BT-102 (GS) – Mathematics-I
• Attempt any five questions.
• All questions carry equal marks.
a)Unit 1Expand (1+x)ˣ by Maclaurin's Theorem.
Expand (1+x)ˣ by Maclaurin's Theorem.
b)Unit 1Find the Maximum value of u=sin x sin y sin(x+y).
Find the Maximum value of u=sin x sin y sin(x+y).
a)Unit 2Find the volume of the solid generated by the revolution of the Cardioid r=a(1+cosθ) about the initial line.
Find the volume of the solid generated by the revolution of the Cardioid r=a(1+cosθ) about the initial line.
b)Unit 2Prove that ∫₀^∞ cos(x²)dx = ½√(π/2).
Prove that ∫₀^∞ cos(x²)dx = ½√(π/2).
a)Unit 3Show that Sequence (xₙ) where |x|<1 converge to 0.
Show that Sequence (xₙ) where |x|<1 converge to 0.
b)Unit 3Find the Fourier Series for the function f(x)=x sin x, (-π<x<π).
Find the Fourier Series for the function f(x)=x sin x, (-π<x<π).
a)Unit 5Show that the following equations are consistent and solve them: x-y+2z=4, 3x+y+4z=6, x+y+z=1.
Show that the following equations are consistent and solve them: x-y+2z=4, 3x+y+4z=6, x+y+z=1.
b)Unit 4If w₁ and w₂ be two subspace of V(F) then Show that w₁ ∩ w₂ also subspace of V(F).
If w₁ and w₂ be two subspace of V(F) then Show that w₁ ∩ w₂ also subspace of V(F).
a)Unit 5Find the Characteristic equation of the matrix and hence find the Eigen values and Eigen vectors.
Find the Characteristic equation of the matrix and hence find the Eigen values and Eigen vectors.
b)Unit 5Show that the following matrix A is Diagonalizable.
Show that the following matrix A is Diagonalizable.
a)Unit 1Find the Maximum and Minimum value of u=a²x²+b²y²+c²z², where x²+y²+z²=1 and lx+my+nz=0.
Find the Maximum and Minimum value of u=a²x²+b²y²+c²z², where x²+y²+z²=1 and lx+my+nz=0.
b)Unit 3Find the Fourier Series for the function f(x)=x+x², (-π<x<π).
Find the Fourier Series for the function f(x)=x+x², (-π<x<π).
a)Unit 2Show that the surfaces area of the solid generated by revolution of the loop of the curve x=t², y=t-t³/3 about the x axis is 3π.
Show that the surfaces area of the solid generated by revolution of the loop of the curve x=t², y=t-t³/3 about the x axis is 3π.
b)Unit 5Investigate for what values of λ and μ the simultaneous equations have solutions.
Investigate for what values of λ and μ the simultaneous equations have solutions.
a)Unit 1Expand log x in power (x-1) by Taylor's theorem and hence find the value log 1.1.
Expand log x in power (x-1) by Taylor's theorem and hence find the value log 1.1.
b)Unit 2Evaluate ∬ xy dx dy where the region of integration is x+y<1 in the positive quadrant.
Evaluate ∬ xy dx dy where the region of integration is x+y<1 in the positive quadrant.