BT-102 (GS) – Mathematics-I
• Attempt any five questions.
• All questions carry equal marks.
a)Unit 1State Lagrange's theorem hence verify for f(x)=x²+2x defined in the interval [-2, 0].
State Lagrange's theorem hence verify for f(x)=x²+2x defined in the interval [-2, 0].
b)Unit 1Find the first six terms of the expansions of the function eˣ cos y in a Taylor series in the neighbourhood of the point (0, 0).
Find the first six terms of the expansions of the function eˣ cos y in a Taylor series in the neighbourhood of the point (0, 0).
a)Unit 1Estimate the extreme values of the function x³+y³-63(x+y)+12xy.
Estimate the extreme values of the function x³+y³-63(x+y)+12xy.
b)Unit 1If u=((y-x)/xy)((z-x)/xz) find the value of x²uₓ+y²u_y+z²u_z.
If u=((y-x)/xy)((z-x)/xz) find the value of x²uₓ+y²u_y+z²u_z.
a)Unit 1Show that the rectangular solid of maximum volume that can be inscribed in a given sphere is a cube.
Show that the rectangular solid of maximum volume that can be inscribed in a given sphere is a cube.
b)Unit 1Find du/dt if u=x²+y², x=a cos t, y=b sin t.
Find du/dt if u=x²+y², x=a cos t, y=b sin t.
a)Unit 2Change the order of integration in ∫₀¹∫ₓ²²⁻ˣ xy dy dx and hence evaluate.
Change the order of integration in ∫₀¹∫ₓ²²⁻ˣ xy dy dx and hence evaluate.
b)Unit 2i) Find the value of Γ(3/2).
ii) Evaluate ∫₀¹ x³(1-√x)² dx.
i) Find the value of Γ(3/2).
ii) Evaluate ∫₀¹ x³(1-√x)² dx.
Unit 3Test the series 1 + x/2 + x²/5 + x³/10 + ... + xⁿ/(n²+1) + ...
Test the series 1 + x/2 + x²/5 + x³/10 + ... + xⁿ/(n²+1) + ...
a)Unit 2Show that β(l,m)=Γ(l)Γ(m)/Γ(l+m).
Show that β(l,m)=Γ(l)Γ(m)/Γ(l+m).
b)Unit 3Expand as a half range f(x)=x sin x series and cosine series for the interval 0<x<2.
Expand as a half range f(x)=x sin x series and cosine series for the interval 0<x<2.
a)Unit 5Transform the matrix into normal form and hence find its rank.
Transform the matrix into normal form and hence find its rank.
b)Unit 5Find the inverse of the matrix by using elementary row transformations.
Find the inverse of the matrix by using elementary row transformations.
a)Unit 5Find the eigen values and eigen vectors of the matrix.
Find the eigen values and eigen vectors of the matrix.
b)Unit 5Test the consistency and hence, solve the following set of equations.
Test the consistency and hence, solve the following set of equations.