JEE Main Mathematics - Continuity
Exam Duration: 60 Mins Total Questions : 30
The value of f(o), so that the function \(f(x)=\frac { 2x-sinx^{ -1 } }{ 2x+tan^{ -1 }x } \)is continuous at each point in its domain and is equal to
- (a)
\(1\over3\)
- (b)
\(-{1\over3}\)
- (c)
\(2\over3\)
- (d)
\(-2\over3\)
Which of the following functions have finite number of point of discontinuity in R (where, [.]represents greatest integer function)?
- (a)
\(|x|\over x\)
- (b)
x.|x|
- (c)
tan x
- (d)
sin [n\(\pi\)x]
The values of p and q for which the function \(f(x)=\begin{cases} \frac { sin(p+1)xsinx }{ x } ,x<0 \\ q,\quad \quad \quad \quad x=0 \\ \frac { \sqrt { x+{ x }^{ 2 } } -\sqrt { x } }{ { x }^{ 3/2 } } ,x>0 \end{cases}\) is continuous for all x in R, are
- (a)
\(p={5\over2},q={1\over2}\)
- (b)
\(p=-{3\over2},q={1\over2}\)
- (c)
\(p={1\over2},q={3\over2}\)
- (d)
\(p={1\over2},q=-{3\over2}\)
Define F(x) as the product of two real functions \(f_2(x)=x,\) \(x\epsilon R\) and \(f_2(x)=\begin{cases} sin\frac { 1 }{ x } ,\quad if\quad x\neq 0 \\ 0,\quad \quad \quad if\quad x=0 \end{cases}\)as \(F(x)=\begin{cases} { f }_{ 1 }(x).{ f }_{ 2 }(x),\quad if\quad x\neq 0 \\ 0,\quad \quad \quad \quad \quad if\quad x=0 \end{cases}\)
Statement I F(x) is continuous on R.
Statement II f1(x) and f2(x) are continuous on R.
- (a)
Statement I is true, Statement II is true; Statement II is the correct explanation for Statement I
- (b)
Statement I is true, Statement II is true; Statement II is not the correct explanation for Statement I
- (c)
Statement I is true, Statement I is true, Statement II is false
- (d)
Statement I is false, Statement I is true, Statement II is true
Suppose a, b, c, d, be non-zero real numbers and ab>0,
and \(\int _{ 0 }^{ 1 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=\int _{ 0 }^{ 2 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=0 } } \\ \)
If \(F(x)=\int { f(x)dx=\int _{ 0 }^{ 1 }{ f(x) } dx } =\int _{ 0 }^{ 2 }{ f(x) } dx=0\), then in which of the following can Rolle's theorem can be applied for F(x), where
\(f(x)=(1+{ e }^{ x^{ 2 } })\)(ax3+bx2+cx+d)
- (a)
[0,1]
- (b)
[0,2]
- (c)
[1,2]
- (d)
all of these
The set of all values of a for which the function
\(f(x)=\left( \frac { \sqrt { a+4 } }{ 1-a } -1 \right) \)x5-3x+log5 decreases for all real x is
- (a)
(-∞,∞)
- (b)
\(\left[ -4,\frac { 3-\sqrt { 21 } }{ 2 } \right] \cup (1,\infty )\)
- (c)
\(\left( -3,5-\frac { \sqrt { 27 } }{ 2 } \right) \cup (2,\infty )\)
- (d)
[1,∞)
If \(f(x)=\frac { a\quad sinx+b\quad cosx }{ c\quad sinx+d\quad cosx } \) is decreasing for all x, then
- (a)
ad-bc>0
- (b)
ad-bc<0
- (c)
ab-cd>0
- (d)
ab-cd<0
If a<0, the function f(x)=eax+e-ax is a monotonically decreasing function for values of x given by
- (a)
x>0
- (b)
x<0
- (c)
x>1
- (d)
x<1
The function f(x)=xx decreases on the interval
- (a)
(0,e)
- (b)
(0,1)
- (c)
(0,1/e)
- (d)
none of these
Let \(f(x)=\int { e^{ x } } \) (x-1)(x-2)dx, then f decreases in the interval
- (a)
(-∾,-2)
- (b)
(-2,-1)
- (c)
(1,2)
- (d)
(2,∾)
For x>1, y=logex satisfies the inequality
- (a)
x-1>y
- (b)
x2-1>y
- (c)
y>x-1
- (d)
\(\frac { x-1 }{ x } <y\)
if \(f(x)=\begin{cases} 1, x\ is\ rational \\ 2, x\ is\ rational \end{cases}\)then
- (a)
f(x) is continuous in R ~ I
- (b)
f(x) is continuous in R ~ Q
- (c)
f(x) is continuous in R but not differentiable in R
- (d)
f(x) is neither continuous nor differentiable in R
If \(f(x)-\int_{-1}^x|t|dt, x\ge-1,\) then
- (a)
f and f' are continuous for x + 1>
- (b)
f is continuous but f' is not so for x + 1> a
- (c)
f and f' are continuous at x = a
- (d)
f is continuous at x = a but f' is not so
The points of discontinuity of the function \(f(x)=\underset{n\rightarrow\infty}{Lim}{(2sin\ x)^{2n}\over 3^n-(2cos\ x)^{2n}}\)are given by
- (a)
\(r\pi\pm{\pi\over 12},r\epsilon I\)
- (b)
\(r\pi\pm{\pi\over 6},r\epsilon I\)
- (c)
\(r\pi\pm{\pi\over 3},r\epsilon I\)
- (d)
none of these
Let f(x) = [cos x + sin xl.O < x < 21t, where [x] denotes the greatest integer less than or equal to x. The number of points of discontinuity of f(x) is
- (a)
6
- (b)
5
- (c)
4
- (d)
3
The value of p for which the function \(f(x)={(4^x-1)^3\over sin(x/p)in\left(1+{x^2\over3}\right)}\)= 12 (ln 4)3, x = 0 may be continuous at x = 0 is
- (a)
1
- (b)
4
- (c)
3
- (d)
2
If f(x) = [√2 sin x], where [x] represents the greatest integer function < x then
- (a)
f(x) is periodic
- (b)
maximum value of f(x) is 1 in the interval [- 2π, 2π]
- (c)
f(x) is discontinuous at \(x={n\pi\over 2}+{\pi\over 4}, n\epsilon I\)
- (d)
f(x) is differentiable at x = n π,n ∈ I
The function f(x) = |x2 - 3x + 2| + cos |x| is not differentiable at x is equal to
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
Which of the following functions are differentiable in (-1,2)?
- (a)
\(\int_x^{2x}(log\ x)^2dx\)
- (b)
\(\int_x^{2x}{sin\ x\over x}2dx\)
- (c)
\(\int_0^{x}{1-t+t^2\over 1+t+t^2}2dx\)
- (d)
none of these
Let \(f(x)=\begin{cases} {a(1-x\ sin\ x)+b\ cos\ x+5\over x^2}, x<0\\ 3, x=0\\ \left\{1+\left(cx+dx^3\over x^2\right)\right\}^{1/x}, x>0 \end{cases}\) If is continuous at x = 0
The value of c is
- (a)
2
- (b)
3
- (c)
0
- (d)
None of these
Let \(f(x)=\begin{cases} {a(1-x\ sin\ x)+b\ cos\ x+5\over x^2}, x<0\\ 3, x=0\\ \left\{1+\left(cx+dx^3\over x^2\right)\right\}^{1/x}, x>0 \end{cases}\) If is continuous at x = 0
The value of ed is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Let \(f(x)=\begin{cases}x+a, x<0\\ |x-1|, x\ge0 \end{cases}\)and \(g(x)=\begin{cases}x+1, x<0\\ (x1)^2+b, x\ge0 \end{cases}\) Where, n is odd natural number> 1and f' (0) > 0 where a and b are non-negative real numbers.
The value of b, if (gof) x is continuous for all real x, is
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
For these values of a and b, (gof)x ∀ x ∈ ( - 1, 1) is
- (a)
even
- (b)
odd
- (c)
neither even nor odd
- (d)
none of these
Let \(f(x)=\begin{cases}x+a, x<0\\ |x-1|, x\ge0 \end{cases}\)and \(g(x)=\begin{cases}x+1, x<0\\ (x1)^2+b, x\ge0 \end{cases}\) Where, n is odd natural number> 1and f' (0) > 0 where a and b are non-negative real numbers.
For these values of a and b, (g0f)x is
- (a)
differentiable at x = - 1
- (b)
differentiable at x = 0
- (c)
differentiable at x = 1
- (d)
non-differentiable at x = 2
If l' (x) = g(x) (x - a)2, where g(a)≠0 arid g is continuous at x = a, then:
- (a)
f is increasing near a if g(a) > 0
- (b)
f is increasing near a if g(a) < 0
- (c)
f is decreasing near a if g(a) > 0
- (d)
f is decreasing near a if g(a) < 0
If f(x) = [x] + [x + 1/3] + [x + 2/3], then ([.] denotes the greatest integer function)
- (a)
f(x) is discontinuous at x = 1, 10, 15
- (b)
f(x) is continuous at x = n/3, where n is any integer
- (c)
\(\int_0^{2/3}f(x)dx=1/3\)
- (d)
\(\underset{x\rightarrow2/3}{lim}f(x)=2\)
Let f(x) = \(=\frac { In(1+ax)-In(1-bx) }{ x } ,x\neq 0.\) If f(x) is continuous at x =0, then f(0) =
- (a)
a - b
- (b)
a + b
- (c)
b - a
- (d)
In a + In b
Let F(x) = { \(\begin{matrix} x+a\sqrt { 2 } sinx,0\le x<\frac { \pi }{ 4 } \\ 2xcotx+b,\frac { \pi }{ 4 } \le x\le \frac { \pi }{ 2 } \\ acos2x-bsinx,\frac { \pi }{ 2 } <x\le \pi \end{matrix}\)
be continuous in [0,\(\pi \)], then a+ b =
- (a)
\(\frac { \pi }{ 12 } \)
- (b)
\(\frac { \pi }{ 6 } \)
- (c)
\(\frac { \pi }{ 4 } \)
- (d)
\(\frac { \pi }{ 3 } \)