Mathematics-IB 2024 — 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

    Find the expansion of f(x) = Sinx in power of x in infinite series.

    Free sample

    Answer: x−x33!+x55!−x77!+⋯x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

  2. II

    The sequence {13n}\left\{ \frac{1}{3^n} \right\} is monotonic _______________.

  3. III

    Find the degree of homogeneity of the function sin⁡−1(y/x)\sin^{-1}(y/x)

  4. IV

    If a square matrix A has eigenvalues 1,3,5 then find the eigen values of the matrix A3A^3.

  5. V

    Find the value of B (1/2 ,1/2).

  6. VI

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

  7. VII

    Test the convergence of the series 1+3+5+⋯1 + 3 + 5 + \cdots

  8. VIII

    If f(x,y,z)=3x2y−y3z2f (x, y, z) = 3x^2y - y^3z^2, then find the value of grad f at (1, -2, -1).

  9. IX

    If AB = B and BA = A, then A is Idempotent (a) true (b) false

    atruebfalse
  10. X

    The value of ∫−∞0dx(2−x)2\int_{-\infty}^{0} \frac{dx}{(2-x)^2} is ____________.

  11. XI

    If Cauchy's mean value theorem is applied on the two functions f(x)=x2f(x) = x^2 , g(x)=xg(x) = x in [-1, 1] then find the value of c..

  12. XII

    A conditionally convergent series is absolutely convergent (a) true (b) false.

    atruebfalse

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

  1. 2

    Find the radius of the curvature of the parabola y2=4xy^2 = 4x at its vertex. Hence find its curvature at that point.

  2. 3

    Verify Rolle’s Theorem for f(x)=x(x−1)(x−2)f(x) = x (x-1)(x-2) in [0,2].

  3. 4

    Test the convergence of the series (2212−21)−1+(3323−32)−2+(4434−43)−3+⋯\left(\frac{2^2}{1^2} - \frac{2}{1}\right)^{-1} + \left(\frac{3^3}{2^3} - \frac{3}{2}\right)^{-2} + \left(\frac{4^4}{3^4} - \frac{4}{3}\right)^{-3} + \cdots

  4. 5

    If θ=tne−r24t\theta = t^n e^{-\frac{r^2}{4t}}, find what value of n will make 1r2∂∂r(r2∂θ∂r)=∂θ∂t\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \theta}{\partial r} \right) = \frac{\partial \theta}{\partial t}.

  5. 6

    Write a Taylor's series formula for the function f(x) = log(1+x), x>-1, about x=2 with Lagrange's form of remainder after three terms.

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

  1. 7(a)

    Show that {xn}\{x_n\} is a convergent sequence, where xn=1+13+132+⋯+13n−1x_n = 1 + \frac{1}{3} + \frac{1}{3^2} + \dots + \frac{1}{3^{n-1}} . Find the limit.

    5 marks
  2. 7(b)

    Check the convergence of the sequence {4n+33n+4}\left\{ \frac{4n+3}{3n+4} \right\}.

    5 marks
  3. 7(c)

    Test the convergence of the series 1+122+2233+3344+4455+…1 + \frac{1}{2^2} + \frac{2^2}{3^3} + \frac{3^3}{4^4} + \frac{4^4}{5^5} + \dots

    5 marks
  4. 8(a)

    Check whether the function xyx2+y2\frac{xy}{x^2 + y^2} is continuous at origin or not.

    5 marks
  5. 8(b)

    In what direction from the point (1, 2, 3) is the directional derivative of f(x,y,z)=x2−y2+2z2f(x, y, z) = x^2 - y^2 + 2z^2 maximum? Also find the maximum value.

    5 marks
  6. 8(c)

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

    5 marks
  7. 9(a)

    Find the radius of the curvature of the curve x1/2+y1/2=a1/2x^{1/2} + y^{1/2} = a^{1/2} at the point where the straight line y=xy = x cuts it.

    5 marks
  8. 9(b)

    Find the evolute of the curve x=a(cos⁡θ+θsin⁡θ),y=a(sin⁡θ−θcos⁡θ)x = a(\cos\theta + \theta\sin\theta), y = a(\sin\theta - \theta\cos\theta).

    5 marks
  9. 9(c)

    Find ∫−∞∞xe−x2dx\int_{-\infty}^{\infty} x e^{-x^2} dx.

    5 marks
  10. 10(a)

    Find the centre of curvature and the equation of circle of curvature of the curve x+y=ax2+by2+cx3x + y = ax^2 + by^2 + cx^3 at (0,0)(0,0).

    7 marks
  11. 10(b)

    Find the equation of the circle of curvature of the curve x=a(cos⁡θ+θsin⁡θ)x = a(\cos\theta + \theta\sin\theta), y=a(sin⁡θ−θcos⁡θ)y = a(\sin\theta - \theta\cos\theta) at the point θ\theta.

    8 marks
  12. 11(a)

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

    7 marks
  13. 11(b)

    Diagonalize the following matrix orthogonally: A=(511−1)A = \begin{pmatrix} 5 & 1 \\ 1 & -1 \end{pmatrix}

    8 marks

About this paper

The MAKAUT Mathematics-IB (BS-M102) 2024 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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Why solve Mathematics-IB PYQs?

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 2024 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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