Graphical Interpretation of Continuity

Q. For ${\sin }^{-1}\left[x\right]$, where $[.]$ is greatest integer function, which of the following is/are true?

1. Domain is $[-1,2)$
2. Product of all integral values of $x$ in the domain is $-1$
3. Sum of all integral values of $x$ in the domain is $0$
4. Domain is $\{-1,0,1\}$

Q. Let $\text{sgn}\left(y\right)$ and $\left\{y\right\}$ denote signum function of $y$ and fractional part function of $y$ respectively.
Which of the following functions is (are) bijective?

1. $f:(-\infty ,0]\to (0,\frac{\pi }{2}]$defined by $f\left(x\right)={\sin }^{-1}\left(e^x\right)$
2. $f:[-1,1]\to \{-1,0,1\}$ defined by $f\left(x\right)=\text{sgn}({\sin }^{-1}|x|-{\cos }^{-1}|x|)$
3. $f:[-3,0]\to [\cos 3,1]$ defined by $f\left(x\right)=\cos x$
4. $f:R-Z\to R$ defined by $f\left(x\right)=\ln \left\{x\right\}$

Q. If $a,b$ and $c$ are the integers that satisfy the domain of ${\sin }^{-1}\left[x\right]$, where $[.]$ is the greatest integer function, then

1. $a+b+c=0$
2. $|a+b+c|=1$
3. $abc=0$
4. $|abc|=1$

Q.

Let R be the feasible region (convex polygon) for a linear programming problem and let,

Z = + be the objective function. When Z has an optimal value (maximum or minimum), where the variables and are subject to constraints described by linear inequalities, this optimal value must occur at ____________of the feasible region.

1. corner point

2. either in the interior or on the boundary line

3. any point in the interior

4. any point on the bounadary line

Q. The relation $R$ defined in $A=\{1,2,3,\}$ by $aR$ $b$ if $|a^2-b^2|\le 5$. Which of the following is false?

1. $R=\left\{\right(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(3,2\left)\right\}$
2. $R^{-1}=R$
3. Domain of $R=\{1,2,3\}$
4. Range of $R=\left\{5\right\}$

Q. For any angle $x\in [-\frac{\pi }{4},\frac{\pi }{3}];\cot x\in$

1. $(-1,\frac{1}{\sqrt{3}})$
2. $(-\infty ,-1]\cup [\frac{1}{\sqrt{3}},\infty )$
3. $[-\infty ,\frac{1}{\sqrt{3}})$
4. $(-\infty ,-1)$

Q. Given the graph of $y=2\tan (x+\frac{\pi }{4})\forall y\in R.$
Then, select the correct statement(s).

1. $\text{For}x\in [-2\pi ,2\pi ],y=0\text{at:}$
   $x=\frac{-5\pi }{4},\frac{-\pi }{4},\frac{3\pi }{4},\frac{7\pi }{4}$
2. Domain: $R-(8n+1)\pi /4;n\in Z$
3. Domain: $R-(4n+1)\pi /4;n\in Z$
4. $\text{For}x\in [-2\pi ,2\pi ],y=0\text{at:}$
   $x=\frac{-5\pi }{4},\frac{-\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4}$

Q.

Among which of these points is $f\left(x\right)=\tan x$ discontinuous?

1. $\frac{-\pi }{4}$

2. 0

3. $\frac{\pi }{4}$

4. $\frac{\pi }{2}$

Q.

Among which of these points is $f\left(x\right)=\tan x$ discontinuous?

1. $\frac{-\pi }{4}$

2. $\frac{\pi }{4}$

3. $\frac{\pi }{2}$

4. 0

Q. Let f : R $\to$ R be a function. Define g:R $\to$ by g(x) = |f(x)| for all x. Then g is

1. Onto if f is onto
2. One-one if f is one-one
3. Continuous if f is continuous
4. None of these

Q. Let f : R $\to$ R be a function. Define g:R $\to$ by g(x) = |f(x)| for all x. Then g is

1. Onto if f is onto
2. One-one if f is one-one
3. Continuous if f is continuous
4. None of these

Q. In which of the following interval(s) of $x,$
$\cos x>0\forall x\in (\pi ,6\pi )$

1. $(\frac{7\pi }{2},\frac{9\pi }{2})$
2. $(\frac{11\pi }{2},6\pi )$
3. $(\frac{3\pi }{2},\frac{5\pi }{2})\cup (\frac{7\pi }{2},\frac{9\pi }{2})\cup (\frac{11\pi }{2},6\pi )$
4. $(\frac{3\pi }{2},\frac{5\pi }{2})$

Q.

The function $f\left(x\right)=\tan x$ where $x\in (\frac{-\pi }{4},\frac{\pi }{4})$.

1. is continuous

2. is discontinuous

3. is increasing

4. is decreasing

Q.

The function $f\left(x\right)=\tan x$ where $x\in (\frac{-\pi }{4},\frac{\pi }{4})$.

1. is continuous

2. is discontinuous

3. is increasing

4. is decreasing

Q. Let $f\left(x\right)={\cot }^{–1}\left(\sin x\right)+{\sin }^{–1}(x–\left\{x\right\})$, where $\left\{x\right\}$ denotes the fractional part function. Then

1. $f$ is one-one
2. $D_f=[-1,2)$
3. $f$ is many one
4. $R_f=\{\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}\}$

Q. Let $f:N\to N$ be defined by $f\left(x\right)=x^2+x+1,x\in N$. Then $f$ is

1. many-one onto
2. one-one but not onto
3. one-one onto
4. none of these

Q. Let $f\left(x\right)={\sin }^{-1}|\sin x|+{\cos }^{-1}\left(\cos x\right),x\in R.$ Which of the following statements is/are TRUE?

1. $f\left(f\right(3\left)\right)=\pi$
2. $f\left(x\right)$ is periodic with fundamental period $2\pi$.
3. $f\left(x\right)$ is neither even nor odd.
4. Range of $f\left(x\right)$ is $[0,2\pi ]$.

Q. If $f\left(x\right)=[9^x-3^x+1],x\in (-\infty ,1)$, then number of integers in the range of $f\left(x\right)$ is (where $[.]$ denotes greatest integer function)

Q. For the curve $f\left(x\right)=\frac{1}{1+x^2}$, let two points on it be $A(\alpha ,f(\alpha \left)\right),B(-\frac{1}{\alpha },f(-\frac{1}{\alpha }))(\alpha >0)$. Find the minimum area bounded by the line segments OA, OB and $f\left(x\right)$, where 'O' is the origin.

1. $\frac{(\pi -1)}{2}$
2. $\frac{\pi }{2}$
3. $\frac{(\pi -2)}{2}$
4. Maximum area is always infinite

Q. $\text{Let}f\left(x\right)=\frac{\sin \pi x}{x^2},x>0.$
$\text{Let}{x}_1<x_2<x_3<...<x_n<...$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<...<y_n<...$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?

1. $x_1<y_1$
2. $x_{n+1}-x_n>2$ for every $n$
3. $x_n\in (2n,2n+\frac{1}{2})$ for every $n$
4. $|x_n-y_n|>1$ for every $n$

Q. ntIntegrate the following with respect to x.n ntx/(x+7x+12)n

Q.

Find the number of discontinuities of the given function between x = 0 and x =2.

___

Q. The number of points of discontinuity and non-differentiability of the function $f\left(x\right)$ in its domain are respectively

1. 2, 1
2. 2, 2
3. 1, 1
4. 1, 2

Q. For the interval $[-2\pi ,2\pi ],\cos x>0$ in the interval

1. $(-2\pi ,-\pi )\cup (\pi ,2\pi )$
2. $(-2\pi ,-\frac{3\pi }{2})\cup (-\frac{\pi }{2},\frac{\pi }{2})\cup (\frac{3\pi }{2},2\pi )$
3. $[-2\pi ,-\frac{\pi }{2}]\cup [\frac{\pi }{2},2\pi ]$
4. $[-\frac{\pi }{2},\frac{\pi }{2}]$

Q. Verify associativity for the following three mappings : f : N → Z₀ (the set of non-zero integers), g : Z₀ → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = e^x.

Q. If $f\left(x\right)=\left[\text{sin}\pi x\right]$ in $(-\frac{1}{2},\frac{1}{2}),$ then which of the following(s) is/are correct:
where $\left[x\right]$ denotes the greatest integer less than or equal to $x.$

1. $f\left(x\right)$ is continuous at $x=0$
2. $f\left(x\right)$ is continuous in $(-1/2,0)$
3. $f\left(x\right)$ is continuous in $(-1/2,1/2)$
4. $f\left(x\right)$ is continuous in $(0,1/2)$

Q. Number of points of extremum of the function $f\left(x\right)=|||x-1|-2|-3|,xϵ[0,\infty )$ is

Q.

The function $f\left(x\right)=\tan x$ where $x\in (\frac{-\pi }{4},\frac{\pi }{4})$.

1. is continuous

2. is discontinuous

3. is increasing

4. is decreasing

Q. $\text{Let}f\left(x\right)=\frac{\sin \pi x}{x^2},x>0.$
$\text{Let}{x}_1<x_2<x_3<...<x_n<...$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<...<y_n<...$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?

1. $x_{n+1}-x_n>2$ for every $n$
2. $|x_n-y_n|>1$ for every $n$
3. $x_1<y_1$
4. $x_n\in (2n,2n+\frac{1}{2})$ for every $n$