BT-102 (GS) – Mathematics-I
• Attempt any five questions.
• All questions carry equal marks.
a)Unit 1Find the points where the function x³+y³-3axy has maximum or minimum value.
Find the points where the function x³+y³-3axy has maximum or minimum value.
b)Unit 1Find the Taylor's expansion of y=sin x about point x=π/2.
Find the Taylor's expansion of y=sin x about point x=π/2.
a)Unit 2The part of the parabola y²=4ax cut off by the latus rectum revolves about the tangent at the vertex. Find the volume of the reel thus generated.
The part of the parabola y²=4ax cut off by the latus rectum revolves about the tangent at the vertex. Find the volume of the reel thus generated.
b)Unit 2Prove that ∫₀¹ dx/√(1-x⁴) = (Γ(1/4))²/(6√(2π)).
Prove that ∫₀¹ dx/√(1-x⁴) = (Γ(1/4))²/(6√(2π)).
a)Unit 3Show that the following series is Convergent: 1/4 - 1/4² + 1/4³ - 1/4⁴ + ...
Show that the following series is Convergent: 1/4 - 1/4² + 1/4³ - 1/4⁴ + ...
b)Unit 3Obtain the Half-Range Sine Series for f(x) = eˣ in 0 < x < 1.
Obtain the Half-Range Sine Series for f(x) = eˣ in 0 < x < 1.
a)Unit 4Show that the set w={(a,b,0):a,b ∈ R} is subspace of R³.
Show that the set w={(a,b,0):a,b ∈ R} is subspace of R³.
b)Unit 4Are the following vectors LD? If so express one of these as a LC of other two: X₁=(1,3,4,2), X₂=(3,-5,2,2), X₃=(-2,1,-3,2).
Are the following vectors LD? If so express one of these as a LC of other two: X₁=(1,3,4,2), X₂=(3,-5,2,2), X₃=(-2,1,-3,2).
a)Unit 5Find a similarity transformation that diagonalise the matrix A.
Find a similarity transformation that diagonalise the matrix A.
b)Unit 5Find the Eigen value and Corresponding Eigen Vectors of the matrix.
Find the Eigen value and Corresponding Eigen Vectors of the matrix.
a)Unit 2Define Beta and Gamma Function and show that relation B(m,n)=Γ(m)Γ(n)/Γ(m+n).
Define Beta and Gamma Function and show that relation B(m,n)=Γ(m)Γ(n)/Γ(m+n).
b)Unit 2Evaluate:
i) ∫₀² x²(1-x)³ dx
ii) ∫₀¹ √(x(1-x)) dx.
Evaluate:
i) ∫₀² x²(1-x)³ dx
ii) ∫₀¹ √(x(1-x)) dx.
a)Unit 2Prove that the surface area of the solid generated by the revolution of the ellipse x²/a²+y²/b²=1 about the major axis is: 2(πab)·{√(1-e²) + sin⁻¹e/e}.
Prove that the surface area of the solid generated by the revolution of the ellipse x²/a²+y²/b²=1 about the major axis is: 2(πab)·{√(1-e²) + sin⁻¹e/e}.
b)Unit 3Show that the sequence (n^(1/n)) converge to 1.
Show that the sequence (n^(1/n)) converge to 1.
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 1Verify Rolle's Theorem for the function y=x²+2, a=-2 and b=2.
Verify Rolle's Theorem for the function y=x²+2, a=-2 and b=2.