Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary


1 (a) (i) The value of limx0xsin1x islimx0xsin1x is (A) 1
(B) π
(C) 0
(D) ∞
1 M
1 (a) (ii) y=log(sinx) then value of d2ydx2= y=log(sinx) then value of d2ydx2= 
(A) sec2 x
(B) -cosec2 x
(C) -cosec xcotx
(D) secxtanx
1 M
1 (a) (iii) Curve: x=t2-1, y=t2-t. For which value of t tangents of the curve are parallel to the x-axis? (A) t=0(B) t=13(C) t=12(D) t=13
1 M
1 (a) (iv) 11x|x|dx=....... (A) 2 (B) 1 (C) 0 (D) None of these
1 M
1 (a) (v) If f(x)=42x,x<1=6x4,x1 then find limx1f(x) (A) -2 (B) 2 (C) 0 (D) doesn't exist.
1 M
1 (a) (vi) If f:R → R, f(x) = 3x+2, then find f-1
(A) Not possible
(B) 3x-2
(C) 2x-3
(D) x23
1 M
1 (a) (vii) What is the solution for following equations ?
2x+3y-5=0, 4x+6y-7=0
(A) No solution
(B) Infinitely many solution
(C) unique solution
(D) { (0,0), (-5, -7) }
1 M
1 (b) (i) limn1n2n= (A) -2 (B) 1 (C) 2 (D) doesn't exist
1 M
1 (b) (ii) 1xcosxdx=....+c(A) 23cos32x(B) 2x(C)2sinx(D) 23cosx
1 M
1 (b) (iii) Find the area of the curve y=x2 + 1 bounded by x-axis and lines x=1 and x=2.
(A) 310(B) 103(C)6(D) 16
1 M
1 (b) (iv) f(x) = [x], x ∈ R then f(x) is
(A) Continuous for all real numbers.
(B) Continuous for all integers
(C) discontinuous for all integers
(D) none of these
1 M
1 (b) (v) Find the vertical asymptotes of the curve y=2xzx21 (A) x=1
(B) x=-1
(C) x= ± 1
(D) y=2
1 M
1 (b) (vi) f(x) =x3-3x2+3x-100 is ....... function.
(A) Increasing
(B) decreasing
(C) constant
(D) none
1 M
1 (b) (vii) Find the value of c using Lagrange's mean value theorem for f(x)=ex, x ∈ [0,1]
(A) log 1
(B) log c
(C) log (1-e)
(D) log (e-1)
1 M

2 (a) (i) Evaluate limx0ex+exx22sin2xx2
2 M
2 (a) (ii) Test for convergence the series 11.2+12.3+13.4+.....
2 M
2 (a) (iii) Expand log sin x in powers of (x-2)
3 M
2 (b) (i) Test for convergence the series 2+32x+43x2+54x3+......
4 M
2 (b) (ii) Trace the curve r=a sin 3θ
3 M

3 (a) (i) (i) Evaluate limx0(ax+x)1x
(ii) Test the convergence of n=032n23n
4 M
3 (a) (ii) Trace the curve x3 + y3 = 3axy.
3 M
3 (b) (i) Test the convergence of n=1[(n+1)x]nnn+1
4 M
3 (b) (ii) Evaluate improper integral 3013xdx
3 M

4 (a) (i) (i) If u=ln(x7+y7+z7x+y+z) show that xux+yuy+zuz=6
(ii) State modified Euler's theorem and find the degree of homogeneous function u(x,y)=1x2+1xy+logxlogyx2
4 M
4 (a) (ii) z=f(x,y), xeu + e-v, y=e-u - ev then prove that
zzzv=xzxyuy
3 M
4 (b) (i) If u=e3xyz show that 3uxyz=(3+27xyz+27x2y2z2)e3xyz
4 M
4 (b) (ii) show that f(x,y)=xyx2+y2; (x,y)e(0,0)=0, (x,y)=(0,0) is continuous at the origin.
3 M

5 (a) (i) Find the extreme values of the function
f(x,y) = x3+y3-3x-12y+20.
4 M
5 (a) (ii) Show that the surface z=xy-2 and x2+y2+z2 =3 have the same tangent plane at (1,1,1).
3 M
5 (b) (i) Find the minimum value of x2 y z3 subject to the condition 2x+y+3z=a using Lagrange's method of undetermined multipliers.
4 M
5 (b) (ii) Expand sin xy in powers of (x-1) and (y- π/2) upto second degree terms.
3 M

6 (a) (i) Sketch the region of integration, reverse the order of integration and evaluate 204x20xe2y4ydydx.
4 M
6 (a) (ii) Evaluate the integral π201sinθ0r2cosθdrdθ
3 M
6 (b) (i) Evaluate the integral 2021yz0xyz dxdydx.
4 M
6 (b) (ii) Evaluate the integral a0a2y20y2x2+y2dydx. by changing into polar co-ordinates
3 M

7 (a) (i) Find the volume bounded by the cylinder x2 + y2 = 4 and the planes y+z=3, z=0.
4 M
7 (a) (ii) Evaluate ∬R x2+y2 dA by changing the variables ,where R is the region lying in first quadrant and bounded by the hyperbola
x2 - y2=1, x2 - y2=9, xy=2 and xy=4.
3 M
7 (b) (i) The region between the curve y=√x, 0≤x≤4 and the x-axis is revolved about the x-axis to generate a solid. Find its volume.
4 M
7 (b) (ii) Test the convergence of the series n=1(1)n+1logn+1
3 M



More question papers from Calculus
SPONSORED ADVERTISEMENTS