BT-102 (GS) – Mathematics-I
• Attempt any five questions.
• All questions carry equal marks.
a)Unit 1State Rolle's theorem hence verify for f(x)=x²+2x defined in the interval [-2, 0].
State Rolle'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 log(1+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 log(1+y) in a Taylor series in the neighbourhood of the point (0, 0).
a)Unit 1The temperature u(x,y,z) at any point in space is u=400xz² find the highest temperature on surface of the sphere x²+y²+z²=1.
The temperature u(x,y,z) at any point in space is u=400xz² find the highest temperature on surface of the sphere x²+y²+z²=1.
b)Unit 1If u=x²tan⁻¹(y/x)-y²tan⁻¹(x/y) find the value of ∂²u/∂x∂y.
If u=x²tan⁻¹(y/x)-y²tan⁻¹(x/y) find the value of ∂²u/∂x∂y.
a)Unit 1Find shortest distance from the origin to the curve x²+4xy+6y²=140.
Find shortest distance from the origin to the curve x²+4xy+6y²=140.
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 2Evaluate ∬ e²ˣ⁺³ʸ dxdy over the triangle bounded by x=0, y=0 and x+y=1.
Evaluate ∬ e²ˣ⁺³ʸ dxdy over the triangle bounded by x=0, y=0 and x+y=1.
a)Unit 3Test the convergence of the series 1 + x/2 + x²/5 + x³/10 + ... + xⁿ/(n²+1) + ...
Test the convergence of the series 1 + x/2 + x²/5 + x³/10 + ... + xⁿ/(n²+1) + ...
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 3Expand f(x)=x sin x, 0<x<2π as a Fourier series.
Expand f(x)=x sin x, 0<x<2π as a Fourier series.
b)Unit 3Find the a₀ and aₙ if the function f(x)=x+x² is expanded in Fourier series defined in (-1, 1).
Find the a₀ and aₙ if the function f(x)=x+x² is expanded in Fourier series defined in (-1, 1).
a)Unit 5i) If A is a skew symmetric matrix then show that A² is a symmetric matrix.
ii) Find eigen values of the matrix [[5,4],[1,2]].
i) If A is a skew symmetric matrix then show that A² is a symmetric matrix.
ii) Find eigen values of the matrix [[5,4],[1,2]].
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 5Verify Cayley Hamilton theorem for the matrix A and hence find A⁻¹.
Verify Cayley Hamilton theorem for the matrix A and hence find A⁻¹.
b)Unit 5Test the consistency and hence, solve the following set of equations: x+2y-z=3, 3x-y+2z=1, 2x-2y+3z=2, x-y+z=-1.
Test the consistency and hence, solve the following set of equations: x+2y-z=3, 3x-y+2z=1, 2x-2y+3z=2, x-y+z=-1.