BT-102 (GS) – Mathematics-I
• Attempt any five questions.
• All questions carry equal marks.
a)Unit 1Prove that a rectangular solid of maximum volume within a sphere is a cube.
Prove that a rectangular solid of maximum volume within a sphere is a cube.
b)Unit 1If u=tan⁻¹((x³+y³)/(x-y)) then prove that x ∂u/∂x + y ∂u/∂y = sin 2u.
If u=tan⁻¹((x³+y³)/(x-y)) then prove that x ∂u/∂x + y ∂u/∂y = sin 2u.
a)Unit 2Show that the surface area of solid generated by revolution of the loop of curve x=t², y=t-t³/3 about the x-axis is 3π.
Show that the surface area of solid generated by revolution of the loop of curve x=t², y=t-t³/3 about the x-axis is 3π.
b)Unit 2Change the order of integration in the following integral and then evaluate ∫₀^∞∫ₓ^∞ e^(-y)/y dy dx.
Change the order of integration in the following integral and then evaluate ∫₀^∞∫ₓ^∞ e^(-y)/y dy dx.
a)Unit 3Find the Fourier Series for f(x)=x+x² in (-π,π).
Find the Fourier Series for f(x)=x+x² in (-π,π).
b)Unit 3Find the half range sine series for f(x)=x(π-x) in (0,π). Hence Deduce that 1-1/3³+1/5³-...=π³/32.
Find the half range sine series for f(x)=x(π-x) in (0,π). Hence Deduce that 1-1/3³+1/5³-...=π³/32.
a)Unit 4Let W₁ and W₂ be subspaces of a vector space V and assume that W₁ ∩ W₂ = {0}. Let w₁∈W₁ and w₂∈W₂ be such that w₁≠0 and w₂≠0. Prove that {w₁, w₂} is linearly independent.
Let W₁ and W₂ be subspaces of a vector space V and assume that W₁ ∩ W₂ = {0}. Let w₁∈W₁ and w₂∈W₂ be such that w₁≠0 and w₂≠0. Prove that {w₁, w₂} is linearly independent.
b)Unit 4i) Prove that W₁ ∩ W₂ is a subspace of V.
ii) Give an example to show that W₁ ∪ W₂ need not be a subspace of V.
iii) Is W₁ ∪ W₂ a subspace of V?
i) Prove that W₁ ∩ W₂ is a subspace of V.
ii) Give an example to show that W₁ ∪ W₂ need not be a subspace of V.
iii) Is W₁ ∪ W₂ a subspace of V?
a)Unit 5Find Eigen Value and Eigen vectors of the matrix.
Find Eigen Value and Eigen vectors of the matrix.
b)Unit 5Diagonalizable the matrix.
Diagonalizable the matrix.
a)Unit 5Reduce the matrix to the normal form, hence find its rank.
Reduce the matrix to the normal form, hence find its rank.
b)Unit 1Find the first 3 terms in the Maclaurin series for
i) sin⁻¹x
ii) xe^(-x).
Find the first 3 terms in the Maclaurin series for
i) sin⁻¹x
ii) xe^(-x).
a)Unit 5Investigate for what values of λ and μ the simultaneous equations have
i) no solution
ii) a unique solution
iii) an infinite number of solutions.
Investigate for what values of λ and μ the simultaneous equations have
i) no solution
ii) a unique solution
iii) an infinite number of solutions.
b)Unit 1Find the Taylor series for the function x⁴+x-2 centered at a=1.
Find the Taylor series for the function x⁴+x-2 centered at a=1.
a)Unit 2Evaluate ∫₀^∞∫ₓ^∞ e^(-y)/y dy dx.
Evaluate ∫₀^∞∫ₓ^∞ e^(-y)/y dy dx.
b)Unit 1If xˣ yʸ zᶻ = C then show that ∂²z/∂x∂y = -(x log ex)⁻¹.
If xˣ yʸ zᶻ = C then show that ∂²z/∂x∂y = -(x log ex)⁻¹.