Range of Quadratic Expression

Q. The inequality $\frac{|x{|}^2-|x|-2}{2|x|-|x{|}^2-2}>0$ holds if and only if

1. $\frac{2}{3}<x<1$
2. none of these
3. $-2<x<2$
4. $-1<x<-\frac{-2}{3}\text{or}\frac{2}{3}<x<1$

Q.

If x be real, then the minimum value of $x^2-8x+17$ is

1. -1

2. 0

3. 1

4. 2

Q.

If x is real, find the range of y from the equation $x^2(y-1)-2x+(2y-1)$ = 0

1. 

2. 

3. 

4. (-, 0 ]

Q. If x is real, then the maximum and minimum values of the expression $\frac{x^2-3x+4}{x^2+3x+4}$ will be

1. 2, 1
2. None of these
3. 7, 1/7
4. 5, 1/5

Q.

Find the range of rational expression $y=\frac{x^2+34x-71}{x^2+2x-7}$ if x is real

1. 

2. 

3. 

4. 

Q. If a and b are the non-zero distinct roots of $x^2+ax+b=0$, then the least value of $x^2+ax+b$ is

1. 
2. 
3. 1
4. -

Q.

The set of all real numbers x for which $x^2-|x+2|+x>0$ is

1. $(-\infty ,-\sqrt{2})\cup (\sqrt{2},\infty )$

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

3. $(\sqrt{2},\infty )$

4. $(-\infty ,-2)\cup (2,\infty )$

Q.

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x-y-z=0-3x+z=0-3x+2y+z=0$
Then the number of such points for which
$x^2+y^2+z^2\le 100$ is ___

Q.

The maximum value of the expression $y=2(a-x)(x+\sqrt{x^2+b^2})$ is

1. $2a^2+b^2$

2. $a^2+2b^2$

3. $a^2+b^2$

4. $|2a^2-b^2|$

Q.

Complete set of values of a such that $\frac{x^2-x}{1-ax}$ attains all real values is

1. [1, 4]

2. [0, 4]

3. $[1,\infty )$

4. $[4,\infty )$