JEE Main Mathematics - Differentiability and Differentiation
Exam Duration: 60 Mins Total Questions : 30
Consider the function f(x)=|logex|,\(\forall\)x>0.Then,
- (a)
LHD does not exist at x=1
- (b)
RHD does not exist at x=1
- (c)
f is differentiable at x=1
- (d)
f is not differential at x=1
Let x=a(cost+logtan\(t\over2\))and y=a sint, then \(dy\over dx\)is
- (a)
cot t
- (b)
tan t
- (c)
-tan t
- (d)
None of the above
Let f:[-5,5]\(\rightarrow\)R be a differentiable function such that f'(x) does not vanish anywhere, then
- (a)
f(-5)>f(5)
- (b)
f(-5)
- (c)
f(-5)=f(5)
- (d)
f(-5)\(\neq\)f(5)
Consider the function f(x) defined by f(x)=x-2|+|x|+|x+2|.Then,
- (a)
f is derivable at x=0,2
- (b)
f is derivable at x=-2,0
- (c)
f is not derivable at x=-2,2
- (d)
f is not derivable at x=-2,0,2
Let the function f defined by \(f(x)=\begin{cases} |1-4x^{ 2 }|,0\le x<1 \\ |{ x }^{ 2 }-2x|,1\le x<2 \end{cases}\)where [.] denotes greatest integer function.Then,
- (a)
f is derivable, \(\forall\ x\epsilon[0,2]\)
- (b)
f is derivable, \(\forall\ x\epsilon[0,2]-\left\{{1\over2},1\right\}\)
- (c)
f is derivable at x=\(1\over2\)
- (d)
None of the above
Let y=f(ax)and f'(sin x)=logex.Then, for \({\pi\over2}
- (a)
\(a^x\ log_e\ a\ log_e(\pi-sin^{-1}\ a^x)\)
- (b)
\(a^x\ log_e\ a\ log_e(\pi+sin^{-1}\ a^x)\)
- (c)
\(a^x\ log_e\ a\ log_e(\sin^{-1}\ a^x)\)
- (d)
None of the above
If f and g are differentiable functions in [0,1] satisfying f(0)=2=g(1), g(0)=0 and f(1)=6 then for some c\(\epsilon\)]o,1[
- (a)
2f'(c)=g'(c)
- (b)
2f'(c)=3g'(c)
- (c)
f'(c)=g'(c)
- (d)
f'(c)=2g'(c)
If function f(x) is differentiable at x=a, then \(\underset{x\rightarrow a}{lim}\)\(x^2\ f(a)-a^2f(x)\over x-a\)is equal to
- (a)
2a f(a)+a2f'(a)
- (b)
-a2 f'(a)
- (c)
a f(a)-a2f'(a)
- (d)
2a f(A)-a2 f'(a)
Let y be implicit function of x defined by x2x-2xx cot y-1=0.Then, y'(1) equals
- (a)
-1
- (b)
1
- (c)
log2
- (d)
-log2
Let \(f(x)=\begin{cases} (x-1)sin\frac { 1 }{ x-1 } ,\quad if\quad x\neq 1 \\ 0,\quad \quad \quad \quad \quad \quad \quad if\quad x=1 \end{cases}\)Then, which one of the following is true?
- (a)
f is differentiable at x=1 but not at x=0
- (b)
f is neither differentiable at x=0 nor at x=1
- (c)
f is differentiable at x=0 and at x=1
- (d)
f is differentiable at x=0 but not at x=1
Consider the function \(f(x)=\begin{cases} { x }^{ 2 }sin\frac { 1 }{ x } ;x\neq 0 \\ 0,\quad \quad otherwise \end{cases}\)Then,
- (a)
f is derivable at x=0
- (b)
f is not derivable at x=0
- (c)
f is derivable at x=0 and f'(0)=0
- (d)
f is derivable at x=0 and f'(0)\(\neq\)0
If y=sin(sin x), then
- (a)
\({d^2y\over dx^2}+tanx.{dy\over dx}+y cos^2x=0\)
- (b)
\({d^2y\over dx^2}-tanx.{dy\over dx}+y cos^2x=0\)
- (c)
\({d^2y\over dx^2}+tanx.{dy\over dx}-y cos^2x=0\)
- (d)
\({d^2y\over dx^2}-tanx.{dy\over dx}-y cos^2x=0\)
If x \(\sqrt { 1+y } +y\sqrt { 1+x } =0,\) then \(\frac { dy }{ dx } \) =
- (a)
\(\frac { x+1 }{ x } \)
- (b)
\(\frac { 1 }{ 1+x } \)
- (c)
\(\frac { -1 }{ (1+x)^{ 2 } } \)
- (d)
\(\frac { x }{ 1+x } \)
If y = tan-1\(\left[ \frac { sinx+cosx }{ cosx-sinx } \right] \)then \(\frac { dy }{ dx } \) is equal to
- (a)
\(\frac { 1 }{ 2 } \)
- (b)
\(\frac { \pi }{ 4 } \)
- (c)
0
- (d)
1
\(\frac { d }{ dx } [sin^{ -1 }(x\sqrt { 1-x } -\sqrt { x } \sqrt { 1-{ x }^{ 2 }) } ]\) is equal to
- (a)
\(\frac { 1 }{ 2\sqrt { x(1-x) } } -\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } \)
- (b)
\(\frac { 1 }{ \sqrt { 1-\{ \sqrt { 1-x } -\sqrt { x(1-{ x }^{ 2 })\} ^{ 2 } } } } \)
- (c)
\(\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } -\frac { 1 }{ 2\sqrt { x(1-x) } } \)
- (d)
\(\frac { 1 }{ \sqrt { x(1-x)(1-x)^{ 2 } } } \)
\(\frac { d }{ dx } \left( tan^{ -1 }\left( \frac { \sqrt { x } -\sqrt { a } }{ 1+\sqrt { xa } } \right) \right) ,x,a>0,\) is
- (a)
\(tan^{ -1 }\sqrt { x } +tan^{ -1 }\sqrt { a } \)
- (b)
\(\frac { 1 }{ 1+x } \)
- (c)
\(\frac { 1 }{ 1+x } +\frac { 1 }{ 1+a } \)
- (d)
\(\frac { 1 }{ 2\sqrt { x } (1+x) } \)
If y=log10x+logey, then \(\frac{dy}{dx}\) is equal to
- (a)
\(\frac { y }{ y-1 } \)
- (b)
\(\frac{y}{x}\)
- (c)
\(\frac { log_{ 10 }e }{ x } \left( \frac { y }{ y-1 } \right) \)
- (d)
None of these
Let f(x) =ex, g(x) = sin-1x and h(x) = f[g(x)],then\(\frac { h'(x) }{ h(x) } \) is equal to
- (a)
\({ e }^{ sin-1 }x\)
- (b)
\(\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } \)
- (c)
sin-1x
- (d)
\(\frac { 1 }{ (1-{ x }^{ 2 }) } \)
Derivative of the function f(x) = log5(log7x),x>7 is
- (a)
\(\frac { 1 }{ x(log5)(log7)(log_{ 7 }x) } \)
- (b)
\(\frac { 1 }{ x(log5)(log7) } \)
- (c)
\(\frac { 1 }{ x(logx) } \)
- (d)
None of these
For what choice of a and b, is the function f(x) = {\(\begin{matrix} { x }^{ 2 }, & x\le c \\ ax+b, & x>c \end{matrix}\) is differentiable at x =c?
- (a)
a = c, b= c
- (b)
a = c, b = -c
- (c)
a =-c2, b= 2c
- (d)
a = 2c, b =-c2
If xy = ex-y, then \(\frac { dy }{ dx } \)is
- (a)
\(\frac { 1+x }{ 1+logx } \)
- (b)
\(\frac { 1-logx }{ 1+logx } \)
- (c)
not defined
- (d)
\(\frac { logx }{ (1+logx)^{ 2 } } \)
If x=a sinθ and y=b cosθ, then \(\frac { { d }^{ 2 }y }{ dx^{ 2 } } \) is equal to
- (a)
\(\frac { a }{ { b }^{ 2 } } \) sec2θ
- (b)
-\(\frac{b}{a}\)sec2θ
- (c)
\(\frac { b }{ { a }^{ 2 } } \)sec3θ
- (d)
-\(\frac { b }{ { a }^{ 2 } } \)sec3θ
The derivative of \({ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \)with respect to \(cos^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 1+x^{ 2 } } \right) is\)
- (a)
-1
- (b)
1
- (c)
2
- (d)
4
If y=tan-1\(\left( \frac { log\left( \frac { e }{ { x }^{ 2 } } \right) }{ logex^{ 2 } } \right) +tan^{ -1 }\left( \frac { 3+2logx }{ 1-6logx } \right) \), then \(\left( \frac { { d }^{ 2 }y }{ dx^{ 2 } } \right) \) is equal to
- (a)
2
- (b)
1
- (c)
0
- (d)
-1
If x = f(i) and y = get(t), then \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \) is equal to
- (a)
\(\frac { g"(t) }{ f"(t) } \)
- (b)
\(\frac { g"(t)f'(t)-g'(t)f"(t) }{ (f'(t){ ) }^{ 3 } } \)
- (c)
\(\frac { g"(t)f'(t)-g'(t)f"(t) }{ (f'(t){ ) }^{ 2 } } \)
- (d)
none of these
Let f(x) satisfy the requirements of Lagrange's mean value theorem in [0,2]. If f(0) =0 and \(f'(x)\le \frac { 1 }{ 2 } \) for all x in [0,2], then
- (a)
|f(x)|\(\le \) 2
- (b)
f(x)\(\le \) 1
- (c)
f(x) = 2x
- (d)
f(x) = 3 for atleast one x in [0, 2]
If (x) is twice differentiable function and f"(0) = a, then \(\lim _{ x\rightarrow 0 }{ \frac { 2f(x)-3f(2x)+f(4x) }{ { x }^{ 2 } } } \) is equal to
- (a)
3a
- (b)
2a
- (c)
5a
- (d)
4a
If f(x)=\(\begin{cases} mx+1 \\ sinx+n \end{cases}\begin{matrix} ifx\le \frac { \pi }{ 2 } \\ ifx>\frac { \pi }{ 2 } \end{matrix}\) is continous at x=\(\frac{\pi}{2}\), then
- (a)
m=1, n=0
- (b)
m=\(\frac{n\pi}{2}\)+1
- (c)
n=\(\frac{m\pi}{2}\)
- (d)
m=n=\(\frac{\pi}{2}\)
Statement-I: If exy + log(xy) + cos(xy) + 5 = 0, then \(\frac { dy }{ dx } =-\frac { y }{ x } \)
Statement-II :\(\frac { dy }{ dx } (xy)=0\Rightarrow \frac { dy }{ dx } =\frac { -y }{ x } \)
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -I.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -I.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.