Q.1(a): Verify Lagrange's mean value theorem for the function f(x) = 2x2-7x+10 in the interval [2, 5]. Q.1(b): If u = f(y-z, z-x, x-y ) prove that du/dx + du/dy + du/dz = 0 Q.2(a): Evaluate ∫ 0,2 ∫ 0,1 ( x^2 + y^2) dx dy Q.2(b): Evaluate ∫0,a∫0,x∫0,x+y e^(x+y+z) dz dy dx Q.3(a): Find rank if the matrix [1 2 3 2, 2 3 5 1, 1 3 4 5] Q.3(b): Solve the system of equations 3x + 3y + 2z =1; x + 2y = 4; 10y + 3z = –2 and 2x – 3y – z = 5 Q.4(a): if u=sin^-1 ((x^2+y^2)/(x+y)) then show that x (du/dx) +y (du/dy) = tanx Q.4(b): Discuss the maximum or minima of the function f(x, y) = x3 - 4xy + 2y2 Q.5(a): Determine whether or not the vectors u(1, 1, 2), V(2,3,1), W(4,5,5) in R3 are linearly dependent. Q.5(b): Let V=R3, show that w is not a subspace of V, where w={(a,b,c): a >= 0 Q.6(a): Find the Eigen values and Eigen vectors for the matrix A : [ 5 4, 1 2] Q.6(b):Show that the following equations are consistent or not. 5x + 3y +14z = 4, y + 2z = 1, x-y+2z = 0 Q.7(a): Evaluate lim n->∞ (1/n+1 + 1/n+2 + ----- + 1/2n) Q.7(b):Prove that Β(m,n)= m! n! / (m+n)! Q.8(a): Find Eigen values of the matrix [2 1 1 , 1 2 1, 0 0 1] Q.8(b): Find the characteristic equation of the matrix [ 2 -1 1, -1 2 -1, 1 -1 2 ]