Let f(x)=cos(2tan−1sin(cot−1x1−x)), 0 < x < 1. Then :
02Medium63×since 2002Q3652
If x=2cosec−1 and y=2sec−1t(∣t∣≥1), then dxdy is equal to :
03Medium63×since 2002Q3653
The derivative of tan−1(sinx+cosxsinx−cosx), with respect to 2x
, where (x∈(0,2π)) is :
04Medium63×since 2002Q3654
The derivative of
tan−1(x1+x2−1) with
respect to tan−1(1−2x22x1−x2) at x = 21 is :
05Medium63×since 2002Q3664
Let y be an implicit function of x defined by x2x−2xxcoty−1=0. Then y′(1) equals
06Medium63×since 2002Q3655
Let f:(−1,1)→R be a differentiable function with f(0)=−1 and f′(0)=1. Let g(x)=[f(2f(x)+2)]2. Then g′(0)=
07Easy63×since 2002Q3656
If g is the inverse of a function f and f′(x)=1+x51, then g′(x) is equal to:
08Easy63×since 2002Q3657
If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of
ƒ(ƒ(ƒ(x))) + (ƒ(x))^2
at x = 1 is :
09Medium63×since 2002Q3658
Let f(x) = log_e(sin x), (0 < x < π) and g(x) = sin^–1
(e^–x
), (x ≥ 0). If α is a positive real number such that
a = (fog)'(α) and b = (fog)(α), then :
10Easy63×since 2002Q3659
Let f : R → R be defined as f(x)=x3+x−5. If g(x) is a function such that f(g(x))=x,∀′x′∈R, then g'(63) is equal to ________________.
11Medium63×since 2002Q3660
Let f(x)=x5+2ex/4 for all x∈R. Consider a function g(x) such that (g∘f)(x)=x for all x∈R. Then the value of 8g′(2) is :
12Hard63×since 2002Q3661
Suppose for a differentiable function h,h(0)=0,h(1)=1 and h′(0)=h′(1)=2. If g(x)=h(ex)eh(x), then g′(0) is equal to:
13Medium63×since 2002Q3662
If y=(x+1+x2)n, then (1+x2)dx2d2y+xdxdy is
14Medium63×since 2002Q3663
Let f(x) be a polynomial function of second degree. If f(1)=f(−1) and a,b,c are in A.P, then f′(a),f′(b),f′(c) are in
15Hard63×since 2002Q3665
If y = [x+x2−1]15+[x−x2−1]15,
then (x² − 1) dx2d2y+xdxdy is equal to :
16Medium63×since 2002Q3666
If f\left( x \right) = \left| {\matrix{
{\cos x} & x & 1 \cr
{2\sin x} & {{x^2}} & {2x} \cr
{\tan x} & x & 1 \cr
} } \right|, then x→0limxf′(x)
17Hard63×since 2002Q3667
If x² + y² + sin y = 4, then the value of dx2d2y at the point (−2,0) is :
18Easy63×since 2002Q3668
If e^y
+ xy = e, the ordered pair (dxdy,dx2d2y) at x = 0 is equal to :
19Easy63×since 2002Q3669
Let x^k + y^k = a^k, (a, k > 0 ) and dxdy+(xy)31=0, then k is:
20Medium63×since 2002Q3670
If y(α)=2(1+tan2αtanα+cotα)+sin2α1,α∈(43π,π)dαdyatα=65πis :