Sign of Quadratic Expression

Q. If both the roots of $a{x}^2+bx+c=0,a\ne 0$ is positive, then

1. $a$ and $b$ are of the same sign.
2. $a,b$ and $c$ are of the same sign.
3. $a$ and $c$ are of the same sign.
4. $b$ and $c$ are of the same sign.

Q. Consider the graph of the quadratic polynomial $y=a{x}^2+bx+c$ as shown below.

Which among the following is/are correct?

1. $\frac{a-b+c}{abc}=0$
2. $abc(9a+3b+c)<0$
3. $\frac{a+3b+9c}{abc}<0$
4. $abc(a-3b+9c)>0$

Q. Consider the graph of the quadratic polynomial $y=a{x}^2+bx+c$ as shown below.

Which among the following is/are correct?

1. $\frac{a-b+c}{abc}=0$
2. $abc(9a+3b+c)<0$
3. $\frac{a+3b+9c}{abc}<0$
4. $abc(a-3b+9c)>0$

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

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 the roots of the equation $a{x}^2+bx+a_1^2+b_1^2+c_1^2-a_1{b}_1-a_1{c}_1-b_1{c}_1=0$ are non real then

1. $2(b-a)+\sum (a_1+b_1{)}^2=0$

2. $2(b-a)+\sum (a_1+b_1{)}^2<0$

3. $2(a-b)+\sum (a_1+b_1{)}^2=0$

4. $2(a-b)+\sum (a_1-b_1{)}^2>0$

Q. Let $f\left(x\right)=a{x}^2+bx+c$ and $g\left(x\right)=A{x}^2+bx+\lambda ,$ where $a\ne 0,A\ne 0,a,b,c,A,\lambda \in R$. Roots of $f\left(x\right)=0$ and $g\left(x\right)=0$ are imaginary, then which of the following may be correct?

1. $f\left(x\right)+g\left(x\right)=0$ for some $x$
2. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0\forall x\in R$
3. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0$ for some $x$
4. Roots of equation $af\left(x\right)+Ag\left(x\right)=0$ are real

Q. If $y=x^2+kx+1$ intersects the $x$-axis at two different points, then the minimum positive integral value of $k$ is

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

Q. If $a{x}^2+(1-\lambda )x+(a-1-\lambda )=0$ where $a\ne 0,$ has real roots for all $\lambda \in R,$ then

1. $a=1$
2. $a>1$
3. $0<a<1$
4. $0<a\le 1$

Q. The maximum value of the function $f\left(x\right)=2\sqrt{x-2}+\sqrt{4-x}$ is $\sqrt{K},$ then $K$ is

Q. Solve the following equations:
$x^4+1-3(x^3+x)=2x^2$.

Q. The values of $a$ for which the quadratic equation $3x^2+2(a^2+1)x+(a^2-3a+2)=0$ has roots of opposite sign, is

1. $a\in (1,2)$
2. $a\in (2,\infty )$
3. $a\in (1,\infty )$
4. $a\in (2,3)$

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 $a{x}^2+bx+c<0,\forall x\in R$, then $a^2{x}^2+abx+ac$ is

1. always positive
2. always negative
3. equal to zero
4. can be positive or 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. 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. $x\in \varphi$
2. $x\in R^{-}$
3. $x\in W$
4. $x\in R^{+}$

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.

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. Negative

2. Positive

3. Zero

4. Depends on sign of b