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

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

Q. If both the roots of the quadratic equation $a{x}^2+bx+c=0$ are negative, then

1. $a,b,c$ can be any negative integers
2. $a,b,c$ can be any positive rational numbers
3. $a,b,c$ can be any real numbers
4. $a,b,c$ can be any integers

Q. The roots of the equation $(b+c)x^2-(a+b+c)x+a=0$, where $a,b,c\in Q$ and $b+c\ne a$, are

1. rational and distinct
2. irrational and equal
3. rational and equal
4. cannot be determined

Q. How many ordered pairs of integers $(x,y)$ satisfy the equation $x^2+y^2=2(x+y)+xy$?
(correct answer + 5, wrong answer 0)

Q.

If $a{x}^2+bx+c=0$ has no real roots and $a+b+c<0$, then

1. $c<0$

2. $c=0$

3. $c>0$

4. Can't say anything

Q. If both the roots of the quadratic equation $a{x}^2+bx+c=0$ are zero, then

1. $a=b=c=0$
2. $b=c=0,a\ne 0$
3. $c=0,a\ne 0$
4. $b=0,a\ne 0$

Q.

If $x^2+4y^2-8x+12=0$ is satisfied by real values of $x\&y,$ then $y\in$

1. $[-1,1]$

2. $[2,6]$

3. $[-2,-1]$

4. $[2,5]$

Q. All the integral values of $a$ for which the quadratic equation $(x-a)(x-10)+1=0$ has integral roots, are

1. $8,12$
2. $8,11$
3. $11,12$
4. $9,12$

Q. If $a,b\in R,a>0$ and the quadratic equation $a{x}^2-bx+1=0$ has imaginary roots, then
$a+b+1$ is

1. positive
2. negative
3. zero
4. depends on the sign of b.

Q. If $2x^2+5x+7=0$ and $a{x}^2+bx+c=0$ have at least one root common such that $a,b,c\in \{1,2,3,\dots ,100\}$, then the difference between the maximum and the minimum possible value of $a+b+c$ is

Q. The value(s) of $m$ such that the roots of the quadratic equation $(m+1)x^2+(m+1)x-m+1=0$ are equal is

1. $-\frac{3}{5}$
2. $-1,\frac{3}{5}$
3. $1$
4. $\frac{3}{5}$

Q. For the quadratic equation $a{x}^2+bx-4=0,a,b\in R$ and $a>0,$ then which of the following is always true?

1. roots are real, distinct and have same sign
2. roots are real and repeated
3. roots are not real
4. roots are real, distinct and have opposite sign

Q.

If a, b, c are positive rational numbers such that a > b > c and the quadratic equation
$(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root in the interval (-1, 0), then

1. b + c > a

2. c + a < 2b

3. both roots of the given equation ae rational

4. the equaiton $a{x}^2+2bx+c=0$ has both negative real roots

Q.

The roots of the equation $4x^4-24x^3+57x^2+18x-45=0$ if one of them is $3+i\sqrt{6}$ , are

1. $3-i\sqrt{6},\pm \frac{\sqrt{3}}{2}$

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

3. $3-i\sqrt{6},\pm \frac{3}{\sqrt{2}}$

4. None of these

Q. The roots of the quadratic equation $x^2-2(a+b)x+2(a^2+b^2)=0(a\ne b)$ are

1. real and distinct
2. rational
3. imaginary
4. real and equal

Q. If the roots of the equation
$(1-q+\frac{p^2}{2})x^2+p(1+q)x+q(q-1)+\frac{p^2}{2}=0$ are equal, then

1. $p^2=8q$
2. $p^2=2q$
3. $p^2=4q$
4. $p^2=q$

Q. The number of distinct positive real roots of the equation $(x^2+6{)}^2-35x^2=2x(x^2+6)$ is

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

Q. Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then

1. $P\left(x\right)=0$ has imaginary roots
2. $P\left(x\right)=0$ has roots of opposite sign
3. $P\left(1\right)=4$
4. $P\left(1\right)=6$

Q. The roots of the equation $\frac{1}{x+1}+\frac{2}{x-4}=2$ are

1. real and equal.
2. rational and distinct.
3. imaginary.
4. irrational and distinct.

Q. The possible value(s) of $a$ for which the equation $x^2+2x-(a^3-3a-3)=0$ has real roots is/are

1. $-1$
2. $4$
3. $2$
4. $6$

Q. If the equation $a{x}^2+2bx+c=0;a,b,c\in R$ has real roots and $m,n$ are real such that $m^2>n>0$, then the equation $a{x}^2+2mbx+nc=0$ has

1. real roots
2. imaginary roots
3. real and distinct roots
4. real and equal roots

Q.

Let $p$ and $q$be real numbers. If $\alpha$ is the root of $x^2+3p^{2}x+5q^2=0,\beta$ is the root of $x^2+9p^{2}x+15q^2=0$and $0<\alpha <\beta ,$ then the equation $x^2+6p^{2}x+10q^2=0$ has root $\gamma$ that always satisfies.

1. $ɣ=\left(\frac{ɑ}{4}\right)+\beta$

2. $\beta <\gamma$

3. $\gamma =\left(\frac{ɑ}{2}\right)+\beta$

4. $\alpha <\beta <\gamma$

Q. If $S$ and $P$ be the arithmetic mean and geometric mean of two non-zero real numbers $a$ and $b(b>a)$ respectively, then the set of values of $k$ for which the quadratic equation $k{x}^2+2Sx+P^2=0$ has real roots is

1. $(-\infty ,-1]\cup [1,\infty )$
2. $(0,1]$
3. $(-\infty ,a]\cup [b,\infty )$
4. $(-\infty ,-1]\cup [\frac{b}{a},\infty )$

Q. Which of the following equation has exactly one root as $0$?
$(a,b,c>0)$

1. $a{x}^2=0$
2. $a{x}^2+c=0$
3. $a{x}^2+bx=0$
4. $bx+c=0$

Q. If $a,b,c$ are rational and $a\ne b,b\ne a+c,$ then the roots of the equation $(a+c-b)x^2+2cx+(b+c-a)=0$ are

1. irrational and distinct
2. distinct integers
3. rational and distinct
4. rational and equal

Q. If the given expression $x^2-(5m-2)x+(4m^2+10m+25)$ can be expressed as a perfect square, then the value(s) of $m$ is/are

1. $8$
2. $-\frac{4}{3}$
3. $\frac{4}{3}$
4. $-8$

Q. If $x^2-2x+{\log }_{\frac{1}{2}}p=0$ does not have two distinct real roots, then the maximum value of $p$ is

1. $1$
2. $\frac{1}{4}$
3. $2$
4. $\frac{1}{2}$

Q. If $y=\frac{(x-a)(x-c)}{x-b}$ assumes all real values for $x\in R-\left\{b\right\},$ then

1. $c>a>b$
2. $a<b<c$
3. $a=b=c$
4. $a>c>b$

Q. The number of integral values of $k$, for which $(x^2-x+1)(k{x}^2-3kx-5)<0$ is