All parts for this question are compulsory a find the 5th


SECTION - A

All parts for this question are compulsory

a) Find the 5th derivative of exx3 .

b) Find the stationary point of f(x,y) = 5x2 + 10y2 + 12xy - 4x - 6y + 1

c) If u= x2-y2 + sin yz where y = ex and z= log x find du/dx

d) If x = r cosθ, y = r sinθ show that ∂(x,y)/∂(r,θ) = r

e) Reduce the matrix 148_matrix.jpg into normal form.

f) Prove that the matrix1902_matrix1.jpg is unitary.

g) Change the order of integration in the double integral 0x e-y/y dxdy and hence find its value.

h) Evaluate the following double integrals 010√(1+x2dxdy/(1+ x2 +y2)

i) Find the value of m if F = mxi^- 5yj^+ 2zk^ is a solenoidal vector.

j) Find the unit normal at the surface z = x2 + y2 at the point (1,2,5).

SECTION - B

Attempt any three parts of the following

a) If y = a cos (log x) + b sin(log x) then prove that x2y2 + xy1 + x = 0 and x2yn+2 + (2n+1) xyn+1 + (n2+1)yn = 0.

b) Prove that ∂(u,v)/∂(x,y) .∂(x,y)/∂(u,v) =1

c) Find the Eigen values and Eigen vectors of the following matrix :

2487_matrix2.jpg

d) To prove the Legendre's duplication formula Γn Γ(n+1/2) = √π/22n-1.Γ(2n)

e) Verify Gauss's divergence theorem for the function

F = x2i+ y2j^+ z2ktaken over the cube 0 ≤ x, y, z ≤ 1.

SECTION - C

Attempt any two parts from each question.

All questions are compulsory.

1. a). Find the nth derivative of 1/(x2 - 5x + 6) .

b). State and prove Euler's theorem for homogeneous functions.

c). If y = [ x + √(1+x2)]m then find (yn)0.

2. a). If u= (x2x3)/x1, u2 = (x3x1)/x2, u3 = (x1x2)/x3 Prove that J(u1u2u3) = 4.

b). The period of a simple pendulum is T = 2π√(l⁄g) Find the maximum error in T due to possible errors up to 1% in l and 2.5% in g.

c). In a plane triangle ABC find the maximum value of u = cos?A cos?B cos?C.

3 a). Find the inverse of 146_matrix3.jpg by elementary operations.

b). Test for consistency of the equations

2x - 3y + 7z = 5
3x + y - 3z = 13
2x + 19y - 47z = 32.

c). Show that the system of three vector

(1,3,2), (1,-7,-8), (2,1,-1) of V3 (R) is linearly dependent.

4 a). Evaluate the integral ?ex+y+z dx dy dz taken over the positive octant such that x + y + z ≤ 1 with the help of Liouvllis Theorem.

b). Show that 02(8 - x3)-1/3 dx = 2π/(3√3).

c). Evaluate ?y dx dy over the area between the parabolas y2 = 4x and x2 = 4y.

5 a). Show that the vector field F = yzi^ + zx+1j^ + xy k^ is conservative field its scalar potential. Also find the work done by F in moving a particle from (1,0,0) to (2,1,4).

b). Prove that Div (? a ¯ )= ? Div a ¯ + (grad ?).a ¯.

c). Find the directional derivative of ? = 5x2y - 5y2x + 5/2z2x at the Point (1,1,1) in the direction of line (x-1)/2 = (y-3)/(-2) = z/1.

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Engineering Mathematics: All parts for this question are compulsory a find the 5th
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