• Attempt any five questions.
• All questions carry equal marks.
Find the points where the function x³+y³-3axy has maximum or minimum value.
Find the Taylor's expansion of y=sin x about point x=π/2.
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.
Prove that ∫₀¹ dx/√(1-x⁴) = (Γ(1/4))²/(6√(2π)).
Show that the following series is Convergent: 1/4 - 1/4² + 1/4³ - 1/4⁴ + ...
Obtain the Half-Range Sine Series for f(x) = eˣ in 0 < x < 1.
Show that the set w={(a,b,0):a,b ∈ R} is subspace of R³.
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).
Find a similarity transformation that diagonalise the matrix A.
Find the Eigen value and Corresponding Eigen Vectors of the matrix.
Define Beta and Gamma Function and show that relation B(m,n)=Γ(m)Γ(n)/Γ(m+n).
Evaluate:
i) ∫₀² x²(1-x)³ dx
ii) ∫₀¹ √(x(1-x)) dx.
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}.
Show that the sequence (n^(1/n)) converge to 1.
Prove that a rectangular solid of maximum volume within a sphere is a cube.
Verify Rolle's Theorem for the function y=x²+2, a=-2 and b=2.
