Discriminant of a Quadratic Equation

Q. If $b_1{b}_2=2(c_1+c_2)$ and $b_1,b_2,c_1,c_2$ are all real numbers, then at least one of the equations $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ has

1. roots of the form $a+ib(a,b\in R,ab\ne 0)$
2. rational roots
3. purely imaginary roots
4. real roots

Q.

Find the value of $p$ for which the quadratic equation $2p{x}^2-8x+p=0$ has equal roots.

Q. If $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ are the two quadratic equations such that $b_1{b}_2=2(c_1+c_2)$ and $b_1,b_2,c_1,c_2\in R,$ then

1. at least one equation has imaginary roots
2. Both the equations will have real roots
3. at least one equation has real roots
4. Both the equations will have imaginary roots

Q. If the roots of the quadratic equation $a{x}^2+bx+c=0$ are opposite in sign and positive root is greater in magnitude, then

1. $b>0,c>0\text{and}a<0$
2. $b<0,c<0\text{and}a>0$
3. $a>0,b>0\text{and}c<0$
4. $a<0,b<0\text{and}c>0$

Q. Find the value of $m$ such that at least one root of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ lies in the interval $(1,2)$

1. $\varphi$
2. $[9,\infty )$
3. $(-\infty ,1]$
4. $(10,\infty )$

Q. $\begin{array}{cc}\text{(I) If}{x}^2+x-a=0\text{has integral roots}& \left(P\right)2\\ \text{and}a\in N,\text{then}a\text{can be equal to}& \\ \text{(II) If the equation}a{x}^2+2bx+4c=16& \left(Q\right)12\\ \text{has no real roots and}a+c>b+4,& \\ \text{then the integral value of}c\text{can be}& \\ \text{(III) If equation}{x}^2+2bx+9b-14=0& \left(R\right)1\\ \text{has only negative roots, then the integral}& \\ \text{values of}b\text{can be}& \\ \text{(IV) If}N\text{be the number of solutions of}& \left(S\right)30\\ \text{the equation}|x-|4-x||-2x=4,\text{then}& \\ \text{the value of}N\text{is}& \end{array}$

Which of the following is the only CORRECT combination?

1. $\left(II\right)\to \left(P\right),\left(Q\right)$
2. $\left(I\right)\to \left(P\right),\left(Q\right),\left(R\right)$
3. $\left(III\right)\to \left(R\right)$
4. $\left(I\right)\to \left(P\right),\left(Q\right),\left(S\right)$

Q.

Find the value of ${}^{\prime }{k}^{\prime }$ if the expression $(4-k)x^2+2(k+2)x+8k+1$ is a perfect square

1. $0$

2. $2$

3. $1$

4. $3$

Q. Find the values of $m$ such that both roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ are greater than $2$

1. $(7,9]$
2. $(10,\infty )$
3. $[9,10)$
4. $[1,9]$

Q. The number of integral values of $m$ for which the equation $(1+m^2)x^2-2(1+3m)x+(1+8m)=0$ has no real root is :

1. $3$
2. infinitely many
3. $1$
4. $2$

Q. If $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ are the two quadratic equations such that $b_1{b}_2=2(c_1+c_2)$ and $b_1,b_2,c_1,c_2\in R,$ then

1. at least one equation has real roots
2. Both the equations will have real roots
3. Both the equations will have imaginary roots
4. at least one equation has imaginary roots

Q. Find the values of m for which the equation $x^4-(1-2m)x^2+(m^2-1)=0$ has three real distinct root

1. $\left\{2\right\}$
2. $\left\{1\right\}$
3. $\varphi$
4. $\{-1\}$

Q. The value(s) of $a$ for which the roots of $2x^2+(a^2-1)x+a^2+3a+4=0$ are reciprocal to each other is/are

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

Q. Which among the following is the correct graphical representation of $y=-x^2-1?$

1. 
2. 
3. 
4. 

Q. The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation, $6x^2-11x+\alpha =0$ are rational numbers is :

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

Q. The discriminant of the quadratic equation $3x^2-4\sqrt{3}x+4=0$ is

Q. The discriminant of the quadratic equation $2x^2-4x+3=0$ is

1. $0$
2. $-8$
3. $-10$
4. $2$

Q. $\text{If}D=0\text{in}y=a{x}^2+bx+c,a\ne 0$
then $x$-axis will be tangent at the vertex for parabola.

1. False
2. True

Q. Let the equation $e^{-x}+2=k,k\in Z$ has atleast one real solution in $R$, then minimum value of $k$ is

Q. If $a,b\in \{1,2,3\}$ and the equation $a{x}^2+bx+1=0$ has real roots, then

Q. $\begin{array}{cc}\text{(I) If}{x}^2+x-a=0\text{has integral roots}& \left(P\right)2\\ \text{and}a\in N,\text{than}a\text{can be equal to}& \\ \text{(II) If the equation}a{x}^2+2bx+4c=16& \left(Q\right)12\\ \text{has no real roots and}a+c>b+4& \\ \text{(III) If equation}{x}^2+2bx+9b-14=0& \left(R\right)1\\ \text{has only negative roots, then the integral}& \\ \text{values of}b\text{can be}& \\ \text{(IV) If}N\text{be the number of solutions of}& \left(S\right)30\\ \text{the equation}|x-|4-x||-2x=4,\text{then}& \\ \text{the value of}N\text{is}& \end{array}$
Which of the following is only CORRECT Combination?

1. $\left(IV\right)\to \left(P\right),\left(Q\right),\left(R\right)$
2. $\left(III\right)\to \left(P\right),\left(Q\right),\left(S\right)$
3. $\left(IV\right)\to \left(P\right),\left(Q\right)$
4. $\left(III\right)\to \left(P\right),\left(Q\right),\left(R\right)$

Q. Consider two quadratic equations, $p{x}^2-2qx+p=0...\left(i\right)$and $q{x}^2-2px+q=0...\left(ii\right)\left(\mathrm{both}p\mathrm{and}q\mathrm{are}\mathrm{real}\right).$If the roots of the equation $\left(i\right)$ are real and unequal, then the roots of the equation $\left(ii\right)$ are

1. Real and equal
2. Real and unequal
3. imaginary
4. Rational and equal

Q. If $x^2+b_{1}x+c_1=0$ and $x^2+b_{2}x+c_2=0$ are the two quadratic equations such that $b_1{b}_2=2(c_1+c_2)$ and $b_1,b_2,c_1,c_2\in R,$ then

1. at least one equation has imaginary roots
2. Both the equations will have imaginary roots
3. at least one equation has real roots
4. Both the equations will have real roots

Q. The number of integral values of $m$ for which the equation $(1+m^2)x^2-2(1+3m)x+(1+8m)=0$ has no real root is :

1. $3$
2. $1$
3. $2$
4. infinitely many

Q. The values of $m$ for which both roots of the quadratic equation $x^2-(m-3)x+m=0;m\in R,$ are greater than $2$ is

1. $(10,\infty )$
2. $[9,10)$
3. $[1,9]$
4. $(7,9]$

Q. If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than $5$, then $k$ lies in the interval

1. $(5,6]$
2. $(-\infty ,4)$
3. $(6,\infty )$
4. $[4,5]$

Q. If $a,b,c,d\in R,$ then the equation $(x^2+ax-3b)(x^2-cx+b)(x^2-dx+2b)=0$ has

1. at least four real roots
2. exactly six real roots
3. at least two real roots
4. no real roots

Q. Consider two quadratic equations, $p{x}^2-2qx+p=0...\left(i\right)$and $q{x}^2-2px+q=0...\left(ii\right)\left(\mathrm{both}p\mathrm{and}q\mathrm{are}\mathrm{real}\right).$If the roots of the equation $\left(i\right)$ are real and unequal, then the roots of the equation $\left(ii\right)$ are

1. imaginary
2. Real and unequal
3. Real and equal
4. Rational and equal

Q. If $a_i$ be the values of $a$ for which the equation $a{x}^2-8x+a=0$ has only one real solution, then the value of $\prod _{i=1}^n{a}_i$ (where $\prod$ stands for product) is

1. $1$
2. $-16$
3. $16$
4. $0$

Q. If $a_i$ be the values of $a$ for which the equation $a{x}^2-8x+a=0$ has only one real solution, then the value of $\prod _{i=1}^n{a}_i$ (where $\prod$ stands for product) is

1. $1$
2. $0$
3. $-16$
4. $16$

Q. Sum of all real $x$ such that $\frac{4x^2+15x+17}{x^2+4x+12}=\frac{5x^2+16x+18}{2x^2+5x+13}$ is

1. $-\frac{17}{4}$
2. $-\frac{13}{4}$
3. $-\frac{15}{3}$
4. $-\frac{11}{3}$