Discriminant of a Quadratic Equation

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

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

1. $a\le b$
2. $a>b$
3. number of possible ordered pairs $(a,b)$ is $3$
4. $a<b$

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

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

Q.

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

1. $1$, two distinct real roots

2. $-1$, no real roots

3. $-1$, two distinct real roots

4. $0$, two equal roots

Q.

If $\alpha$ is a real root of the equation $x^3+p{x}^2+qx+r=0$, where p, q and r are real. If $p^2-4q-2p\alpha -3{\alpha }^2\ge 0$ then other roots are ________.

1. Imaginary conjugate

2. Irrational conjugate

3. Real numbers

4. None of these

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

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

1. 
2. 
3. 
4. 

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

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

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

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

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. $\frac{2}{3}$

2. $3$

3. $1$

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

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

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

Q. The number of integral value(s) of $y$ for which $(y^2-5y+3)(x^2+x+1)-2x<0$ for all $x\in R$ is

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

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

1. $D=96$
2. $D=0$
3. $D=-2$
4. $D=4$

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

Q. The number of integral value(s) of $y$ for which $(y^2-5y+3)(x^2+x+1)-2x<0$ for all $x\in R$ is

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

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

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. One of the root is independent of the coefficients
3. Both the roots are independent of the coefficients.
4. Roots of the equation are $-1,-2.$

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

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

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

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

Q.

If $\alpha$ is a real root of the equation $x^3+p{x}^2+qx+r=0$, where p, q and r are real. If $p^2-4q-2p\alpha -3{\alpha }^2\ge 0$ then other roots are ________.

1. Imaginary conjugate

2. Irrational conjugate

3. None of these

4. Real numbers

Q.

Reduce the expression $y=\frac{x^2-x+1}{x^2+x+1}$ to the form $a{x}^2+bx+c$ and give condition for $x$ to be real.

1. $(y+1{)}^2-4(y-1{)}^2\ge 0$

2. $(y-1{)}^2-2\ge 0$

3. $(y-1{)}^2+4(y+1{)}^2\ge 0$

4. $(y-1{)}^2-4(y+1{)}^2\ge 0$