Fractional Part Function

Q.

Fill in the blank:

$\_\_\_\_\_÷48=-1$

Q.

If $\left[x\right]$ stands for the greatest integer function then the value of $\left[\frac{1}{5}+\frac{1}{1000}+\frac{1}{5}+\frac{1}{2000}+......+\frac{1}{5}+\frac{999}{1000}\right]$ is:

1. $199$

2. $201$

3. $202$

4. $200$

Q.

Write the following numbers in the form $\frac{\mathrm{p}}{\mathrm{q}}$ where $\mathrm{q}$ is a positive integer.

$4\frac{2}{5}$

Q. Let the solution of the equation $5\left\{x\right\}=2\left[x\right]+x$ be $a/4.$ Then the value of $a$ is where $\{.\}$ and $[.]$ represents fractional part function and greatest integer function respectively.

Q.

Write the converse of a conditional statement.

"If Left hand limit = Right hand limit, then we say that limit exists".

(i.e. ${lim}_{x\to a}f\left(x\right)=f\left(a\right)$)

Q. Let the solution of the equation $5\left\{x\right\}=2\left[x\right]+x$ be $a/4.$ Then the value of $a$ is where $\{.\}$ and $[.]$ represents fractional part function and greatest integer function respectively.

Q.

Add or subtract.

$-0.\overline{63}+\frac{5}{99}$.

Q.

Convert the following into vulgar fractions: $0.\overline{56}$

Q.

Add or subtract.

$\frac{11}{6}-0.\overline{27}$.

Q.

Convert the following into vulgar fractions: $0.\overline{4}$

Q. The value of {0.23} + {-0.23} + {-1} + {0} + {2.4} + {-2.4} is equal to

1. \N
2. -1.2
3. 1
4. 2

Q. The domain of the function $f\left(x\right)=\frac{1}{1-\left\{x\right\}}$ is
(where $\{.\}$ denotes the fractional part of $x$)

1. $\varphi$
2. $[0,1)$
3. $R$
4. $R-Z$

Q. The total number of solutions of $[x{]}^2=x+2(x\},$ where [.] and {.} denote the greatest integer function and the fractional part function, respectively, is equal to

1. 2
2. 3
3. 4
4. 6

Q. Find the value of {0.23} + {-0.23} + {-1} + {0} + {2.4} + {-2.4}
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Q.

How many real numbers satisfy the relation [x] = $\frac{3}{2}$ {x}.

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Q. The domain of the function $f\left(x\right)=\frac{1}{[x{]}^2-7[x]+10}$ is
(where $[.]$ denotes the greatest integer function)

1. $R$
2. $R-[5,6)$
3. $R-[2,3)$
4. $R-[2,3)\cup [5,6)$

Q. The domain of $f\left(x\right)=\sqrt{{\log }_x\left\{x\right\}}$ is
$\left(\right\{x\}$ represents the fractional part of $x)$

1. $R$
2. $[0,1]$
3. $(0,1)$
4. $(0,1]$

Q.

$Therangeofthefunctionf\left(x\right)=\frac{x+3}{|x+3|},x\ne -3is$

1. R-{-3}

2. all positive integers

3. {-1, 1}

4. {3, -3}

Q.

Which of the following functions are identical to f(x) =
$f\left(n\right)=\{\begin{array}{cc}x,& 1\le x<2\\ x^2,& 2\le x<3\end{array}$

(1 $\le$ x < 3)

1. 

2. 

3. 

4. 

Q. $f:R\to R$ is a function defined by $f\left(x\right)=3\left\{x\right\}-2|x|.$ Then the value of $f(-0.7)-f(-2)$ is
(Here, $\{.\}$ denotes the fractional part function)

Q. The number of real numbers that satisfy the relation $\left[x\right]=\frac{3}{2}\left\{x\right\}$ is

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

Q.

The functions that is identical to f(x) =
$f\left(x\right)=\{\begin{array}{cc}x,& 1\le x<2\\ x^2,& 2\le x<3\end{array}$

for (1 $\le$ x < 3) is

1. $[x{]}^x$
2. $x^{\left[x\right]}$
3. $x^2$
4. $x^3$

Q. Find the value of {0.23} + {-0.23} + {-1} + {0} + {2.4} + {-2.4}
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Q. This section contains 1 Assertion-Reason type question, which has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (a), (b), (c) तथा (d) दिये गये हैं, जिनमें से केवल एक सही है।

A : f : R → R, $f\left(x\right)=x^2\log (|x|+1)$ is an into function.
A : f : R → R, $f\left(x\right)=x^2\log (|x|+1)$ एक अन्तर्क्षेपी फलन है।

R : $f\left(x\right)=x^2\log (|x|+1)$ is a continuous even function.
R : $f\left(x\right)=x^2\log (|x|+1)$ एक सतत सम फलन है।

1. Both (A) and (R) are true and (R) is the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
3. (A) is true but (R) is false
   
   (A) सही है लेकिन (R) गलत है
4. (A) is false but (R) is true
   
   (A) गलत है लेकिन (R) सही है

Q. Number of integers in range of $x$ which satisfies the following

Q.

How many real numbers satisfy the relation [x] = $\frac{3}{2}$ {x}.

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Q. Consider $f\left(x\right)=\{x+[{\log }_2(2+x)\left]\right\}+\{x+[{\log }_2(2+x^2)\left]\right\}+...+\{x+[{\log }_2(2+x^{10})\left]\right\},$ then correct statement is
(where {} and [] denotes the fractional part function and greatest integer function respectively)

1. $\left[f\right(e\left)\right]=7$
2. $f\left(\pi \right)=20\pi -60$
3. Number of solutions of the equation $f\left(x\right)=x$ is $9$
4. Number of solutions of the equation $f\left(x\right)=x$ is $10$