Mathematics - Differential Calculus
Exam Duration: 45 Mins Total Questions : 30
The period of the function \(\\ \\ \\ f(x)={ sin }^{ 4 }x+{ cos }^{ 4 }x\)is:
- (a)
\(\frac { \pi } { 4 }\)
- (b)
\(\frac { \pi } { 2 }\)
- (c)
\({ \pi }\)
- (d)
None of these
The domain of the definition of the function \(f(x)=\sqrt { { sin }^{ -1 }(\log _{ 2 }{ x } ) } +\log _{ 2 }{ (\log _{ 3 }{ (\log _{ 4 }{ x)) } } } \)is:
- (a)
[1,2]
- (b)
\([4,\infty ]\)
- (c)
\([1,\infty ]\)
- (d)
None of these
Which of the following functions is inverse of itself:
- (a)
\(f(x)=\frac { 1-x }{ 1+x } \)
- (b)
\(f(x)={ 5 }^{ \log { x } }\)
- (c)
\({ 2 }^{ x }(x-1)\)
- (d)
None of these
For \(x\epsilon R,\underset { x->\infty }{ lim } ({ \frac { x-3 }{ x+2 } ) }^{ x }=\)
- (a)
e
- (b)
\({ e } ^ { -1 }\)
- (c)
\({ e } ^ { -5 }\)
- (d)
\({ e } ^ { 5 }\)
The function \(f\left( x \right) =1+\left| \sin { x } \right| \) is
- (a)
continuous nowhere
- (b)
differentiable nowhere
- (c)
continuous and differentiable everywhere
- (d)
non differentiable at an infinite number of points.
The set of all points where the fuction \(f\left( x \right) =\)\(\frac { x }{ 1+\left| x \right| } \), is differentiable is
- (a)
\(\left( -\infty ,\infty \right) \)
- (b)
\(\left( 0,\infty \right) \sim \left\{ 0 \right\} \)
- (c)
\(\left( -\infty ,0 \right) \cup \left( 0,\infty \right) \)
- (d)
\(\left( 0,\infty \right) \)
The derivative with respect to x of the function \(\tan { ^{ -1 }{ \left( \frac { \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } } \right) } } \) is
- (a)
\(\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } \)
- (b)
\(\frac { 1 }{ \sqrt { 1-{ x }^{ 4 } } } \)
- (c)
\(-\frac { 1 }{ \sqrt { 1-{ x }^{ 4 } } } \)
- (d)
NONE OF THESE
If f(a)=2,g(a)=-1,\(f^{ ' }\left( a \right) =1\)and \(g^{ ' }\left( a \right) =2\),the value of \(\underset { x->a }{ lim } [\frac { f(x)g(a)-f(a)g(x) }{ x-a } ]\)is:
- (a)
5
- (b)
\(\frac { 1 } { 5 }\)
- (c)
\(-\frac { 1 } { 5 }\)
- (d)
-5
The integer n for which \(\underset { x->0 }{ lim } \frac { (cosx-1)(cosx-{ e }^{ x }) }{ { x }^{ n } } \), is a finite nm-zero number is,
- (a)
4
- (b)
3
- (c)
2
- (d)
1
If \(y=\log _{ \cos { x } }{ \sin { x } } \), then \(\frac { dy }{ dx } \) is equal to
- (a)
\(\left( \cot { x } \log { \cos { x } } +\tan { x } \log { \sin { x } } \right) /{ \left( \log { \cos { x } } \right) }^{ 2 }\)
- (b)
\(\left( \tan { x } \log { \cos { x } } +\cot { x } \log { \sin { x } } \right) /{ \left( \log { \cos { x } } \right) }^{ 2 }\)
- (c)
\(\left( \cot { x } \log { \cos { x } } +\tan { x } \log { \sin { x } } \right) /{ \left( \log { \sin { x } } \right) }^{ 2 }\)
- (d)
NONE OF THESE
If \({ x }^{ y }={ e }^{ x-y }\), and \(\frac { dy }{ dx } =\frac { \log { x } }{ D } \) then D is equal to
- (a)
(1+log x)
- (b)
(1 - log x)
- (c)
(1 - log x)2
- (d)
NONE OF THESE
If \(G\left( x \right) =-\sqrt { 25-{ x }^{ 2 } } \), then value of \(\underset { x\rightarrow 1 }{ lim } \left[ \frac { G\left( x \right) -G\left( 1 \right) }{ x-1 } \right] \) is
- (a)
1
- (b)
not defined
- (c)
\(\frac { 1 }{ 2\sqrt { 6 } } \)
- (d)
NONE OF THESE
The derivative of the function \(\sin ^{ -1 }{ \left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) } \) at x = 1 is
- (a)
1
- (b)
-1
- (c)
0
- (d)
NONE OF THESE
A function f(x) is defined as follows \(f(x)=\frac { \sin { (a+1)x+\sin { x } } }{ x } ;forx<0\quad f(x)=c;forx=0=\frac { \sqrt { x+bxz } }{ bx\sqrt { y } } \);for x>0 If f(x) is continuous at x=0, then
- (a)
\(a=\frac { -3 } { 2 },b=0,c=\frac { 1 } { 2 }\)
- (b)
\(a=\frac { -3 } { 2 },b=1,c=\frac { -1 } { 2 }\)
- (c)
\(a=\frac { -3 } { 2 },c=\frac { 1 } { 2 },b\)can have any value
- (d)
None of these
In order that the function \(f(x)={ (x+1) }^{ \cot { x } }\)is continuous at x=0, f(0) must be defined as
- (a)
f(0)=0
- (b)
f(0)=e
- (c)
f(0)=\(\frac { 1} { e }\)
- (d)
None of these
If the function \(g\left( x \right) \) is inverse of \(f\left( x \right) \) and \(f^{ ' }\left( x \right) =\frac { 1 }{ 1+{ x }^{ n } } \) then \(g^{ ' }\left( x \right) \), is
- (a)
\(\frac { 1 }{ 1-{ \left[ g\left( x \right) \right] }^{ n } } \)
- (b)
\(\frac { 1 }{ 1+{ \left[ g\left( x \right) \right] }^{ n } } \)
- (c)
\(\frac { 1 }{ 1+g\left( x \right) } \)
- (d)
\(\frac { 1 }{ 1-g\left( x \right) } \)
If \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } =a{ e }^{ \tan ^{ -1 }{ \left( y/x \right) } }\); a > 0, y (0) > 0, then y''(0) equals
- (a)
\(\frac { a }{ 2 } { e }^{ \pi /2 }\)
- (b)
\(a{ e }^{ \pi /2 }\)
- (c)
\(-\frac { 2 }{ a } { e }^{ -\pi /2 }\)
- (d)
\(-\frac { a }{ 2 } { e }^{ -\pi /2 }\)
The equation of one of the tangents to the curve that is parallel to \(y=\cos { \left( x+y \right) } ,-2\pi \le x\le 2\pi \) the line x + 2y = 0, is
- (a)
x + 2y = 1
- (b)
x + 2y =\(\frac { \pi }{ 2 } \)
- (c)
x + 2y =\(\frac { \pi }{ 4 } \)
- (d)
NONE OF THESE
The chord joining the points where x = p and x = q on the curve y = ax2 + bx + c is parallel to the tangent at the point on the curve whose abscissa is
- (a)
\(\frac{1}{2}\) (p + q)
- (b)
\(\frac{1}{2}\) (p - q)
- (c)
\(\frac{pq}{c}\)
- (d)
NONE OF THESE
The angle of intersection of the curves x2 + y2 = 2a2 and x2 - y2 = a2 is
- (a)
30\(°\)
- (b)
45\(°\)
- (c)
60\(°\)
- (d)
90\(°\)
The equation of the tangents at the origin to the curve y2 = x2 (1 + x) are
- (a)
y = \(\pm \) x
- (b)
x = \(\pm \) y
- (c)
y = \(\pm \) 2x
- (d)
NONE OF THESE
The length of the longest interval in which the function \(3\sin { x } -4\sin ^{ 3 }{ x } \), is increasing, is
- (a)
\(\frac { \pi }{ 3 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\frac { 3\pi }{ 3 } \)
- (d)
\( \pi \)
The value of b for which the function \(f\left( x \right) =\sin { x } -bx+c\), is decreasing in the interval \(\left( -\infty ,\infty \right) \), is given by
- (a)
b < 1
- (b)
b \(\le \) 1
- (c)
b > 1
- (d)
b \(\ge \) 1
The function, \(f\left( x \right) =\cos { \left( \frac { \pi }{ x } \right) } \), is decreasing in the intervel
- (a)
(2k + 1, 2k)
- (b)
\(\left( \frac { 1 }{ 2k+1 } ,2k \right) \)
- (c)
\(\left( \frac { 1 }{ 2k+2 } ,\frac { 1 }{ 2k+1 } \right) \)
- (d)
NONE OF THESE
If \(f\left( x \right) =\frac { x }{ \sin { x } } ,g\left( x \right) =\frac { x }{ \tan { x } } \), when \(0\le x\le 1\), then in this interval
- (a)
both \(f\left( x \right) \) and \(g\left( x \right) \) are increasing functions
- (b)
both \(f\left( x \right) \) and \(g\left( x \right) \) are decreasing functions
- (c)
\(f\left( x \right) \) is an increasing function
- (d)
\(g\left( x \right) \) is an increasing function
The function \(f\left( x \right) \) = x - [x], where [ ] denotes the greatest integer function, on the interval [0, 1), is
- (a)
strictly increasing
- (b)
strictly decreasing
- (c)
neither increasing nor decreasing
- (d)
monotonically increasing
The maximum value of \(\frac { \log { x } }{ x } \), is
- (a)
1
- (b)
\(\frac { 2 } { e } \)
- (c)
e
- (d)
\(\frac { 1 } { e } \)
The function \(f\left( x \right) ={ x }^{ 2 }\log { x } \), in the [1, e] has
- (a)
a point of maximum
- (b)
a point of minimum
- (c)
points of maximum as well as minimum
- (d)
neither a point of maximum nor minimum
The minimum value of \(\left| x \right| +\left| x+\frac { 1 }{ 2 } \right| +\left| x-3 \right| +\left| x-\frac { 5 }{ 2 } \right| \) is
- (a)
0
- (b)
2
- (c)
4
- (d)
6
Let \(P\left( x \right) ={ a }_{ 0 }+{ a }_{ 1 }{ x }^{ 2 }+{ a }_{ 2 }{ x }^{ 4 }+...+{ a }_{ n }{ x }^{ n }\) be a polynomial in a real variable x with 0012<...n. The function P(x) has
- (a)
neither a maximum nor a minimum
- (b)
only one maximum
- (c)
only one minimum
- (d)
NONE OF THESE