JEE Main Mathematics - Differential Calculus
Exam Duration: 60 Mins Total Questions : 30
The value of the function \(f(x)={ cos }^{ 2 }x+{ cos }^{ 2 }(\frac { \pi }{ 3 } +x)-\cos { x\cos { (\frac { \pi }{ 3 } +x) } } \)is:
- (a)
0
- (b)
\(\frac { 3 } { 4 }\)
- (c)
1
- (d)
\(\frac { 4 } { 5}\)
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 period of the function \(f(x)=\left| sinx \right| \),is
- (a)
\(\frac { \pi } { 2 }\)
- (b)
\(\pi\)
- (c)
\(2\pi\)
- (d)
\(4\pi\)
The real valued function \(f(x)=\frac { { a }^{ x }-1 }{ { x }^{ n }({ a }^{ x }+1) } \)is even; then n equals
- (a)
2
- (b)
\(\frac { 2 } { 3 }\)
- (c)
\(\frac { 1 } { 4 }\)
- (d)
\(-\frac { 1 } { 3 }\)
The domain of the function y(x)given by the equation\({ 2 }^{ x }+{ 2 }^{ y }=2\)is:
- (a)
0<x<1
- (b)
\(0\le x\le 1\)
- (c)
\(-\infty <x<0\)
- (d)
\(-\infty <x<1\)
Domain of definition of the function of the function \(f(x)=\frac { 3 }{ 4-{ x }^{ 2 } } +\log _{ 10 }{ ({ x }^{ 3 }-x) } \)is:
- (a)
(1,2)
- (b)
\((-1,0)\cup (1,2)\)
- (c)
\((1,2)\cup (2,\infty )\)
- (d)
\((-1,0)\cup (1,2)\cup (2,\infty )\)
The domain of the function \(f(x)={ sin }^{ -1 }(\log _{ 2 }{ \frac { { x }^{ 2 } }{ 2 } } )\)is:
- (a)
[-2,2]
- (b)
[-2,-1]\(\cup \)[1,2]
- (c)
[-1,1]~{0}
- (d)
None of these
The domain of the function \(f(x)=\sin { [\log { (\frac { \sqrt { 4-{ x }^{ 2 } } }{ 1-x } ) } ] } \)is:
- (a)
(-2,1)
- (b)
(1,2)
- (c)
(2,1)
- (d)
None of these
Range of function \(f(x)=\frac { 1 }{ 2-\cos { 3x } } \)
- (a)
[-1,1]
- (b)
\([\frac { 1 }{ 3 } ,1]\)
- (c)
\([\frac { 2 }{ 3 } ,1]\)
- (d)
None of these
\(\underset { x->1 }{ lim } (1-x{ )\tan { (\frac { \pi x }{ 2 } ) } }\)equals
- (a)
\(\frac {\pi} { 2 }\)
- (b)
\(\pi\)+2
- (c)
\(\frac { 2 } { \pi }\)
- (d)
None of these
Let f:R->R, such that f(1)=3 and \(f^{ ' }\left( 1 \right) =6\).then \(\underset { x->0 }{ lim } { (\frac { f(1+x) }{ f(1) } ) }^{ 1/x }\)equals
- (a)
1
- (b)
\({ e } ^ { \frac { 1 } { 2 } }\)
- (c)
\({ e } ^ { 2 }\)
- (d)
\({ e } ^ { 3 }\)
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 \(y={ e }^{ ax }\cos { bx } \), then \(\frac { dy }{ dx } \) is equal to
- (a)
\(\frac { { e }^{ ax } }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 } } } \left( a\cos { bx } +b\sin { bx } \right) \)
- (b)
\({ e }^{ ax }\left( a\cos { bx } +b\sin { bx } \right) \)
- (c)
\(\frac { { e }^{ ax } }{ { a }^{ 2 }+{ b }^{ 2 } } \left( a\cos { bx } +b\sin { bx } \right) \)
- (d)
NONE OF THESE
The value of \(\underset { x->0 }{ lim } \frac { \int _{ 0 }^{ { x }^{ 2 } }{ \sec ^{ 2 }{ t\quad dt } } }{ x\quad \sin { x } } \)is:
- (a)
3
- (b)
2
- (c)
1
- (d)
0
If \(f(x)=\sqrt { { x }^{ 2 }+x } +\frac { \tan ^{ 2 }{ \alpha } }{ \sqrt { { x }^{ 2 }+x } } ;\alpha \epsilon (0,\frac { \pi }{ 2 } )\),is always greater than or equal to
- (a)
\(2\tan { \alpha } \)
- (b)
1
- (c)
2
- (d)
\(\sec ^{ 2 }{ \alpha } \)
The function \(\{ f(x)=\frac { \left| x \right| }{ x } ;x\neq 0and=1;x=0\)is discontinuous at
- (a)
x=0
- (b)
x=1
- (c)
x=2
- (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
\(\underset { x->1 }{ lim } \frac { \sqrt { 1-\cos { 2(x-1) } } }{ x-1 } \)
- (a)
exists and is equal to \(\sqrt { 2 }\)
- (b)
exists and is equal to -\(\sqrt { 2 }\)
- (c)
does not exist because (x-1)->0
- (d)
does not exist because left handed limits is not equal to right handed limit
The derivative of the function \(f\left( x \right) =\left| 1nx \right| \) at x = 1 is
- (a)
- 1
- (b)
1
- (c)
0
- (d)
Does not exist
If \(y=a{ e }^{ -kt }\cos { \left( pt+c \right) } \) and \(\frac { { d }^{ 2 }y }{ d{ t }^{ 2 } } +2k\frac { dy }{ dt } +{ n }^{ 2 }y=0\), then n2 equals
- (a)
p2 - k2
- (b)
p2
- (c)
k2
- (d)
p2+ k2
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 normal to the curve \(x=a\left( t+\sin { t } \right) ,y=a\left( 1-\cos { t } \right) ,\left( 0
- (a)
2\(\pi \)
- (b)
\(\pi \)
- (c)
\(\frac { \pi }{ 2 } \)
- (d)
0
The conics, ax2 + by2 = 1 and a1x2 + b1y2 = 1 intersect orthogonally; then
- (a)
\(\frac { 1 }{ a } +\frac { 1 }{ { a }_{ 1 } } =\frac { 1 }{ b } +\frac { 1 }{ { b }_{ 1 } } \)
- (b)
\(\frac { 1 }{ a } -\frac { 1 }{ { a }_{ 1 } } =\frac { 1 }{ b } -\frac { 1 }{ { b }_{ 1 } } \)
- (c)
\(\frac { 1 }{ a } +\frac { 1 }{ b } =\frac { 1 }{ { a }_{ 1 } } -\frac { 1 }{ { b }_{ 1 } } \)
- (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
Let \(h\left( x \right) =f\left( x \right) -{ \left[ f\left( x \right) \right] }^{ 2 }+{ \left[ f\left( x \right) \right] }^{ 3 }\) for every real number x. Then
- (a)
h is increasing whenever fis increasing and decreasing whenever f is decreasing
- (b)
h is decreasing whenever f is increasing
- (c)
h is increasing whenever f is decreasing.
- (d)
nothing can be said in general
If 2a + 3b + 6c = 0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval
- (a)
(0, 1)
- (b)
(1, 2)
- (c)
(2, 3)
- (d)
NONE OF THESE
For \(x\ge 0\), the value of \(f\left( x \right) =1+xIn\left( x+\sqrt { { x }^{ 2 }+1 } \right) -\sqrt { 1+{ x }^{ 2 } } \)
- (a)
\(\le 0\)
- (b)
\(\ge 0\)
- (c)
0
- (d)
Not defined
The real number x when added to its inverse gives the mininum value of the sum at x equal to
- (a)
- 2
- (b)
- 1
- (c)
1
- (d)
2
A circular plate expands under the influence of heat so that its radius increases from 5 cm to 5.06 cm. The approximate increse in the area is
- (a)
0.88 cm2
- (b)
1.88 cm2
- (c)
2.88 cm2
- (d)
NONE OF THESE