Location of Roots

Q. Consider the quadratic equation $(c-5)x^2-2cx+(c-4)=0$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0,2)$ and another root lies in the interval $(2,3)$. The number of elements in $S$ is

1. $18$
2. $12$
3. $11$
4. $10$

Q. If both the roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $m$ lies in the interval :

1. $(4,5]$
2. $(5,6)$
3. $(4,5)$
4. $(3,4)$

Q. The number of real roots of the equation $\sqrt{x}+\sqrt{x-\sqrt{1-x}}=1$ is

Q. $\alpha$ and $\beta$ are roots of the quadratic equation $a{x}^2+bx+c=0$. The equation has real roots which are of opposite signs, then the equation $\alpha (x-\beta {)}^2+\beta (x-\alpha {)}^2=0$

1. has no real roots
2. has two distinct real roots
3. has real equal roots
4. has real roots which are opposite in signs

Q. If $\alpha ,\beta$ are roots of $4x^2-16x+c=0,c>0$ such that $1<\alpha <2<\beta <3$, then the number of integral value of $c$ is

Q. If both roots of the equation $x^2+ax+2=0$ lie in the interval $(0,3),$ then the exhaustive range of values of $a$ is

1. $(-\frac{11}{3},\infty )$
2. $(-\infty ,-2\sqrt{2}]\cup [2\sqrt{2},\infty )$
3. $(-\frac{11}{3},-2\sqrt{2}]$
4. $(-6,0)$

Q. The range of $a$ for which the equation $x^2+ax-4=0$ has its smaller root in the interval $(-1,2)$ is

1. $(-\infty ,-3)$
2. $(0,3)$
3. $(0,\infty )$
4. $(-\infty ,-3)\cup (0,\infty )$

Q. If the roots of the quadratic equation $(4p-p^2-5)x^2-(2p-1)x+3p=0$ lie on either side of unity, then the number of integral values of $p$ is

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

Q. If one root of $x^2-2p(x-4)-15=0$ is less than $1$ and the other root is greater than $2,$ then the range of $p$ is

1. $R$
2. $(-\frac{7}{3},\frac{7}{3})$
3. $(\frac{7}{3},\infty )$
4. $(-\infty ,\frac{7}{3})$

Q. If $a,b,c$ are rational numbers $(a>b>c>0)$ 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 which of the following statements is (are) CORRECT?

1. $a+c<2b$
2. Both roots are rational
3. $a{x}^2+2bx+c=0$ has both roots negative
4. $c{x}^2+2bx+a=0$ has both roots negative

Q.

The quadratic equation, whose roots are three times the roots of $3a{x}^2+3bx+c=0$, is

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

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

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

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

Q. The roots of the equation $x^2+2(a-3)x+9=0$ lie between $-6$ and $1.$ If $2,h_1,h_2,\dots ,h_{20},\left[a\right],$ where $[.]$ represents the greatest integer function, are in harmonic progression, then the value of $10h_9$ is

Q. The complete set of values of $a$ for which the inequality $(a-1)x^2-(a+1)x+(a-1)\ge 0$ is true for all $x\ge 2$ is

1. $(1,\frac{7}{3}]$
2. $(\frac{7}{3},\infty )$
3. $(-\infty ,\frac{7}{3}]$
4. $[\frac{7}{3},\infty )$

Q. The complete set of values of $a$ for which the inequality $a{x}^2-(3+2a)x+6>0,a\ne 0$ holds good for exactly three integral values of $x$ is

1. $(-3,-\frac{3}{2})$
2. $(-3,-\frac{3}{2}]$
3. $[-3,-\frac{3}{2}]$
4. $(-3,\frac{3}{2})$

Q. If $x_1$ and $x_2$ are the roots of $a{x}^2+bx+c=0,b\ne 0$ and they are of opposite sign, then the roots of $x_1(x-x_2{)}^2+x_2(x-x_1{)}^2=0$ are

1. real
2. non real
3. of same sign
4. of opposite sign

Q. If atleast one of the root of the equation $x^2-(k+2)x+\frac{7k}{4}=0$ is negative, then $k$ lies in the interval

1. $(-\infty ,0)$
2. $(-\infty ,1]\cup [4,\infty )$
3. $R-\left\{0\right\}$
4. $(-\infty ,0)\cup [4,\infty )$

Q. Let $x_1,x_2(x_1\ne x_2)$ be the roots of the equation $x^2+2(m-3)x+9=0$. If $-6<x_1,x_2<1,$ then ${}^{\prime }{m}^{\prime }$ lies in the interval

1. $(6,\frac{27}{4}]$
2. $(\frac{27}{4},9)$
3. $(2,\frac{27}{4})$
4. $(6,\frac{27}{4})$

Q. If both the roots of $x^2+2(a+2)x+9a-1=0$ are negative, then ${}^{\prime }{a}^{\prime }$ lies in

1. $(\frac{1}{9},1]\cup [4,\infty )$
2. $[\frac{1}{9},1]\cup [4,\infty )$
3. $(\frac{1}{9},\frac{5-\sqrt{5}}{2}]\cup [\frac{5+\sqrt{5}}{2},\infty )$
4. $(-2,\infty )$

Q. If $3\text{and}4$ lies between the roots of the equation $x^2+2kx+9=0$ then $k$ lies in the interval

1. $(-\infty ,-3)$
2. $(-\infty ,-\frac{25}{8})\cup (3,\infty )$
3. $(-\infty ,-\frac{25}{8})$
4. $(-\frac{25}{8},\infty )$

Q. If $a\ne 0$ and equation $a{x}^2+bx+c=0$ has two roots $\alpha$ and $\beta$ such that $\alpha <-3$ and $\beta >2,$ which of the following is always true ?

1. $a(a+|b|+c)>0$
2. $a(a+|b|+c)<0$
3. $9a-3b+c>0$
4. $(9a-3b+c)(4a+2b+c)<0$

Q. Number of integral values of $\lambda$ for which $x^2-2\lambda x<41-6\lambda \forall x\in (1,6],$ is

Q.

If $x^2$ + 2(a - 1)x + a + 5 =0 has real roots belonging to the interval (1, 3) then a$ϵ$

1. $(-\infty ,\frac{-8}{7}]$

2. $(4,\infty )$

3. $(-\infty ,-\frac{48}{3})$

4. $(\frac{-8}{7},-1)$

Q. The values of $a$ for which the number $6$ lies in between the roots of the equation $x^2+2(a-3)x+9=0,$ belong to

1. $(\frac{3}{4},\infty )$
2. $(-\infty ,-\frac{3}{4})$
3. $(-\infty ,0)\cup (6,\infty )$
4. $(-\infty ,0)\cup (3,\infty )$

Q.

Suppose that the three quadratic equations $a{x}^2-2bx+c=0,b{x}^2-2cx+a=0$ and $c{x}^2-2ax+b=0$ all have only positive roots. Then,

1. $b^2=ca$
2. $c^2=ab$
3. $a^2=bc$
4. $a=b=c$

Q. The number of integral values of $m$ such that the roots of $x^2-(m-3)x+m=0$ lie in the interval $(1,2),$ is

Q. Let $f\left(x\right)=({\lambda }^2+\lambda -2)x^2+(\lambda +2)x$ be a quadratic polynomial. The sum of all integral values of $\lambda$ for which $f\left(x\right)<1\forall x\in R$, is

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

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 set of values of $k$ for which the equation $(k+2)x^2-2kx-k=0$ has two roots on the number line symmetrically placed about $1$ is

1. $(-\infty ,-2)\cup (1,\infty )$
2. $(1,\infty )$
3. $\{0,2\}$
4. $\varphi$

Q. If both the roots of $x^2+2(k+1)x+9k-5=0$ are less than $0$ and $k<100,$ then the number of integral values of $k$ is

1. $93$
2. $94$
3. $92$
4. $95$

Q. If both the roots of $x^2+2ax+a=0$ are less than $2$, then the set values of ${}^{\prime }{a}^{\prime }$ is

1. $(-\infty ,0]\cup [1,\infty )$
2. $[-\frac{4}{5},0)\cup (1,\infty )$
3. $(-\frac{4}{5},0]\cup [1,\infty )$
4. $(-2,0]\cup [1,\infty )$