Mathematics – I — Winter 2021

Find the extreme values of the function f(x,y) = x3 +⁡y3 −3x−12y+20

If u = log⁡(tanx +tany+tanz)⁡then show that 𝜕u 𝜕u 𝜕u sin2x +sin2y +sin2z = 𝜕x 𝜕y 𝜕z

Evaluate ex −e−x −2log(1+x) lim x→0 xsinx

Find the Fourier series of f(x) = 2x −⁡x2 in the interval (0,3).

Use Ratio test to check the convergence of the series ∞ 2n +1 ∑ 3n +1 n=1

Find the Maclaurin’s series of cosx and use it to find the series of sin2x.

Find the Fourier series of f(x) = x2 in the interval (0,2𝜋) and 𝜋2 1 1 1 hence deduce that = − + −⋯

Find the inverse of the following matrix by Gauss-Jordan method:123 A = [2 5 3]108

Find the directional derivative of f(x,y,z) = xyz at the point P(−1,1,3) in the direction of the vector a = i−2j+2k.

0 1 −3 −1 Find the rank of the matrix [ ⁡⁡⁡⁡ ] by reducing to 1 1 −2 0 row echelon form.

Find the eigenvalues and corresponding eigenvectors of the matrix401 A = [−2 1 0] −2 0 1

𝜕u 𝜕u 𝜕u If u = f(x−y,y−z,z−x), then prove that +⁡ + = 𝜕x 𝜕y 𝜕z0

Verify Cayley-Hamilton theorem for the following matrix and use it to find A−1211 A = [0 1 0]112

Expand 2x3 +7x2 +x −1 in powers of (x−2)

2 1 0 If 1 is an eigenvalue of the matrix [−1 0 1] then find its001 corresponding eigen vector.

Solve following system by using Gauss Jordan method x+2y+z−w =⁡−2 2x +3y−z+2w = x +y+3z−2w =⁡−6 x+y+z+w =

Use integral test to show that the following infinite series is convergent ∞1 ∑ n(1+⁡log2n) n=1

For the odd periodic function defined below, find the Fourier series −1, −1 < x < 0 f(x) = { 1, 0 < x < 1

Determine the radius and interval of convergence of the following infinite series x2 x3 x4 x − +⁡ −⁡ +⁡⋯4916

Show the following limit does not exist using different path approach 2x2y2 lim (x,y)→(0,0)x4 +y4

Evaluate the following integral along the region R ∬(x+y)dydx R where Ris the region bounded by x = 0,x = 2,y = x,y = x+2. Also, sketch the region.

Change the order of integration and hence evaluate the same. Do sketch the region. ∞ ∞ e−y ∫ ∫ dydx y 0 x

The following integral is an improper integral of which type? Evaluate ∞ dx ∫ x2 +10

If x = rsin𝜃cos𝜑,y = rsin𝜃sin𝜑,z = rcos𝜃, then find the jacobian 𝜕(x,y,z) 𝜕(r,𝜃,𝜑)

Find the volume of the solid generated by rotating the region bounded by y = x2 −2x and y = x about the line y = 4.