BT-102 Mathematics-I - Jun 2024 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aExpand $(1+x)^m$ by Maclaurin's Theorem.
   How to answer

   • State Maclaurin's theorem with remainder
   • Compute derivatives of (1+x)^m at x=0
   • Form series m x + m(m-1)/2! x^2 + ...
   • Mention condition |x|<1 for convergence
2. 7 Marks
   bFind the Maximum value of $u = \sin x \sin y \sin(x+y)$.
   How to answer

   • Set partial derivatives u_x=0, u_y=0
   • Solve for stationary points x=y=pi/3
   • Use second derivative test for maxima
   • Compute maximum value 3√3/8
3. 7 Marks
   aFind the volume of the solid generated by the revolution of the Cardioid $r = a(1+\cos\theta)$ about the initial line.
   How to answer

   • Use volume by revolution about initial line: V = 2/3 π ∫ r^3 sinθ dθ
   • Substitute r = a(1+cosθ)
   • Integrate from 0 to π and double for symmetry
   • Simplify to 8/3 π a^3
4. 7 Marks
   bProve that $$\int_0^{\infty} \cos(x^2)dx = \frac{1}{2}\sqrt{\frac{\pi}{2}}$$
   How to answer

   • Consider Fresnel integral via contour / Gamma function
   • Use ∫0^∞ cos(x^2)dx with substitution t=x^2
   • Evaluate using ∫0^∞ e^{-st} /√t dt and Laplace
   • Obtain (1/2)√(π/2)
5. 7 Marks
   aShow that Sequence $\langle x^n \rangle$ where $|x| < 1$ converge to 0.
   How to answer

   • Recall definition of convergence of sequence
   • For |x|<1, estimate |x^n-0|=|x|^n
   • Given ε>0 choose N > log ε / log|x|
   • Conclude limit is 0
6. 7 Marks
   bFind the Fourier Series for the function $f(x) = x\sin x$, $(-\pi < x < \pi)$.
   How to answer

   • Note f(x)=x sin x is even on (-π,π)
   • Compute a0, an via integration by parts
   • bn=0 due to evenness
   • Write Fourier cosine series
7. 7 Marks
   aShow that the following equations are consistent and solve them. $$x - y + 2z = 4$$ $$3x + y + 4z = 6$$ $$x + y + z = 1$$
   How to answer

   • Form augmented matrix and find rank of A and [A|B]
   • Show rank = 3 = number of unknowns so consistent
   • Solve by elimination / Cramer's rule
   • Verify solution satisfies all equations
8. 7 Marks
   bIf $w_1$ and $w_2$ be two subspace of V(F) then Show that $w_1 \cap w_2$ also subspace of V(F).
   How to answer

   • Recall subspace axioms: non-empty, closed under addition and scalar multiplication
   • Show 0 ∈ w1∩w2
   • Show closure under linear combination
   • Conclude w1∩w2 is subspace of V(F)
9. 7 Marks
   aFind the Characteristic equation of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & -4 \\ 1 & 0 & -1 \end{bmatrix}$ and hence find the Eigen values and Eigen vectors.
   How to answer

   • Form |A-λI|=0 to get characteristic polynomial
   • Solve cubic for eigenvalues
   • For each λ solve (A-λI)X=0
   • Write eigenvectors
10. 7 Marks
    bShow that the following matrix A is Diagonalizable. $$A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{bmatrix}$$
    How to answer

    • Find eigenvalues of A
    • Check algebraic vs geometric multiplicity
    • Find three linearly independent eigenvectors
    • Conclude diagonalizable via P^{-1}AP=D
11. 7 Marks
    aFind the Maximum and Minimum value of $u = a^2x^2 + b^2y^2 + c^2z^2$, where $x^2 + y^2 + z^2 = 1$ and $lx + my + nz = 0$.
    How to answer

    • Use Lagrange multipliers with two constraints
    • Form auxiliary equations for x,y,z
    • Solve for stationary values of u
    • Identify max and min values
12. 7 Marks
    bFind the Fourier Series for the function $f(x) = x + x^2$ $(-\pi < x < \pi)$.
    How to answer

    • Identify odd part x and even part x^2
    • Compute a0, an, bn using standard formulas
    • Evaluate integrals over (-π,π)
    • Combine sine and cosine series
13. 7 Marks
    aShow that the surfaces area of the solid generated by revolution of the loop of the curve $x = t^2$, $y = t - 1/3t^3$ about the $x$ axis is $3\pi$.
    How to answer

    • Parametrize loop: find t limits where y=0
    • Use surface formula S=2π∫ y ds
    • Compute dx/dt, dy/dt and ds
    • Evaluate integral to get 3π
14. 7 Marks
    bInvestigate for what values of $\lambda$ and $\mu$ the simultaneous equations. $$X + Y + Z = 6$$ $$X + 2Y + 3Z = 10$$ $$X + 2Y + \lambda Z = \mu$$
    How to answer

    • Write augmented matrix for three equations
    • Compare ranks for λ=3 and λ≠3
    • Discuss μ=10 vs μ≠10 when λ=3
    • State unique / infinite / no solution cases
15. 7 Marks
    aExpand $\log x$ in power $(x-1)$ by Taylor's theorem and hence find the value $\log 1.1$.
    How to answer

    • State Taylor's theorem about x=1
    • Compute derivatives of log x at x=1
    • Form series (x-1)-(x-1)^2/2+...
    • Put x=1.1 to approximate log1.1
16. 7 Marks
    bEvaluate $\iint xy \, dx \, dy$ where the region of integration is $x + y < 1$ in the positive quadrant.
    How to answer

    • Sketch triangle x≥0,y≥0,x+y<1
    • Set limits: x from 0 to 1-y, y from 0 to 1
    • Integrate ∫∫ xy dx dy iteratively
    • Obtain value 1/24