Roots under Different Values of Coefficients

Q. The number of solutions of $\sqrt{3x^2+x+5}=x-3$ is

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

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

1. Roots are always negative
2. Roots are of opposite sign
3. Roots are always positive
4. Roots are of same sign

Q. Let \(a, b, c\) be real numbers such that \(a + b + c < 0\) and the quadratic equation \(ax^2 + bx + c = 0\) has imaginary roots. Then

Q. If $\alpha$ and $\beta (\alpha <\beta )$ are the roots of the equation $x^2+bx+c=0$, where $c<0<b$, then

1. $\alpha <0<\beta <|\alpha |$
2. $\alpha <0<|\alpha |<\beta$
3. $\alpha <\beta <0$
4. $0<\alpha <\beta$

Q. Number of values of $n\in Z$ for which $n^2+n+2$ is a perfect square is

Q.

In a quadratic equation with leading coefficient 1, a student reads the coefficient 16 of x wrongly as 19 and obtain the roots as -15 and -4, then the correct roots are

1. 6, 10

2. -7, -9

3. -6, -10

4. 15, 4

Q. The graph of quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ is shown below


Which of the following statement(s) is/are correct?

1. $f\left(x\right)>0;\forall x>\beta$
2. $ac>0$
3. $\frac{c}{a}<-1$
4. $\frac{c}{a}(\alpha -\beta )>2$

Q. Let $a,b,c$ be real numbers such that $a+b+c<0$ and the quadratic equation $a{x}^2+bx+c=0$ has imaginary roots. Then

1. $a>0,c<0$
2. $a>0,c>0$
3. $a<0,c<0$
4. $a<0,c>0$

Q. Let the quadratic equation be $(b+c-a)x^2+(c+a-b)x+(a+b-c)=0$, where $a,b,c\in R$ and $a\ne c$. If $a+b+c=0$, then the roots are

1. real, equal in magnitude but opposite in sign
2. not real
3. real and equal
4. real and distinct

Q. If $a:b=1:6,a\ne 0,b\ne 0$ then the roots of the equation $a{x}^2+bx+5a=0$ is/are

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

Q. The roots of the equation $3x^2-2x-6=0$ are

1. both negative.
2. imaginary roots.
3. both of opposite signs.
4. both positive.

Q. If one root of the equation $2x^2+bx+2=0$ is $-2$, then the other root is:

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

Q.

If a + b + c = 0, the correct statements for the equation $a{x}^2+bx+c=0$ are

1. One of the roots is +ve

2. $\frac{(a+b)}{a}$ is a root of the equation

3. $\frac{c}{a}$ is a root of the equation.

4. $\frac{-(a+b)}{a}$ is a root of the equation

Q. The equation $2x^2+5x-7=0$ has:

1. $2$ Complex Roots
2. $1$ Rational & $1$ Integral Roots
3. $2$ Integral Roots
4. $2$ Irrational Roots

Q. For $2x^2+5x-7=0$, roots are:

1. $-1$
2. $\frac{-7}{2}$
3. $1$
4. $\frac{2}{7}$

Q.

Let $x_1,x_2$ be the roots of $a{x}^2+bx+c=0$ and $x_1.x_2<0$, $x_1+x_2$ is non - zero. Roots of $x_1(x-x_2{)}^2+x_2(x-x_1{)}^2=0$ are

1. Real and of opposite signs

2. Negative

3. positive

4. non real

Q. Product of roots of the equation
$\sqrt{13-x^2}=x+5$

1. $7$
2. $6$
3. $-7$
4. $-6$

Q. For the equation $x^2+bx+c=0,$ if $1+b+c=0$ for all $b,c\in R,$ then roots are

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

Q. Suppose the quadratic polynomial $P\left(x\right)=a{x}^2+bx+c$ has positive coefficients $a,b,c$ in arithmetic progression in that order. If $P\left(x\right)=0$ has integer roots $\alpha$ and $\beta$, then $\alpha +\beta +\alpha \beta$ equals

1. $14$
2. $5$
3. $3$
4. $7$

Q. For the equation $x^2+bx+c=0,$ if $1+b+c=0$ for all $b,c\in R,$ then roots are

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

Q. What can be said about the roots of the equation $4x^2+7x=0?$

1. $2$ Integral Roots
2. $2$ Irrational roots
3. $2$ Complex roots
4. $1$ Integral Root and $1$ Rational Roots

Q.

Consider the equation $x^2+2x-n=0$, where $n\in [5,100]$. Total number of different values of ${}^{\prime }{n}^{\prime }$ so that the given equation has integral roots is:

1. $4$

2. $3$

3. $6$

4. $8$

Q. If $-1$ is one of the roots of the equation $(m-2)x^2+8x+(m+4)=0,$ then $m=$

1. $-5$
2. $3$
3. $-3$

Q. The sum of integral values of $n$ for which $n^2+17n+75$ is a perfect square is

1. $-17$
2. $-14$
3. $0$
4. $-15$

Q. The equation $2x^2+5x-7=0$ has:

1. $2$ Complex Roots
2. $1$ Rational & $1$ Integral Roots
3. $2$ Irrational Roots
4. $2$ Integral Roots

Q. If one root of the equation $2x^2+bx+2=0$ is $-2$, then the other root is:

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

Q. The number of values of $c\in N$ for which the the quadratic equation $x^2-13x+6c=0$ has distinct integer roots are:

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

Q. The value(s) of $x$ satisfying the equation $x=\sqrt{20+\sqrt{20+\sqrt{20+\cdots }}}$ is

1. both $-4$ and $5$
2. $0$
3. $-4$
4. $5$

Q.

Consider the equation $x^2+2x-n=0$, where $n\in [5,100]$. Total number of different values of ${}^{\prime }{n}^{\prime }$ so that the given equation has integral roots is:

1. $4$

2. $8$

3. $6$

4. $3$

Q. Let the quadratic equation be $(b+c-a)x^2+(c+a-b)x+(a+b-c)=0$, where $a,b,c\in R$ and $a\ne c$. If $a+b+c=0$, then the roots are

1. not real
2. real and distinct
3. real and equal
4. real, equal in magnitude but opposite in sign