Discriminant of a Quadratic Equation

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

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

Q. Let $a$ and $c$ be odd prime numbers and $b$ be an integer. If the quadratic equation $a{x}^2+bx+c=0$ has rational roots, then which of the following statement(s) is/are correct?

1. Roots of the equation are $-1,-\frac{c}{a}.$
2. Both the roots are independent of the coefficients.
3. Roots of the equation are $-1,-2.$
4. One of the root is independent of the coefficients

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

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

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

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

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

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

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. $1$

3. $2$

4. $3$

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

Q.

Find the discriminant of the quadratic equation $3x^2-5x+2=0$ and hence, find the nature of the roots.

1. $0$, two equal roots

2. $-1$, no real roots

3. $-1$, two distinct real roots

4. $1$, two distinct real roots

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. at least two real roots
3. no real roots
4. exactly six real roots

Q.

For the equation $3x^2+px+3=0,p>0$ one of the root is square of the other, then $p$ is equal to

1. $3$

2. $\frac{2}{3}$

3. $\frac{1}{3}$

4. $1$

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

Q. The roots of $\sqrt{2}{x}^2+8x+\sqrt{2}=0$ are

1. equal and opposite in sign
2. non real
3. reciprocals of each other
4. equal

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

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

Q. If the inequality $(m-2)x^2+8x+m+4>0$ is satisfied for all $x\in R$, then the least integral $m$ is

1. $2$
2. $4$
3. $5$
4. $7$

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