Mathematics – I — Winter 2023

Evaluate lim x lnx x→0+

Define beta and gamma functions. What is the relationship between beta and gamma functions?

Solve the following system of linear equations using Gauss-Jordan elimination: x +x +x = 0345 −x −x +2x −3x +x = 0 x +x −2x −x = 0 2x +2x −x +x = 0

1

Define rank of the matrix. Find rank(A) if A = [ ]24 (b ) Test the convergence of the series ∑∞ ( 1 ) n n=1 1+n −5 2

Find eigenvalues and eigenvectors of the matrix A = [ ] 2 −2

Find the Fourier series of the function f(x) = x+𝜋 if −𝜋 < x < 𝜋 and f(x+2𝜋) = f(x)

If w = x2 +y2,x = r−s,y = r+s, using chain rule, prove that 𝜕w = 4s. 𝜕s

Find the directional derivative of f(x,y,z) == 2x2 +3y2 +z2 at point (2,1,3) in the direction of the vector i−2k.

Find local extreme values of the function f(x,y) = 4x2 +9y2 + 8x −36y+24.

2x2y

Determine whether lim exists and find it if exists. (x,y)→(0,0)x3+y3

Find the equation of the tangent plane to z = 3x2 −xy at the point (1,2,1).

Find the maximum and minimum values of the function f(x,y) = 3x+ 4y on the circle x2 +y2 = 1.

) Calculate ∬ f(x,y)dA for f(x,y) = 100−6x2y and R:0 ≤ x ≤ R 2,−1 ≤ y ≤ 1 (b ) Sketch the region of integration for the integral ∫ 2 ∫ 2x (4x +2)dydx 0 x2 and write an equivalent integral with the order of integration reversed. sinx

Calculate ∬ dA where Ris the triangle in the xy-plane bounded Rx by the x-axis, the line y = x and the line x = 1.

) Evaluate ∫ ln8 ∫ lny ex+ydxdy.

Find the area of the region Rbounded by y = x and y = x2 in the first quadrant. (c ) Evaluate ∫ 1 ∫ √1−x2 (x2 +y2)dydx by changing into polar00 coordinates.

Find the Maclaurin series for cosx. (b ) Determine the convergence or divergence of the series ∑∞ ne−n2 . n=1 (c ) Test the convergence of the series ∑∞ (−1)n−1 .07 n=1 n2

) Define monotonic sequence. Is the sequence { 1 } monotonic? n2 (b ) Investigate the convergence of the series ∑∞ 2n+5 . n=0 3n x2 x3

Find the interval of convergence of the series x− + −⋯23