Objective Question (MCQ) Mark
[ marks]07 1. cos2n The sequence { } converges to n
[ marks]1 (b) 0 (c) 2 (d) −1 2. Sum of the series ∑∞ 4 is n=0 2n
[ marks]4 (b) 2 (c) 8 (d) 3. The coefficient of x4 in the expansion of cos x is
[ marks](b) − (c) (d) − 4! 4! 4 4. 1 x lim (1+ ) = _________________ x→∞ x
[4 marks]0 (b) 1 (d) e (d) ∞ 5. The curve x3 +y3 = 3xy is symmetric about
[ marks]x-axis (b) y-axis (c) line y = x (d) origin 6. Asymptote parallel to x-axis of the curve 3x3 +xy2 +xy = 0 is
[ marks]y=3 (b) y=1 (c) y=0 (d) not possible 7. The curve r2 = a2cos2𝜃 is not symmetric about 𝜋 𝜋
[ marks]initial line (b) line 𝜃 = (c) line 𝜃 = (d) pole42
[ marks]07 1. If u = ytan−1(x/y)+xcot−1(y/x), then xu +yu =_________ x y
[ marks]0 (b) u (c) 2u (d) 3u 2. For an implicit function f(x,y) = c, the value of dy is dx
[ marks]fx
[ marks]fy
[ marks]− fx
[ marks]− fy fy fx fy fx 3. y2tan−1x lim =_______________ (x,y)→(0,1) x
[ marks]0 (b) 1 (c) -1 (d) ∞ 4. ∫ 1 ∫ y ex/ydxdy =__________00 e−1
[ marks](b) e−1 (c) e (d) e+12 5. 𝜕(x,y) If u = x−y and v = x+y, then the value of J = is ______ 𝜕(u,v)
[ marks]1 (b) −1 (c) 1/2 (d) 1/4 6. The series ∑∞ 1 is convergent when n=1 np
[ marks]p = 1 (b) p < 1 (c) p > 1 (d) p = 7. The series ∑∞ n−1 is__________ n=1 n+1
[ marks]divergent
[ marks]convergent and sum
[ marks]convergent and sum
[2 marks]none of these
[ marks]Test the convergence of the series ∑∞ n+1 n=1 n3−3n+2
[3 marks]Test the convergence of the series ∑∞ 4n+1 n=1 5n 2x−xcos x−sinx 1 1
[4 marks]Evaluate (i) lim ; (ii) lim( − ). x→0 2x3 x→0 x2 sin2x
[7 marks]If u = log (x2 +y2), verify that 𝜕2u = 𝜕2u .03 𝜕x𝜕y 𝜕y𝜕x
[ marks]If u = rm, prove that u +u +u = m(m+1)rm−2, where xx yy zz r2 = x2 +y2 +z2.
[4 marks]Find the maxima and minima of the function f(x,y) = x2y−xy2 +4xy−4x2 −4y2.
[7 marks]If u = f( y−x , z−x ), show that x2𝜕u +y2𝜕u +z2𝜕u = 0.03 xy xz 𝜕x 𝜕y 𝜕z
[ marks]Evaluate ∬(6x2 +2y)dxdy over the region Rbounded between y = x2 and y = 4.
[4 marks]4 2√x Change the order of integration and evaluate ∫ ∫ dydx. 0 x2/4
[7 marks]If z = xy2 +x2y, x = at2, y = 2at, find dz . dt
[3 marks]If u = ex2+y2−xy, then prove that 𝜕u 𝜕u
[4 marks]x +y = 2u ln u; 𝜕x 𝜕y (ii) x2𝜕2u +2xy 𝜕2u +y2𝜕2u = 2u ln u(2lnu +1).
[ marks]𝜕x2 𝜕x𝜕y 𝜕y2
[7 marks]Check the absolute and conditional convergence of the series ∑∞ (−1)n+1 . n=1 √n (ii) Find the radius and interval of convergence of the power series ∑∞ (−5)nxn . n=0 n!
[3 marks]Evaluate ∬xdA, over the region Rbounded by the parabolas y2 = 4x and x2 = 4y.
[3 marks]𝜋 Expand cos( +x) in powers of x by Taylor series. Hence find the4 value of cos 46°. 4 (y/2)+1 2x−y
[4 marks]Evaluate ∫ ∫ dxdy by applying the transformations 0 y/2 2x−y y u = ,v = .22
[2 marks]Evaluate the improper integral ∫ ∞ 1 dx. −∞1+x2
[3 marks]Find the volume of the solid generated by revolving the region bounded by y = √x and the lines y = 2,x = 0 about the line y = 2.
[4 marks]Trace the curve y2(a−x) = x3, a > 0.
[7 marks]