Mathematics-IB 2025 — MAKAUT · BS-M102 · Mid Sem PYQ with solutions

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Maulana Abul Kalam Azad University of Technology

Solutions

30 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

    The series 1+r+r2+r3+r4+..........1+r+r^{2}+r^{3}+r^{4}+.......... ..is convergent if r is.......

    Free sample

    Answer: ∣r∣<1|r| < 1

    Explanation

    This is a geometric series with common ratio rr. A geometric series converges if and only if the absolute value of its common ratio is strictly less than 1.

  2. II

    Write the value of Γ(2)\Gamma(2).

  3. III

    What is the rank of the 5th order identity matrix?

  4. IV

    Prove that B(m,n)=B(n,m)B(m,n)=B(n,m)

  5. V

    If f(x)f(x) satisfies the conditions of Rolle's theorem in [a,b][a,b] then we get a point on the curve where the tangent is parallel to

  6. VI

    The series 1−13+132−133+.....1-\frac{1}{3}+\frac{1}{3^{2}}-\frac{1}{3^{3}}+..... is

  7. VII

    The singularity of the integral ∫02dx(x−1)is..........\int_{0}^{2}\frac{dx}{(x-1)}is..........

  8. VIII

    limx→01−cos xsin x=?lim_{x\rightarrow0}\frac{1-cos~x}{sin~x}=?

  9. IX

    The greatest value of the function f(x)=x(x−1)2f(x)=x(x-1)^{2} in the interval 0≤x≤20\le x\le2 is

  10. X

    The series ∑1np\sum\frac{1}{n^{p}} is convergent if p is .

  11. XI

    What is Fourier series?

  12. XII

    The value of x which makes the vector xi^+2j^+8k^x\hat{i}+2\hat{j}+8\hat{k} and 2i^+3j^−k^2\hat{i}+3\hat{j}-\hat{k} perpendicular is

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

  1. 2

    ∫0π2tan xdx\int_{0}^{\frac{\pi}{2}}\sqrt{tan~x}dx

  2. 3

    Given the function f(x,y)={xy(x2−y2)x2+y2,(x,y)≠(0,0)0,(x,y)=(0,0)f(x,y)=\begin{cases}\frac{xy(x^{2}-y^{2})}{x^{2}+y^{2}} & ,(x,y)\ne(0,0)\\ 0 & ,(x,y)=(0,0)\end{cases}, find from definition fxy(0,0)f_{xy}(0,0) and fyx(0,0)f_{yx}(0,0). Verify whether fxy(0,0)=fyx(0,0)f_{xy}(0,0)=f_{yx}(0,0).

  3. 4

    Prove that the series x−x22+x33−⋅⋅⋅(−1)n+1xnn+⋅⋅⋅x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdot\cdot\cdot(-1)^{n+1}\frac{x^{n}}{n}+\cdot\cdot\cdot is absolutely convergent when ∣x∣<1|x|<1 and conditionally convergent when x=1x=1

  4. 5

    Prove that ∣bc−a2ca−b2ab−c2ca−b2ab−c2bc−a2ab−c2bc−a2ca−b2∣=(a3+b3+c3−3abc)2.\begin{vmatrix}bc-a^{2}&ca-b^{2}&ab-c^{2}\\ ca-b^{2}&ab-c^{2}&bc-a^{2}\\ ab-c^{2}&bc-a^{2}&ca-b^{2}\end{vmatrix}=(a^{3}+b^{3}+c^{3}-3abc)^{2}.

  5. 6

    Show that limx→π2(sin x)tan x=1lim_{x\rightarrow\frac{\pi}{2}}(sin~x)^{tan~x}=1

Group CLong Answer Type Question · answer any 3

  1. 7(a)

    Show that ∫0∞e−x2x9dx=12\int_{0}^{\infty}e^{-x^{2}}x^{9}dx=12

    5 marks
  2. 7(b)

    Find the value of ∫0π2sin4x cos5xdx.\int_{0}^{\frac{\pi}{2}}sin^{4}x~cos^{5}xdx.

    5 marks
  3. 7(c)

    Evaluate, if possible, ∫−11dxx\int_{-1}^{1}\frac{dx}{x} Also find its Principal value

    5 marks
  4. 8(a)

    Verify Rolle's theorem. for f(x)=/x , -1<x<1.

    5 marks
  5. 8(b)

    State Cauchy's mean value theorem and deduce it from Lagrange mean value theorem.

    5 marks
  6. 8(c)

    Expand cosx in an infinite series (in power of x) in Lagrange's form of remainder.

    5 marks
  7. 9(a)

    Discuss the convergence of the series 1+2p2!+3p3!+4p4!+⋅⋅⋅1+\frac{2^{p}}{2!}+\frac{3^{p}}{3!}+\frac{4^{p}}{4!}+\cdot\cdot\cdot

    5 marks
  8. 9(b)

    Test the convergence of the series 1−122+132−142+⋅⋅⋅1-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\cdot\cdot\cdot

    5 marks
  9. 9(c)

    Show that the series ∑n=1∞cos nxn2\sum_{n=1}^{\infty}\frac{cos~nx}{n^{2}} is absolutely convergent.

    5 marks
  10. 10(a)

    Find the maximum value of x3y2x^{3}y^{2} subject to the constraint x+y=1x+y=1, using the method of Lagrange's multiplier.

    8 marks
  11. 10(b)

    Find the angle between the surfaces x2+y2+z2=9x^{2}+y^{2}+z^{2}=9 and x2+y2−3=zx^{2}+y^{2}-3=z at the point (2,−1,2)(2,-1,2).

    7 marks
  12. 11(a)

    Expanding the determinant by Laplace's method prove that [0abc−a0de−b−d0f−c−e−f0]=(af−be+cd)2\begin{bmatrix}0&a&b&c\\ -a&0&d&e\\ -b&-d&0&f\\ -c&-e&-f&0\end{bmatrix}=(af-be+cd)^{2}

    8 marks
  13. 11(b)

    Determine the condition for which the system x+y+z=1x+y+z=1; x+2y−z=bx+2y-z=b; 5x+7y+az=b25x+7y+az=b^{2} admits

    • (i)
      one solution,
    • (ii)
      no solution,
    • (iii)
      many solutions.
    Show answer to question 11(b)7 marks

About this paper

The MAKAUT Mathematics-IB (BS-M102) 2025 Mid Sem previous year question paper, set in semester 1. Download the verified PDF or read it online, then check your answers against the worked solutions above.

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Frequently asked questions

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What is the subject code for Mathematics-IB?

BS-M102 is the MAKAUT subject code for Mathematics-IB.

Which exam is this Mathematics-IB question paper from?

The MAKAUT 2025 Mid Sem exam, semester 1. A student shared it, and it was checked before publishing.

Which branches is this Mathematics-IB paper for?

It's a common paper for 6 B.Tech branches: Civil Engineering, Electronics and Communication Engineering, Electrical Engineering, Mechanical Engineering, Computer Science and Engineering (Artificial Intelligence and Machine Learning) and Computer Science and Engineering (Data Science). Every branch gets the same questions.

Where can I find more BS-M102 question papers?

On the Mathematics-IB subject page, which collects every year's papers along with notes and the syllabus.

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