1988

1) If f(x)=tanx prove that f′′(0)=nc2fn−2(0)+nc4fn−4(0)…..=sinnπ2.

2) Find the minimum value of x2+y2+z2 when x+y+z=k.

3) Find the a symptotes of the cubic x3−xy2−2xy+2x−y=0 and show that they cut the curve again in points which lie on the line 3x−y=0.

4) Evaluate ∬x3/2y2(1−x2−y2)dxdy over the positive quadrant of the circle x2+y2=1.

1987

1) If x1=12(x+9x) and for n>0,xn+1=12(xn+9xn), find the value of limn→xxn(x>0 is assumed).

2) If x=−a,h=2a,f(x)=x13, find θ from the mean value theorem f(x+h)=f(x)+h f^{\prime}(x+\theta h)

3) If u=x+y−z,v=x−y+z and w=x2+(y−z)2, examine whether or not there exists any functional relationship between u,v and w, and find the relation, if any.

4) If u=cosec−1(x1/(n+1)+y1/(n+1)x+y)1/2, show that x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2 = 14n2tanu(2n+sec2u).

5) Show, by means of a suitable substitution that ∫π/20sin2x−1θcos2y−1θdθ=12∫∞0tx−1(1+t)x+ydt,x, y,>0.Establishtheinequality\dfrac{1}{2}<\int_{0}^{1} \dfrac{d x}{\left(4-x^{2}+x^{3}\right)^{\dfrac{1}{2}}}<\dfrac{\pi}{6}$$.

6) Find the volume of the solid generated by revolving the curve y2=x32a−x;a>0, about its asymptote x=2a.

7) Evaluate ∬Dx1/2y1/2(1−x−y)2/3dxdy, where D is the domain bounded by the lines x=0, y=0, x+y=1.

1986

1) A function f(x) is defined as follows:

Examine whether or not f(x) is differentiable at x=0.

2) If f′(x) exists and is continuous, find the value of Lx→01x2∫x0(x−3y)f(y)dy

3) Evaluate ∫20∫x0{(x−y)2+2(x+y)+1}−12 dxdy by using the transformation x=u(1+v),y=v(1+u). Assume u, v are positive in the region concerned.

4) Use Rolle’s theorem to establish that under suitable conditions (to be stated) |f(a)f(b)g(a)g(b)|=(b−a)|f(a)f′(ξ)g(a)g′(ξ)|, a<ξ<b. Hence or otherwise deduce the inequality nbn−1(a−b)<an−bn<nan−1(a−b) where a>b and n>1.

5) If u=(ax3+by3)n3n(3n−1)+xf(yx) find the value of x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2.

6) Without evaluating the involved integrals, show that ∫x1tdt1+t2+∫1/21dtt(1+t2)=0.

7) If f(x) is periodic of period T, show that ∫a+Taf(t)dt is independent of a.

1985

1) If f(x,y)={xyx2+y2;(x,y)≠(0,0)0;(x,y)=(0,0) Show that both the partial derivatives fx and fy exist at (0,0) but the function is not continuous there.

2) If for all values of the parameter λ and for some constant ′n′,F(λx,λy)=λnF(x,y) identically, where F is assumed differentiable, prove that x∂F∂x+y∂F∂y=nF.

3) Prove the relation between beta and gamma functions β(m,n)=Γ(m)Γ(n)Γ(m+n).

4) If a function f defined on [a,b] is continuous on [a, b] and differentiable on (a,b) and f(a)=f(b), then prove that ∃ atleast one real number “ c′,a<c <b such that f′(c)=0.

5) Use Maclaurin’s expansion to show that log(1+x)=x−x22+x33−x44+…….... Hence find the value of log(1+x+x2+x3+x4).

1984

1) Show that tanx is not continuous at x=π2.

2) Let f(x,y)=(x+y)sin(1x+1y),x↑0,y↑0 f(x,0)=f(0,y)=0 Examine whether (i) f(x,y) is continuous at (0,0) and (ii) Ltf(x,y), for (y↑0) and Ltx→0f(x,y) for (x↑0), exist.

TBC

3) If B(p,q) be the Beta function, show that pB(n,q)=(q−1)B(p+1,q−1), where p,q are real p>0,q>1. Hence or otherwise find B(p,n) when ′n′ is an integer (>0).

4) If u=x+y1−xy and v=tan−1x+tan−1y, find ∂(u,v)∂(x,y). Are u and v functionally related? If so, find the relationship.