Sign of Quadratic Expression

Q. The range of $a$ for which $x^2-ax+1-2a^2$ is always positive for all real values of $x$, is

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

Q.

Find all the roots of the equation $x^7$ = 1

1. 

2. 

3. 

4. 

Q.

The origin and the roots of the equation $x^2+ax+b$ = 0 form an equilateral triangle, if :

1. a²=b

2. b²=3a

3. b²=a

4. a²=3b

Q.

Which of the following equation has imaginary roots?

1. x² + 4x + 1 = 0

2. 2x² + 5x + 3 = 0

3. 3x² + 7x + 9

4. x² + 2√2 x + 1

Q. Find the values of y for which the following will be positive, negative or zero. $y=x-6\sqrt{x}+8$

1. $y>0\forall xϵ(-\infty ,2)\cup (4,\infty )y<0\forall xϵ(2,4)y=0\forall xϵ\{2,4\}$
2. $y>0\forall xϵ(-\infty ,4)\cup (16,\infty )y<0\forall xϵ(4,16)y=0\forall xϵ\{4,16\}$
3. $y>0\forall xϵR$
4. $y>0\forall xϵ(0,4)\cup (16,\infty )y<0\forall xϵ(4,16)y=0\forall xϵ\{4,16\}$

Q.

If a, b $\epsilon$ R, a $\ne$ 0 and the quadratic equation $a{x}^2-bx+2$ =0 has imaginary roots, then a + b + 2 is

1. Depends on sign of b

2. Positive

3. Zero

4. Negative

Q. $\text{Let}f\left(\alpha \right)={\alpha }^2-a\alpha +1,f\left(\beta \right)=b\beta -a{\beta }^2-\frac{3}{a},\text{where}a,b\in I^{-}\text{and}|f\left(\alpha \right)+f\left(\beta \right)|=|f\left(\alpha \right)|+|f\left(\beta \right)|\text{is satisfied}\forall \alpha ,\beta \in R,\text{then which of the following is/are true?}$

1. Sum of all values of $a$ is $-3$
2. Sum of all values of $a$ is $-2$
3. Sum of all values of $b$ is $-6$
4. Sum of all values of $b$ is $-3$

Q. Find the sign of the quadratic polynomial.
$f\left(x\right)=x^2+5|x|+6$

1. $y>0\forall xϵ(-\infty ,2)\cup (3,\infty )y<0\forall xϵ(2,3)y=0\forall xϵ2,3$
2. $y>0\forall xϵ(-\infty ,-3)\cup (-2,2)\cup (3,\infty )y<0\forall xϵ(-3,-2)\cup (2,3)y=0\forall xϵ\{-3,-2,2,3\}$
3. $y>0\forall xϵR$
4. $y>0\forall xϵ(-\infty ,-3)\cup (-2,\infty )y<0\forall xϵ(-3,-2)y=0\forall xϵ\{-3,-2\}$

Q. Let $f\left(x\right)=x^2+2bx+2c^2$ and $g\left(x\right)=-x^2-2cx+b^2,b,c\ne 0$. If the minimum of $f\left(x\right)$ is greater than maximum value of $g\left(x\right)$, then the range of $\frac{b}{c}$ is

1. $(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$
2. $(0,\frac{1}{\sqrt{2}})$
3. $(-\frac{1}{\sqrt{2}},0)$
4. $(\frac{1}{\sqrt{2}},\infty )$

Q. If $(\lambda -2)x^2+(\lambda -1)x+3<0,\forall x\in R,$ then the range of $\lambda$ is

1. $(\frac{14-\sqrt{96}}{2},2)$
2. $(\frac{14-\sqrt{96}}{2},\frac{14+\sqrt{96}}{2})$
3. $(-\infty ,2)$
4. $\varphi$

Q.

If the roots of the equation $4x^3-12x^2+11x+k=0$ are in A.P. Then $k=$

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

Q. The set of real values of $x$, for which $h\left(x\right)=1+2x^2+4x^4+6x^6+\cdots +100x^{100}$ is concave downward is

1. $x\in R^{-}$
2. $x\in W$
3. $x\in R^{+}$
4. $x\in \varphi$