Absolute Value Function

Q.

Range of the function $f\left(x\right)=\frac{x}{1+x^2}$ is

1. $(-\infty ,\infty )$

2. $[-1,1]$

3. $[\frac{-1}{2},\frac{1}{2}]$

4. $[-\sqrt{2},\sqrt{2}]$

Q. if x is real, then the expression (x^2+34x-71)/(x^2+2x-7) (1) lies between 4 and 7 (2) lies between 5 and 9 (3) Has no value between 4 and 7 (4) Has no value between 5 and 9

Q.

Range of $f\left(x\right)=\frac{x^2+34x-71}{x^2+2x-7}$ is

1. $[5,9]$

2. $(5,9]$

3. $(-\infty ,5]\cup [9,\infty )$

4. $(-\infty ,5)\cup (9,\infty )$

Q. Number of solution(s) of equation $|x-1|-|x-2|=10$ is

1. $0$
2. $1$
3. $2$
4. $4$

Q.

If $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\sqrt{(\mathrm{x}+1)({\mathrm{e}}^{\mathrm{x}}-1)(\mathrm{x}-4)(\mathrm{x}+5)(\mathrm{x}-6)}}$, then the domain of $f\left(x\right)$ is

1. (-∞, -5) ∪ (-1, 4) ∪ (6, 8)

2. (-∞, -5) ∪ (-1, 0) ∪ (0, 4) ∪ (6, ∞)

3. (-5, 1) ∪ (0, 4) ∪ (6, ∞)

4. None of these

Q.

Does a table represent a linear or nonlinear function? Explain.

Q.

If t - 1 and -t -1, t ∈ R, are the roots of (a + 2) $x^2$ + 2ax - 1 = 0 then complete set of values of 'a' is :

1. ( 0, ∞)

2. ∅

3. ( -∞, ∞)

4. ( - ∞, 0)

Q. Graph of $y=|x-2|-5$ is

1. 
2. 
3. 
4. 

Q. If $\left|\frac{(x^2-x-6)(x-5)}{(x^2+1)(x-4)}\right|=-\frac{(x^2-x-6)(x-5)}{(x^2+1)(x-4)}$, then $x$ lies in

1. $x\in [-2,3]\cup [4,\infty )$
2. $x\in [-2,3]\cup (4,5]$
3. $x\in (-\infty ,3]\cup (4,5]$
4. $x\in [-2,5]$

Q. Write the value of $\underset{x\to -\infty }{\lim }\left(3x+\sqrt{9x^2-x}\right).$

Q. If $\underset{x\to 1}{\lim }\frac{x+x^2+x^3+...+x^n-n}{x-1}=5050$ then n equal

(a) 10
(b) 100
(c) 150
(d) none of these

Q. If $\left|\frac{|x|-1}{|x|-2}\right|=4$, then $x\in$

1. $\{-\frac{9}{5},-\frac{7}{3},\frac{7}{3},\frac{9}{5}\}$
2. $\{\frac{7}{3},\frac{9}{5}\}$
3. $\{-\frac{7}{3},\frac{7}{3}\}$
4. $\{-\frac{9}{5},\frac{9}{5}\}$

Q. Prove that:
(i) $\frac{n!}{(n-r)!}$ = n (n − 1) (n − 2) ... (n − (r − 1))

(ii) $\frac{n!}{(n-r)!r!}+\frac{n!}{(n-r+1)!(r-1)!}=\frac{(n+1)!}{r!(n-r+1)!}$

Q.

Does a table represents a linear or nonlinear function $?$ Explain.

Q. Number of solutions of the equation $|x-1|+|x-2|+|x-3|=7$ is

1. $1$
2. $2$
3. $3$
4. $0$

Q. Which of the following options has (have) the correct combination of the function and its graph?

1. $y=-3(x-\frac{1}{2}),x\le 2$
   
2. $y=3$
   
3. $y=2x-3,x>-4$
   
4. $y=-5x$
   

Q. Write the eccentricity of the ellipse 9x² + 5y² − 18x − 2y − 16 = 0.

Q. Write the value of $\underset{n\to \infty }{\lim }\frac{n!+\left(n+1\right)!}{\left(n+1\right)!+\left(n+2\right)!}.$

Q. If $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P., prove that:
(i) $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P.
(ii) a (b +c), b (c + a), c (a +b) are in A.P.

Q. Let f : R → R be defined by f(x) = 2x + |x|. Then f(2x) + f(−x) − f(x) =

(a) 2x
(b) 2|x|
(c) −2x
(d) −2|x|

Q. The range of the function $f\left(x\right)=\frac{x^2-x}{x^2+2x}$ is
(a) R
(b) R − {1}
(c) R − {−1/2, 1}
(d) None of these

Q. The correct statement among the following is

1. Distance between two points $a,b$ along the number line is given by $|a|-|b|$
2. $|a|=|b|\Rightarrow a=b$
3. $|x-5|=5-x\Rightarrow x<5$
4. Distance between two points $a,b$ along the number line is given by $||a|-|b||$

Q. Which among the following has the greatest absolute value?

1. $3$
2. $-\frac{9}{2}$
3. $-\sqrt{36}$
4. $-4$

Q. 18. If f(3x + 2y, 2x - 3y) = x+y then find f(x, y)

Q.

Find the value of |x+b| + |x + a| + |a-b| ___

Q. $\underset{x\to 0}{\lim }\frac{{\left(1+x\right)}^6-1}{{\left(1+x\right)}^2-1}$

Q.

The function $f\left(x\right)=ax+\frac{b}{x}$ , $a,b,x>0$ takes on the least value at $x=$

1. $b$

2. $\sqrt{a}$

3. $\sqrt{b}$

4. $\sqrt{\frac{b}{a}}$

Q. (i) If $\left(\frac{a}{3}+1,b-\frac{2}{3}\right)=\left(\frac{5}{3},\frac{1}{3}\right)$, find the values of a and b.
(ii) If (x + 1, 1) = (3, y − 2), find the values of x and y.

Q. The value of $\sum _{r=1}^n\left\{(2r-1)\mathit{}a+\frac{1}{b^r}\right\}$ is equal to

(a) $a{n}^2+\frac{b^{n-1}-1}{b^{n-1}(b-1)}$

(b) $a{n}^2+\frac{b^n-1}{b^n(b-1)}$

(c) $a{n}^3+\frac{b^{n-1}-1}{b^n(b-1)}$

(d) none of these

Q. If a, b, c are in A.P., prove that:
(i) (a − c)² = 4 (a − b) (b − c)
(ii) a² + c² + 4ac = 2 (ab + bc + ca)
(iii) a³ + c³ + 6abc = 8b³.