Mathematics - Continuity
Exam Duration: 45 Mins Total Questions : 30
\(f(x)=\begin{cases} |x-a|sin\frac { 1 }{ x-a } ,if\quad x\neq 2 \\ 0,\quad \quad \quad \quad \quad \quad if\quad x=a \end{cases}\)
- (a)
continuous at x=a
- (b)
discontinuous at x=a
- (c)
discontinuous for all \(x \epsilon R\)
- (d)
None of the above
For what value of k, the function \(f(x)=\begin{cases} \frac { \sqrt { 1+kx } -\sqrt { 1-kx } }{ x } ,if-1\le x<0 \\ \frac { 2x+1 }{ x-2 } ,\quad \quad \quad \quad if\quad 0\le x\le 1 \end{cases}\)is continuous at x=0?
- (a)
\(1\over2\)
- (b)
1
- (c)
\(-{3\over2}\)
- (d)
\(-{1\over2}\)
If \(f(x)=\frac { { \left( { 3 }^{ x }-1 \right) }^{ 2 } }{ sinx.log_{ e }(1+x) } ,x\neq 0\) is continuous at x=o, then f(o) is
- (a)
loge3
- (b)
2loge3
- (c)
(loge3)2
- (d)
None of the above
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]
For what value of k, the function\(f(x)=\begin{cases} \frac { x }{ |x|+2{ x }^{ 2 } } ,x\neq 0 \\ k,\quad \quad \quad x=0 \end{cases}\) continuous at x=0?
- (a)
\(1\over2\)
- (b)
1
- (c)
\(3\over2\)
- (d)
No value
The number of points at which the function \(f(x)={1\over x-[x]}\)([.] denotes the greatest integer function) is not continuous, is
- (a)
1
- (b)
2
- (c)
3
- (d)
None of these
Suppose a, b, c, d, be non-zero real numbers and ab>0,
and \(\int _{ 0 }^{ 1 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=\int _{ 0 }^{ 2 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=0 } } \\ \)
Which statement is correct?
- (a)
ac>0
- (b)
ac<0
- (c)
ad<0
- (d)
None of these
Suppose a, b, c, d, be non-zero real numbers and ab>0,
and \(\int _{ 0 }^{ 1 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=\int _{ 0 }^{ 2 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=0 } } \\ \)
Rolle's theorem can be applied for ax3+bx2+cx+d in the interval
- (a)
[0,1]
- (b)
[0,2]
- (c)
[1,2]
- (d)
none 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,∞)
On which of the following intervals is the function x100+sinx-1 decreasing?
- (a)
(0,\(\pi \)/2)
- (b)
(0,1)
- (c)
\(\left( \frac { \pi }{ 2 } ,\pi \right) \)
- (d)
None of these
I the function f(x)=cos\(\left| x \right| \)-2 ax+b increases along the entire number scale, the range of value of a is given by
- (a)
a≤b
- (b)
a=b/2
- (c)
a≤-1/2
- (d)
a≥-3/2
Let f'(x)>0 and g'(x)<0 for all x \(\in \) R, then
- (a)
f{g(x)} > f(g(x+1)}
- (b)
f{g(x)} > f(g(x-1)}
- (c)
g{f(x)} > g{f(x+1)}
- (d)
g{f(x)} > g{f(x-1)}
Let the function f(x)=sinx+cosx, be defined in [0,\(\pi \)], then f(x)
- (a)
increases in \((\pi /4,\pi /2)\)
- (b)
decreases in \((\pi /4,5\pi /4)\)
- (c)
increases in \(\left[ 0,\frac { \pi }{ 4 } )\cup (\frac { 5\pi }{ 4 } ,2\pi \right] \)
- (d)
decreases in \(\left[ 0,\frac { \pi }{ 4 } )\cup (\frac { \pi }{ 2 } ,2\pi \right] \)
Let f be a function satisfying f(x + y) = f(x) + f(y) and f(x) = X2 g(x) for all x and y, where g(x) is a continuous function, then f' (x) is equal to
- (a)
g'(x)
- (b)
g(0)
- (c)
g(0)+g'(x)
- (d)
0
If f(x) is a twice differentiable function, then between two consecutive roots of the equation f' (x) = 0, there exists
- (a)
at least one root of f(x) = a
- (b)
at most one root of f(x) = a
- (c)
exactly one root of f(x) = a
- (d)
at most one root of f"(x) = a
The function defined by f(x) = (-1)[x3]([.] denotes greatest integer function) satisfies
- (a)
discontinuous for x = n1/3, where n is any integer
- (b)
f(3/2) = 1
- (c)
f' (x) = 0 for - 1 < x < 1
- (d)
none of the above
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 jump of the function at the point of the discontinuity of the function \(f(x)={1-k^{1/x}\over 1+k^{1/x}}(k>0)\)is
- (a)
4
- (b)
2
- (c)
3
- (d)
none of these
If \(f(x)=\begin{cases}[cos\ \pi x],x<1\\ |x-2|, 1\le x\le2\end{cases}\)([-] denotes the greatest integer function), then f(x) is
- (a)
continuous and non-differentiable at x = -1 and x = 1
- (b)
continuous and differentiable at x = 0
- (c)
discontinuous at x = 1/2
- (d)
continuous but not differentiable at x = 2
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
Let \(f(x)={-tan\ x\over 4x-\pi},x\neq\pi/4\ and\ x\epsilon[0,\pi/2)=\lambda,x=\pi/4\) If f(x) is continuous in (0, π/2], then ⋋ is
- (a)
1
- (b)
1/2
- (c)
-1/2
- (d)
none of these
Let f: R ⇒ R be a differential function satisfying \(f\left(x+y\over 3\right)={2+f(x)+f(y)\over 3}\) for all real x and y and f' (2) = 2.
The value of f(1) is
- (a)
0
- (b)
2
- (c)
4
- (d)
none of these
Let f(x) be a real valued function not identically zero such that
f(x + yn) = f(x) + {f(y)}n∀x, y ∈ R
The value of f(2) is
- (a)
0
- (b)
-1
- (c)
2
- (d)
none of these
Let f(x) be a real valued function not identically zero such that
f(x + yn) = f(x) + {f(y)}n∀x, y ∈ R
The function f(x) 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.
The value of a, if (gof)x is continuous for all real x, is
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
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 f(x) = tan-1 cot x, then
- (a)
f(x) is periodic with period π
- (b)
x) is discontinuous at x = π/2, 3π/2
- (c)
f(x) is not differentiable at x = π, 99π, 100π
- (d)
f(x) = - 1, for 2 nπ < x < (2n + 1)π
Give, a real valued function f such that \(f(x)=\begin{cases}{tan^2x\over (x^2-[x])^2}\ for\ x>0\\ 1\ for\ x=0\\ \sqrt{\{x\}cot\{x\}}\ for\ x<0\end{cases}\)where, [x] is the integral part and {x} is the fractional part of x, then
- (a)
\(\underset{x\rightarrow0}{lim}f(x)=1\)
- (b)
\(\underset{x\rightarrow0^-}{lim}f(x)=cot\ 1\)
- (c)
\(cot^{-1}\left(\underset{x\rightarrow0}{lim}f(x)\right)^2=1\)
- (d)
f is continuous at x = 0
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