Mathematics - Continuity
Exam Duration: 45 Mins Total Questions : 30
If \(f(x)=\frac { 3x+4tanx }{ x } \) is to be defined to make continuous at x=0, then the defined function should be
- (a)
\(f(x)=\begin{cases} \frac { 3x+4\quad tan\quad x }{ x } ,x\neq 0 \\ 7\quad \quad \quad \quad \quad ,x=0 \end{cases}\)
- (b)
\(f(x)=\begin{cases} \frac { 3x+4\quad tan\quad x }{ x } ,x\neq 0 \\ 6\quad \quad \quad \quad \quad ,x=0 \end{cases}\)
- (c)
\(f(x)=\begin{cases} \frac { 3x+4\quad tan\quad x }{ x } ,x= 0 \\ 7\quad \quad \quad \quad \quad ,x\neq0 \end{cases}\)
- (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
If \(f(x)={a\ sinx+sin2x\over x^3},x\neq0\)and f(x)is continuous at x=0, then
- (a)
a=2
- (b)
f(0)=1
- (c)
f(0)=-1
- (d)
a=1
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)
ab<0
- (c)
ad>0
- (d)
None of these
Suppose a, b, c, d, be non-zero real numbers and ab>0,
and \(\int _{ 0 }^{ 1 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=\int _{ 0 }^{ 2 }{ (1+e^{ x^{ 2 } })({ ax }^{ 3 }+{ bx }^{ 2 }+cx+d)dx=0 } } \\ \)
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
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 \(f(x)=(ab-b^{ 2 }-2)x+\int _{ 0 }^{ x }{ (cos^{ 4 }\theta } +sin^{ 4 }\theta )d\theta \) is decreasing function of x for all x\(\in \)R and b\(\in \)R, being independent of x, then
- (a)
\(a\in (0,\sqrt { 6 } )\)
- (b)
\(a\in (-\sqrt { 6 } ,\sqrt { 6 } )\)
- (c)
\(a\in (-\sqrt { 6 } ,0)\)
- (d)
none of these
For all x\(\in \)(0,1)
- (a)
ex<1+x
- (b)
loge(1+x)<x
- (c)
sin x>x
- (d)
loge x>x
Which of the following functions are decreasing on (0,\(\pi \) /2)?
- (a)
cos x
- (b)
cos 2x
- (c)
cos 3x
- (d)
tan x
If ф(x)=f(x)+f(2a-x) and f''(x)>0, a>0, 0≤x≤2a, then
- (a)
ф(x) increases in (a, 2a)
- (b)
ф(x) increases in (a, a)
- (c)
ф(x) decreases in (a, 2a)
- (d)
ф(x) decreases in (a, a)
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) 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
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
\(f(x)=\begin{cases}X-[X],for2n\le x<2n+1,n\epsilon N,\\ where[x] = Integral\ part\ of\ x \ge x \\ {1\over 2}for 211+ 1\le x < 2n+ 2 \end{cases}\)the function
- (a)
is discontinuous at x = 1, 2
- (b)
is periodic with period 1
- (c)
is periodic with period 2
- (d)
\(\int_0^2f(x)dx\)exists
f(x) = 1 + x (sin x) [cosx],O < x < π/2 ([.] denotes the greatest integer function)
- (a)
is continuous in (0, π/2)
- (b)
is strictly decreasing in (0, π/2)
- (c)
is strictly increasing in (0, π/2)
- (d)
has global maximum value 2
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 function f(x) is
- (a)
discontinuous at one point
- (b)
continuous everywhere but non-differentiable at some points
- (c)
discontinuous at three points
- (d)
continuous and differentiable everywhere
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
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
\(f(x)={sin(\pi[x-\pi])\over 1+[x^2]}\)where [.] denotes the greatest integer function. Then f(x) is
- (a)
continuous at integral points
- (b)
continuous everywhere but not differentiable
- (c)
differentiable once but lf' (x), f" (x), ... do not exist
- (d)
differentiable for all x
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
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: R ⟶ R be a function such that \(f\left(x+y\over 3\right)={f(x)+f(y)\over3}\) f(0) = 3 and f'(0) = 3, then
- (a)
\(f(x)\over x\)is differentiable in R
- (b)
f(x) is continuous but not differentiable in R
- (c)
f(x) is continuous in R
- (d)
f(x) is bounded in R
If f(x) = \(\frac { \sqrt { 4+x } -2 }{ x } \), x \(\neq \) 0 be continuous at x =0, then f(0) =
- (a)
\(\frac { 1 }{ 2 } \)
- (b)
\(\frac { 1 }{ 4 } \)
- (c)
2
- (d)
\(\frac { 3 }{ 2 } \)
If f (x) = \(\sqrt { 1+{ cos }^{ 2 }{ (x }^{ 2 }), } \) then the value of \(f'\left( \frac { \sqrt { \pi } }{ 2 } \right) \)is
- (a)
\(\frac { \sqrt { \pi } }{ 6 } \)
- (b)
\(-\left( \sqrt { \frac { \pi }{ 6 } } \right) \)
- (c)
\(\frac { 1 }{ \sqrt { 6 } } \)
- (d)
\(\frac { \pi }{ \sqrt { 6 } } \)