Mathematics-IB 2026 — MAKAUT · BS-M102 · Mid Sem PYQ with solutions

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Maulana Abul Kalam Azad University of Technology

Solutions

29 answers across 3 groups

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

  1. I

    The nth derivative of (ax+b)10(ax+b)^{10} when n>10 is

    Free sample

    Answer: 0

    Explanation

    The derivative of a polynomial of degree mm is zero for all derivatives of order n>mn > m. Since the highest power of xx is 10, any derivative of order n>10n > 10 is zero.

  2. II

    Write the value of Γ(1/2)\Gamma(1/2).

  3. III

    If f(x) satisfies the conditions of Rolle's theorem in an interval [a,b][a,b] then f′(x)f^{\prime}(x) becomes zero at.

  4. IV

    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

  5. V

    If f(x,y)=x3+y2f(x,y)=x^{3}+y^{2} then fxy(x,y)=f_{xy}(x,y)=.

  6. VI

    What is a trace of a matrix? Explain with an example

  7. VII

    Write a relation between Beta and Gama functions.

  8. VIII

    Lagrange's Mean Value theorem is obtained from Cauchy's Mean value theorem for two functions f(x)f(x) and g(x)g(x) by putting g(x)=...........g(x)=...........

  9. IX

    Is a conditionally convergent series is absolutely convergent? Justify.

  10. X

    If φ=x2−y2\varphi=x^{2}-y^{2} then ∇⃗2φ\vec{\nabla}^{2}\varphi is equal to

  11. XI

    If eigenvalues of a 3×33\times3 matrix A are -1, 1 and 0, then what is the trace of [A100+I][A^{100}+I]?

  12. XII

    The sequence {13n}\{\frac{1}{3^{n}}\} is

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

  1. 2

    Solve by using Matrix inversion method: 2x+3y+4z=9; 3x-4y+5z=4; 4x-5y+6z=5

  2. 3

    If xyz=abcxyz=abc by using Lagrange's undetermined multipliers prove that the minimum value of bcx+cay+abzbcx+cay+abz is 3abc, where a, b, c>0.

  3. 4

    Find values of ∫0∞1(1+x)xdx\int_{0}^{\infty}\frac{1}{(1+x)\sqrt{x}}dx

  4. 5

    Show that lim⁡x→π2(1−sin⁡x)tan⁡x=0\lim_{x\rightarrow\frac{\pi}{2}}(1-\sin x)\tan x=0

  5. 6

    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

Group CLong Answer Type Question · answer any 3

  1. 7(a)

    Find the Fourier series of the function f(x)={π+2x,−π<x<0π−2x,0≤x≤πf(x)=\begin{cases}\pi+2x, & -\pi<x<0\\ \pi-2x, & 0\le x\le\pi\end{cases} and hence deduce that 112+132+152+...=π28\frac{1}{1^{2}}+\frac{1}{3^{2}}+\frac{1}{5^{2}}+...=\frac{\pi^{2}}{8}

    8 marks
  2. 7(b)

    Find the half range cosine series of the function f(x)=x,0<x<2f(x)=x, 0<x<2, and then find the sum.

    7 marks
  3. 8(a)

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

    5 marks
  4. 8(b)

    Find the value of ∫0π2sin⁡4xcos⁡5xdx\int_{0}^{\frac{\pi}{2}}\sin^{4}x \cos^{5}x dx

    5 marks
  5. 8(c)

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

    5 marks
  6. 9(a)

    Evaluate: lim⁡x→0(1/x2−1/sin⁡2x)\lim_{x\rightarrow0}(1/x^{2}-1/\sin^{2}x)

    5 marks
  7. 9(b)

    Use Mean Value Theorem to prove: x/(1+x)<log⁡(1+x)<x; if x>0x/(1+x)<\log(1+x)<x; \text{ if } x>0

    5 marks
  8. 10(a)

    If y=tan⁡−1xy=\tan^{-1}x then prove that i) (1+x2)y1=1(1+x^{2})y_{1}=1 ii) (1+x2)yn+1+2nxyn+n(n−1)yn−1=0(1+x^{2})y_{n+1}+2nxy_{n}+n(n-1)y_{n-1}=0.

    5 marks
  9. 10(b)

    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).

    5 marks
  10. 10(c)

    Define curl and divergence of vector valued function. When a vector valued function is called solenoidal and irrotational?

    5 marks
  11. 11(a)

    Expanding the determinant by Laplace's method in terms of minors of 2nd order formed from the first two, prove that ∣0abc−a0de−b−d0f−c−e−f0∣=(af−be+cd)2.\begin{vmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \end{vmatrix}=(af-be+cd)^{2}.

    7 marks
  12. 11(b)

    Find the eigenvalues and the eigenvectors corresponding to the smallest eigenvalue of the matrix A=(211121001)A=\begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 0 & 1 \end{pmatrix}

    8 marks

About this paper

The MAKAUT Mathematics-IB (BS-M102) 2026 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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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 2026 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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