Roots under Different Values of Coefficients

Q.

If one root is square of the other root of the equation $x^2+px+q=0$, then the relation between $p\text{and}q$ is

1. $p^3–(3p–1)q+q^2=0$

2. $p^3–q(3p+1)+q^2=0$

3. $p^3+q(3p-1)+q^2=0$

4. $p^3-q(3p-1)+q^2=0$

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

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

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

1. of opposite signs
2. both negative
3. both positive
4. none of these

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

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. $\frac{c}{a}<-1$
3. $ac>0$
4. $\frac{c}{a}(\alpha -\beta )>2$

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

2. None of these

3. $\sqrt{5}$

4. $3-\sqrt{5}$

Q. Let one root of $a{x}^2+bx+c=0$ where $a,b,c\in Z,a\ne 0$ be $3+\sqrt{5}$, then the other root of the equation will be

1. $\sqrt{5}$
2. $3$
3. $-3+\sqrt{5}$
4. $3-\sqrt{5}$

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

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

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 $a,c$ have the same sign but $b$ is of opposite 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 all three of $a,b,c$ have the same sign and $\Delta >0$

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

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

Q. The number of values of $c\in N$ for which the the quadratic equation $x^2-13x+4c=0$ has distinct integer roots are:

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

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

Q. Which of the following is set of the roots of the equation, $3x^2-5=0?$

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

Q. The number of values of $\alpha \in N$ for which the the quadratic equation $6x^2-11x+\alpha =0$ has rational roots are:

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

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. $6$

2. $4$

3. $8$

4. $3$

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

1. Negative
2. Positive

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

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

Q. If $\left(\begin{array}{cc}2& 4\\ -1& k\end{array}\right)$ is a nilpotent matrix of index $2,$ then $k$ equals to

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

Q. If one root of a quadratic equation $a{x}^2+bx+c=0$ is $0$, and the value of the coefficients $a,b$ are $8,7$ respectively, then the other root is:

1. $\frac{8}{7}$
2. $-\frac{7}{8}$
3. $-\frac{8}{7}$
4. $\frac{7}{8}$

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

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

Q. If one root of a quadratic equation $a{x}^2+bx+c=0$ is $0$, and the value of the coefficients $a,b$ are $8,7$ respectively, then the other root is:

1. $-\frac{8}{7}$
2. $-\frac{7}{8}$
3. $\frac{7}{8}$
4. $\frac{8}{7}$

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. 15, 4

3. 6, 10

4. -7, -9

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. real and equal
3. not real
4. real and distinct

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. 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 both the roots of $a{x}^2+bx+c=0,a\ne 0$ is positive, then

1. $a$ and $b$ are of the same sign.
2. $a,b$ and $c$ are of the same sign.
3. $a$ and $c$ are of the same sign.
4. $b$ and $c$ are of the same sign.

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