(n+1)} converges or not.

  3. III

    Is the function f(x,y)={xyxy+x−y,(x,y)≠(0,0)0,(x,y)=(0,0)f(x,y) = \begin{cases} \frac{xy}{xy+x-y}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} continuous at (1,2)?

  4. IV

    Evaluate: ∫0π/2∫02r dr dθ\int_0^{\pi/2} \int_0^2 r \, dr \, d\theta

  5. V

    Write the general solution of the differential equation p=cos⁡(y−xp)p = \cos(y - xp) ,where p=dydxp = \frac{dy}{dx}.

  6. VI

    Is the function f(x,y)={x2+y2+xy,(x,y)≠(2,3)10,(x,y)=(2,3)f(x,y) = \begin{cases} x^2 + y^2 + xy, & (x,y) \neq (2,3) \\ 10, & (x,y) = (2,3) \end{cases} continuous at (0,0) ?

  7. VII

    Determine whether the series ∑n=1∞1n2+n+1\sum_{n=1}^\infty \frac{1}{n^2+n+1} converges or not.

  8. VIII

    Find the degree of the homogeneous function f(x,y)=x+yx−yf(x,y) = \frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}

  9. IX

    Find the value of ∬dxdy\iint dxdy over the first quadrant of the circle having centre at the origin and radius 4 c.m.

  10. X

    Evaluate: 1(D−2)(D−3)e2x\frac{1}{(D-2)(D-3)} e^{2x}

  11. XI

    If a graph has 5 vertices and 7 edges then write the size of its adjacency matrix.

  12. XII

    Give an example of a bounded sequence which is not convergent.

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

  1. 2

    Let f(x,y)={x3+y3x−y,x≠y0,x=yf(x,y) = \begin{cases} \frac{x^3+y^3}{x-y}, & x \neq y \\ 0, & x = y \end{cases}. Prove that f(x,y)f(x,y) is not continuous at (0,0) but ∂f∂x,∂f∂y\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} exist at (0,0).

  2. 3

    Using the transformation x+y=u,y=uvx + y = u, y = uv. Show that ∫01dx∫01−xeyx+ydy=12(e−1)\int_0^1 dx \int_0^{1-x} e^{\frac{y}{x+y}} dy = \frac{1}{2}(e - 1).

  3. 4

    Solve: x2d2ydx2−xdydx−3y=x2log⁡xx^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} - 3y = x^2 \log x

  4. 5

    Prove that the maximum degree of any vertex in a simple graph with nn vertices is n−1n - 1.

    Prove that the maximum number of edges in a connected simple graph with nn vertices is 12n(n−1)\frac{1}{2}n(n - 1).

    Show answer to question 5
  5. 6

    Using the transformation x−y=u,x+y=vx - y = u, x + y = v prove that ∬Rcos⁡x−yx+ydx dy=sin⁡12\iint_R \cos\frac{x-y}{x+y} dx \, dy = \frac{\sin 1}{2} where R is the region bounded by x+y=1x + y = 1, x=0,y=0x = 0, y = 0.

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

  1. 7(a)

    Prove that the minimum number of edges in a connected graph with nn vertices is n−1n - 1.

    5 marks
  2. 7(b)

    Prove that a complete graph with nn vertices consists of 12n(n−1)\frac{1}{2}n(n - 1) number of edges.

    Show that a bipartite graph cannot contain a cycle of odd length.

    Show answer to question 7(b)5 marks
  3. 7(c)

    Prove that there exist no simple graph with five vertices having degrees 4,4,4,2,2 .

    Draw, if possible, a simple graph with five vertices having degrees 2,3,3,3,3 .

    Show answer to question 7(c)5 marks
  4. 8(a)

    Show that the sequence {5,5+5,5+5+5,… }\left\{\sqrt{5}, \sqrt{5 + \sqrt{5}}, \sqrt{5 + \sqrt{5 + \sqrt{5}}}, \dots\right\} tends to a definite finite limit. Also find the limit.

    5 marks
  5. 8(b)

    Discuss the convergence of the power series 1+37x+3⋅67⋅10x2+3⋅6⋅97⋅10⋅13x3+⋯(x>0)1 + \frac{3}{7}x + \frac{3\cdot 6}{7\cdot 10}x^2 + \frac{3\cdot 6\cdot 9}{7\cdot 10\cdot 13}x^3 + \cdots \quad (x > 0)

    5 marks
  6. 8(c)

    Discuss the convergence of the series x−x33!+x55!−⋯ ,xx - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots , x be any number.

    5 marks
  7. 9(a)

    Let f(x,y)={1,xy≠00,xy=0f(x,y) = \begin{cases} 1, & xy \neq 0 \\ 0, & xy = 0 \end{cases} . Show that the two repeated limits exist at (0,0) and are equal but the simultaneous limit lim⁡(x,y)→(0,0)f(x,y)\lim_{(x,y)\to(0,0)} f(x,y) does not exist.

    5 marks
  8. 9(b)

    Let f(x,y)={x+ysin⁡1x,x≠00,x=0f(x,y) = \begin{cases} x + y\sin\frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases} . Verify that the simultaneous limit lim⁡(x,y)→(0,0)f(x,y)\lim_{(x,y)\to(0,0)} f(x,y) exists and the repeated limit lim⁡x→0lim⁡y→0f(x,y)\lim_{x\to 0} \lim_{y\to 0} f(x,y) exists but the repeated limit lim⁡y→0lim⁡x→0f(x,y)\lim_{y\to 0} \lim_{x\to 0} f(x,y) does not exist.

    5 marks
  9. 9(c)

    Let f(x,y)={ysin⁡1x+xsin⁡1y,xy≠00,xy=0f(x,y) = \begin{cases} y\sin\frac{1}{x} + x\sin\frac{1}{y}, & xy \neq 0 \\ 0, & xy = 0 \end{cases} . Show that at (0,0) the double limit exists but the repeated limits do not exist.

    5 marks
  10. 10(a)

    Verify Stoke's theorem for F⃗=(2x−y)i^−yz2j^−y2zk^\vec{F} = (2x - y)\hat{i} - yz^2\hat{j} - y^2z\hat{k} where SS is the upper half surface of the sphere x2+y2+z2=1x^2 + y^2 + z^2 = 1 and CC is its boundary.

    5 marks
  11. 10(b)

    Verify Green’s theorem in the plane for ∮C[(xy+y2)dx+x2dy]\oint_C [(xy + y^2) dx + x^2 dy] where CC is the closed curve of the region bounded by y=xy = x and y=x2y = x^2.

    5 marks
  12. 10(c)

    Verify the Divergence Theorem for the vector function F⃗=(x2−yz)i^+(y2−zx)j^+(z2−xy)k^\vec{F} = (x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k}, taken over the rectangular parallelopiped 0≤x≤a,0≤y≤b,0≤z≤c0 \le x \le a, 0 \le y \le b, 0 \le z \le c.

    5 marks
  13. 11(a)

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

    5 marks
  14. 11(b)

    Show that F⃗=(6xy+z3)i^+(3x2−z)j^+(3xz2−y)k^\vec{F} = (6xy + z^3)\hat{i} + (3x^2 - z)\hat{j} + (3xz^2 - y)\hat{k} is irrotational. Find a scalar function φ\varphi such that F⃗=∇φ\vec{F} = \nabla\varphi.

    5 marks
  15. 11(c)

    If r=∣r⃗∣r = |\vec{r}|, where r⃗=xi^+yj^+zk^\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}, then prove that ∇⋅(r⃗r)=2r\nabla \cdot \left(\frac{\vec{r}}{r}\right) = \frac{2}{r}.

    5 marks

About this paper

The MAKAUT Mathematics-III (Differential Calculus) (BSC-301) 2024 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.

Common paper for 3 branches

Every branch listed gets the same questions. Pick a branch to see its other papers.

Why solve Mathematics-III (Differential Calculus) PYQs?

Frequently asked questions

5 questions

What is the subject code for Mathematics-III (Differential Calculus)?

BSC-301 is the MAKAUT subject code for Mathematics-III (Differential Calculus).

Which exam is this Mathematics-III (Differential Calculus) question paper from?

The MAKAUT 2024 Mid Sem exam, semester 3. A student shared it, and it was checked before publishing.

Which branches is this Mathematics-III (Differential Calculus) paper for?

It's a common paper for 3 B.Tech branches: Computer Science and Engineering, Computer Science and Engineering (Data Science) and Information Technology. Every branch gets the same questions.

Where can I find more BSC-301 question papers?

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

Are there solutions for this paper?

Yes. Every question has a worked answer on this page, grouped like the paper. The first answer is free to read; sign in (it's free) to unlock the rest.

Have notes or the answer key for Mathematics-III (Differential Calculus)?

Share it and help the next batch of students.

Related question papers

Mathematics-III (Differential Calculus)BSC-301 · 2023 · Mid SemMathematics-III (Differential Calculus)BSC-301 · 2025 · Mid SemSolved

Explore more MAKAUT resources