Objective Question (MCQ) Mark
[ marks]Choose the appropriate answer for the following questions: 1. n2+4 The Series ∑∞ is convergent if n=1 np
[7 marks]p < 3 (b) p > 3 (c) p = 3 (d) p > −1 2. The series 1+ 3 + 9 + 27 + … is
[ marks]Convergent (b) Divergent
[ marks]Oscillates finitely (d) Oscillates infinitely 3. The Maclaurin series expansion of sinx is
[ marks]∞ x2n+1 (b) ∞ x2n ∑ ∑ (2n+1)! (2n)! n=0 n=0
[ marks]∞ (−1)nx2n+1 (d) ∞ (−1)nx2n ∑ ∑ (2n+1)! (2n)! n=0 n=0 4. x2y The value of lim is (x,y)→(0,0)x2 +y2
[ marks]1 (b) 1/2 (c) 0 (d) 5. 1 2 ∫ ∫ ∫ dz dy dx = _________000
[3 marks]12 (b) 3 (c) 0 (d) 6. y 𝜕(u,v) If u = , v = xy then =_______ x 𝜕(x,y)
[6 marks]2y (b) 2x (c) 2y (d) 2x − − x y x y 7. 𝜕u If u = xy then is 𝜕x
[ marks]0 (b) yxy (c) xylnx (d) y xy−1
[ marks]Choose the appropriate answer for the following questions: 1. If u = sin−1( x ) + tan−1( y ) then the value of x 𝜕u + y 𝜕u is y x 𝜕x 𝜕y
[7 marks]u (b) −u (c) 0 (d) 2. The value of lim x e−x is x→∞
[ marks]1 (b) 0 (c) e (d) 1/e Page 1 of 3. sinx The value of lim is x→0 x
[3 marks]1 (b) 0 (c) 𝜋 (d) ∞ 4. Which of the following curve passes through the origin?
[ marks]xy2 = 4a(3a−x) (b) y2(a−x) = x2(x +2)
[ marks]y2(a−x) = x+5 (d) None of these 5. If in the equation of a curve, x occurs only as an even power then the curve is symmetric about__________.
[ marks]Origin (b) x-axis (c) y-axis (d) None 6. The curve r = a(1+cos𝜃),a > 0 represents______.
[ marks]Circle (b) Cardioid
[ marks]Lemniscate of Bernoulli (d) None of these 7. ∬r3drd𝜃 over the region included between the circles r = 2sin𝜃 and r = 4sin𝜃 is
[ marks]𝜋 2sin𝜃 (b) 𝜋 4sin𝜃 ∫ ∫ r3drd𝜃 ∫ 2 ∫ r3drd𝜃 0 sin𝜃 0 2sin𝜃
[ marks]𝜋 4sin𝜃 (d) 𝜋 4sin𝜃 ∫ ∫ r3drd𝜃 ∫ ∫ r3drd𝜃 −𝜋 2sin𝜃 0 2sin𝜃
[ marks]Test the convergence of the series: ∑∞ 2n+3 n=1 (n+1)2
[3 marks]2n Test the convergence of the series: ∑∞ n=1 n3+1
[4 marks]Find the interval of convergence for the series,345 2x + x2 + x3 + x4 + …
[7 marks]∞ 1 Evaluate:∫ dx x2 +10
[3 marks]5sinx−7sin2x +3sin3x Evaluate:lim x→0 tanx−x
[4 marks]Trace the curve: xy2 = 4a2(a−x)
[7 marks]Expand 3x3 +8x2 +x −2 in powers of x−3.
[3 marks]𝜋 sin𝜃 Evaluate:∫ ∫ r drd𝜃00
[4 marks]Find the volume generated by revolving the area cut off from the parabola 9y = 4(9−x2) by the line 4x +3y = 12 about x −axis.
[7 marks]𝜕3u If u = e3xyz, show that = (3+27xyz+27x2y2z2)e3xyz 𝜕x𝜕y𝜕z
[3 marks]Find the equation of the tangent plane and normal line to the surface x2 +y2 +z2 = 1 at (2,2,1).
[4 marks]If u =1 log( x3+y3 ), find the value of x2𝜕2u +2xy 𝜕2u +y2𝜕2u 3 x2+y2 𝜕x2 𝜕x𝜕y 𝜕y2 (ii) Expand ex+y in power of x −1 and y+1 by using Taylor’s series. Page 2 of
[3 marks]𝜕2u 𝜕2u If u = log(x2 +y2), prove that = . 𝜕x𝜕y 𝜕y𝜕x
[3 marks]𝜕z 𝜕z If z = eax+by f(ax −by), prove that b +a = 2abz 𝜕x 𝜕y
[4 marks]Find the maximum and minimum distances of the point (3,4,12) from the sphere x2 +y2 +z2 = 1.
[7 marks]2 2 yz Evaluate: ∫ ∫ ∫ xyz dx dy dz010
[3 marks]04 Evaluate: ∬ (x+y)2dxdy R Where Ris the region bounded by x+y = 0,x+y = 1,2x−y = 0, 2x −y = 3 using transformations u = x+y and v = 2x −y.
[ marks]03
[ marks]Evaluate: ∬ r3drd𝜃; where Ris the region bounded between the R circles r = 2cos𝜃 and r = 4cos𝜃. (ii) By changing the order of integration, Evaluate the integral04 𝜋 𝜋 siny ∫∫ dydx y 0 x Page 3 of
[3 marks]