Roots under Different Values of Coefficients

Q.

If one root of $a{x}^2+bx+c=0$ where $a,b,c$ are integers be $3+\sqrt{5}$, then the other root is

1. $3-\sqrt{5}$

2. 3

3. None of these

4. $\sqrt{5}$

Q. For the quadratic equation $a{x}^2+bx+c=0;a,b,c\in R,$
which of the following is/are true ? (where $\Delta =b^2-4ac$, )

1. roots are negative, if all three of $a,b,c$ have the same sign and $\Delta >0$
2. $x=0$ is a repeated root, if $b=c=0$
3. if $c=0,$ then atleast one root is zero.
4. roots are negative, if $a,c$ have the same sign but $b$ is of opposite sign and $\Delta >0$

Q. If x2+5x+7=0andax2+bx+c=0 have a common root and a, b, c ∈ N, then the minimum value of a+b+c is .

1. 1
2. 7
3. 13
4. 5

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

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

Q.

Which one of the following is the equation whose roots are respectively three times the roots of the equation $a{x}^2+bx+c=0$?

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

2. $a{x}^2+3bx+9c=0$

3. $a{x}^2-3bx+9c=0$

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

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

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

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

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

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

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

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. -7, -9

2. -6, -10

3. 6, 10

4. 15, 4

Q. The number of pairs of positive integers $(p,q)$ such that $GCD(p,q)=1$ and $\frac{p}{q}+\frac{14q}{9p}$ is an integer are
(correct answer + 2, wrong answer - 0.50)

1. $4$
2. $9$
3. $12$
4. infinite

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

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

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 equal
3. real and distinct
4. real, equal in magnitude but opposite in sign

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

Q. The number of distinct roots of the quadratic equation $a{x}^2+(a-2)x+a^2-4=0$ when $a=2$ is

Q.

If $\alpha ,\beta$ are the roots of the equations $x^2+4x+3=0$, then the equation whose roots are $2\alpha +\beta$ and $\alpha +2\beta$ is

1. $x^2-12x-33=0$

2. $x^2-12x+35=0$

3. $x^2+12x-33=0$

4. $x^2+12x+35=0$

Q.

The coefficient of $x$ in the equation $a{x}^2+bx+c=0$ was wrongly taken as $17$ instead of $13$ & its roots were $-2,-15$. Actual roots are _____.

1. $-2,15$

2. $-4,-9$

3. $-3,-10$

4. $-5,-6$

Q.

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

Q. If the roots of the equation $a{x}^2+bx+1=0$ reamians unchanged if each coefficient of the equation is increased by $1$ units, then

1. $a\ne b$
2. $a=b\ne 1$
3. $a=b=1$

Q. Both the roots of $a{x}^2+bx+c=0,a\ne 0$ are of same sign then nature of $ac$ is ?

1. Positive
2. Negative

Q. Which of the following is/are the roots of the equation $4x^2+7x=0$ ?

1. $\frac{-4}{7}$
2. $\frac{-7}{4}$
3. $0$
4. $\frac{-7}{4}$

Q.

If $a{x}^2+bx+c,a,b,c\in R,a<0$ has no real zeros and $a-b+c<0$, then the value of $ac$

1. $>0$

2. $<0$

3. $=0$

4. $<-1$

Q. If $p,q$ are the roots of the equation $x^2+px+q=0$ then

1. $p=-2,q=0$
2. $p=0,q=1$
3. $p=-2,q=1$
4. $p=1,q=-2$

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

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

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

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

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. non real

3. Negative

4. positive

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

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