Mathematics-IA 2025 — MAKAUT · BS-M101 · Mid Sem PYQ with solutions

  • Solved
  • Verified

Maulana Abul Kalam Azad University of Technology

Solutions

27 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

    f(x)=|x| satisfies Roll's theorem [-1,1]. State true or false with reasons.

    Free sample

    Answer: False

    Explanation

    To satisfy Rolle's Theorem on the interval [−1,1][-1, 1], a function f(x)f(x) must be continuous on the closed interval [−1,1][-1, 1] and strictly differentiable on the open interval (−1,1)(-1, 1). The given function f(x)=∣x∣f(x) = |x| is continuous everywhere, including on [−1,1][-1, 1]. However, we must check its differentiability at x=0x = 0, which lies within the open interval (−1,1)(-1, 1). The left-hand derivative at x=0x=0 is −1-1, and the right-hand derivative at x=0x=0 is 11. Because the left-hand and right-hand derivatives are not equal, f(x)=∣x∣f(x) = |x| is not differentiable at x=0x = 0. Since it fails the necessary condition of differentiability on the open interval, it does not satisfy Rolle's theorem.

  2. II

    Every square matrix A can be expressed as a sum of symmetric and skew symmetric matrix and it can be expressed as=

  3. III

    Collection of vectors containing null vectors is linearly __________.

  4. IV

    The eigen value of the matrix A are a and b; then what are the eigen values of A2A^2 ?

  5. V

    The value of Γ(5/2)=\Gamma(5/2) = ?

  6. VI

    lim⁡x→01−cos⁡xsin⁡x=?\lim_{x \to 0} \frac{1 - \cos x}{\sin x} = ?

  7. VII

    Find the value of C for which the vector (1,2) and (3,C) linearly independent.

  8. VIII

    A set of vectors in a vector space V over a field F is orthonormal if ∣∣α∣∣=||\alpha|| = ________.

  9. IX

    What is eigen value of a square matrix? Find the eigen value of the matrix A=(100020003)A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}

  10. X

    Define Beta and Gamma functions.

  11. XI

    The product of two Orthogonal matrix is __________ matrix.

  12. XII

    The product of any matrix and its transpose is ___________ matrix.

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

  1. 2

    Check whether the vectors ( ( 1, 1, 0), (1, 1, 1), (0, 1, -1) is a basis in R3R^3 or not.

  2. 3

    State Lagrange's mean value theorem. Prove that x1+x<log⁡(1+x)<x,x>0\frac{x}{1+x} < \log(1+x) < x, x > 0

  3. 4

    Solve the following equations by matrix inversion method, if possible: 2x+3y+z= 6; 3y+5z= 8; -x+y=0

  4. 5

    Define subspace of a vector space. Let V={M2 set of all 2×2 matrices of the form (abcd),a,b,c,d∈R}V=\{M_2 \text{ set of all } 2 \times 2 \text{ matrices of the form } \begin{pmatrix} a & b \\ c & d \end{pmatrix}, a,b,c,d \in R\} and S is the subset of V such that S={(abcd):a+b=0}S=\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a+b=0\} Prove that S is a subspace of V

  5. 6

    Prove that B(m, n) = B(n, m). Also write the relation between Beta and Gamma functions and find B(3,4).

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

  1. 7(a)

    Define kernel of a linear transformation between two vector spaces. Is f(x)=sin⁡xf(x) = \sin x a Linear transformation? Justify your answer.

    5 marks
  2. 7(b)

    State Cayley Hamilton theorem for matrices. What is the Characteristic equation of a square matrix? Explain with an example.

    4 marks
  3. 7(c)

    What is eigen vector? Find the eigen values and eigen vectors of matrix (2−114)\begin{pmatrix} 2 & -1 \\ 1 & 4 \end{pmatrix}.

    6 marks
  4. 8

    Show that the set of all complex numbers is a vector space over real field w.r.t the usual addition between two complex number and multiplication between a real number and a complex number. Find a basis of this space and hence the dimension.

  5. 9

    What is diagonalization of matrix? Diagonalise the matrix A=(64−2412−4−2−413)A = \begin{pmatrix} 6 & 4 & -2 \\ 4 & 12 & -4 \\ -2 & -4 & 13 \end{pmatrix}. Write the matrix P, such that P−1APP^{-1}AP is diagonal matrix. Also show that P−1AP=DP^{-1}AP = D.

  6. 10(a)

    Prove that ∫0∞1(1+x2)5dx=35π/256\int_0^\infty \frac{1}{(1+x^2)^5} dx = 35\pi/256

    7 marks
  7. 10(b)

    Obtain the evolute of the parabola y2=4axy^2=4ax.

    8 marks
  8. 11(a)

    Verify Lagrange's MVT for f(x)=x(x−1)(x−2)f(x)= x(x-1)(x-2) in [0,1/2][0, 1/2].

    5 marks
  9. 11(b)

    Expand the following functions in power of x in infinite series stating in each case the condition under which the expression is valid: f(x)= \cos x

    5 marks
  10. 11(c)

    Evaluate Γ(1/2)Γ(3/2)Γ(5/2)Γ(7/2)\Gamma(1/2) \Gamma(3/2) \Gamma(5/2) \Gamma(7/2).

    5 marks

About this paper

The MAKAUT Mathematics-IA (BS-M101) 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.

Common paper for 2 branches

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

Why solve Mathematics-IA PYQs?

Frequently asked questions

5 questions

What is the subject code for Mathematics-IA?

BS-M101 is the MAKAUT subject code for Mathematics-IA.

Which exam is this Mathematics-IA 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-IA paper for?

It's a common paper for 2 B.Tech branches: Computer Science and Engineering and Information Technology. Every branch gets the same questions.

Where can I find more BS-M101 question papers?

On the Mathematics-IA 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-IA?

Share it and help the next batch of students.

Related question papers

Mathematics-IABS-M101 · 2026 · Mid SemSolvedMathematics-IABS-M101 · 2024 · Mid SemSolvedMathematics-IABS-M101 · 2023 · Mid Sem

Explore more MAKAUT resources