Modulus Function

Q. The range of the function $f\left(x\right)=\frac{e^x-e^{|x|}}{e^x+e^{|x|}}$ is

1. $(-\infty ,\infty )$
2. $[0,1)$
3. $(-1,0]$
4. $(-1,1)$

Q. if alpha and beta are the roots of equation ax^2+bx+c=0, then write alpha^5+beta^5 in terms of a, b, c.

Q.

If $f\left(x\right)=\frac{x+1}{x-1},$ show that $f\left[f\right(x\left)\right]=x.$

Q.

If $l\left(x\right)$ is the least integer not less than $x$ and $g\left(x\right)$ is the greatest integer not greater than $x$, then $\underset{x\to e+\mathrm{\pi }}{\lim }\left\{l\left(x\right)+g\left(x\right)\right\}=?$

1. $9$

2. $13$

3. $1$

4. None of these

Q. The values of $k$ for which the equation $||x-2|-5|=k$ has $4$ distinct solutions is

1. $k\in [0,5]$
2. $k\in [0,5)$
3. $k\in (0,5)$
4. $k\in (1,5]$

Q. f(x) = [x], the greatest integer less then or equal to x and k is an integer. Then, ${lim}_{x\to k}f\left(x\right)=$___

1. k
2. k - 1
3. k + 1
4. does not exist

Q. While doing the transformation $f\left(x\right)\to |f\left(x\right)|$, we take the mirror image of the graph lying below x-axis and leave the other parts as it is to get the graph of f(x) from |f(x)|

1. True

2. False

Q. let f(x)=1/root x+|x| then the domain of f is

Q. The number of integers that satisfy the relation $|x-1|\le 2$ is

1. 2
2. 3
3. 4
4. 5

Q. Let $f\left(x\right)$ be a function defined as $f\left(x\right)=a|x|+b.$ If $f\left(6\right)=5$ and $f(-3)=4,$ then which of the following is/are correct?

1. $f\left(x\right)=3|x|+\frac{1}{3}$
2. $ab=1$
3. Domain of $f\left(x\right)$ is $R$
4. Range of $f\left(x\right)=[3,\infty )$

Q. Match the domain of the following functions:

1. $\left(-1,1\right]$
2. $R-\left(n\pi +\frac{\pi }{2},n\in Z\right\}$
3. $\left(-1,1\right]$
4. $\left(-\infty ,+\infty \right)$

Q.

If $f\left(x\right)=\frac{x-1}{x+1},x\ne -1$ then show that $f\left(f\right(x\left)\right)=\frac{-1}{x}$, where $x\ne 0)$.

Q. The number of integers satisfying the condition $|x-3|>5$ and $|x+1|$ $\le$ 4 is

1. 2
2. 3
3. 4
4. 5

Q. If $|ln|x||=|k-1|-4$ has exactly two distinct real roots for $x\in R-\left\{0\right\}$, then the sum of all possible integral values of k is

1. -5
2. -3
3. 2
4. Infinite

Q.

The relation R ={(x, $\sqrt{x}$):x is a natural number less than 100}. Write the relation in roster form. All the elements of Relation are integers.

1. {(1, -1), (4, -2), (9, -3), (16, -4), (25, -5), (36, -6), (49, -7), (64, -8), (81, -9)}

2. {(1, 1), (4, 2), (9, 3), (16, 4), (25, 5), (36, 6), (49, 7), (64, 8), (81, 9)}

3. {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25), (6, 36), (7, 49), (8, 64), (9, 81)}

4. {(-1, 1), (-2, 4), (-3, 9), (-4, 16), (-5, 25), (-6, 36), (-7, 49), (-8, 64), (-9, 81)}

Q.

Identify the function f(x) from the description given below.

1. x - 1 $<f\left(x\right)\le x$

2. Its domain is R and Range is I(set of integers)

3. f(x) = 3 $\Rightarrow 3\le x<4$

4. f(x) = -3 $\Rightarrow -3\le x<-2$

1. Signum function

2. Greatest integer function

3. Absolute value function

4. Least integer function

Q.

How many of the following statements are correct?
1. Graph of $-f\left(x\right)$ is obtained by taking the reflection of $f\left(x\right)$ about x-axis.
2. Graph of $f(-x)$ is obtained by taking the reflection of $f\left(x\right)$ about x-axis.

3. Graph of $-f(-x)$ is obtained by taking the mirror image of the graph of $f\left(x\right)$ about x-axis and then about y-axis.

Q.

How many integers satisfy the relation |x - 1|$\le$ 2 ?

Q.

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

Q.

Find the number of integers satisfying the condition $|x-3|$$>$ 5 and $|x+1|$ $\le$ 4

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Q.

If I < x < I +1 , find [-x] where I is an integer.

Q. Find k so that $\underset{x\to 2}{\lim }f\left(x\right)$ may exist, where $f\left(x\right)=\left\{\begin{array}{ll}2x+3,& x\le 2\\ x+k,& x>2\end{array}\right..$

Q. If Alpha and beta are the roots of quadratic equation ax²+bx+c=0 , then write Alpha ⁵+ beta⁵.

Q.

Write the following as intervals:

(i) {x : x $\in R,-4<x\le 6$}

(ii) {x : x$\in R,-12<x<-10$}

(iii) {x : x $\in R,0\le x<7$}

(iv) {x : x $\in R,3\le x\le 4$}

Q. If x satisfies $\left(x-1\right|+\left(x-2\right|+\left(x-3\right|\ge 6$, then the values of x lie in the range

1. $0\le x\le 4$
2. $x\le -2orx\ge 4$
3. $x\le 0orx\ge 4$
4. $x\le 1orx\ge 4$

Q.

Find the number of integers satisfying the condition |x-3| $>$ 5 and |x + 1 | $\le$ 4

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Q. The domain of the function f(x) = sqrt 7 IxI-x^2-6 is

Q.

Find the number of integers satisfying the condition |x-3| $>$ 5 and |x + 1 | $\le$ 4

__

Q.

How many integers satisfy the relation |x - 1|$\le$ 2

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Q. The integral value(s) of $x$ satisfying $|x^2-9|+|x^2-4|=5$ is/are

1. $-3$
2. $2$
3. $-2$
4. $3$