Location of Roots

Q. If both the roots of the equation $x^2+2(k+1)x+9k-5=0$ are negative, then the minimum integral value of $k$ is

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

Q. If the roots of $x^2-6kx+(2-2k+9k^2)=0$ are greater than $3$, then the range of $k$ is

1. $(1,\infty )$
2. $(\frac{11}{9},\infty )$
3. $(-\infty ,1)\cup (\frac{11}{9},\infty )$
4. $(1,\frac{11}{9})$

Q.

The number of roots of the equation log(-2x) = 2 log(x+1) are

1. 3

2. 2

3. 1

4. 0

Q. If both the distinct roots of the equation $x^2-ax-b=0(a,b\in R)$ are lying in between $-2$ and $2$, then

1. $|a|<2-\frac{b}{2}$
2. $|a|>2-\frac{b}{2}$
3. $|a|<4$
4. $|a|>\frac{b}{2}-2$

Q. cos x = b, for what value b do the roots of the equation form an AP

1. 
2. 
3. 
4. 

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

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 the quadratic equation $a{x}^2+bx+c=0$ with real coefficients has real and distinct roots in $(1,2)$ then $a$ and $5a+2b+c$

1. have same sign.
2. have opposite sign.
3. are equal.
4. are not real.

Q. If at least one of the roots of the equation $x^2-(a-3)x+a=0$ lies in the interval $(1,2),$ then $a$ lies in the interval

1. $[9,\infty )$
2. $(10,\infty )$
3. $[9,10)$
4. $(5,7)\cup (10,\infty )$

Q. The number of integral value(s) of $m$ for which the quadratic expression,
$(1+2m)x^2-2(1+3m)x+4(1+m),x\in R,$ is always positive, is

Q.

Let $x^2$ - (m - 3) x + m = 0, m ∈ R be a quadratic equation. The values of m for which both roots lie in between 1 and 2 is given by

1. m < 10

2. m [1, 9]

3. m (5, 7)

4. m

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 real number k for which the equation $x^3+3x+k=0$ has two distinct real roots in [0, 1]

1. lies between 1 and 2
2. lies between 2 and 3
3. lies between -1 and 0
4. does not exist.

Q. If all the roots of the equation $p{x}^4+q{x}^2+r=0,p\ne 0,q^2\ge 9pr$ are real, then which of the following option(s) is/are correct?

1. $q>0,p>0,r>0$
2. $q>0,p<0,r>0$
3. $q>0,p<0,r<0$
4. $q<0,p>0,r>0$

Q. If atleast one of the root of the equation $x^2-(a-3)x+a=0$ is lies in the interval $(1,2)$, then $a$ lies in the interval

1. $[9,\infty )$
2. $(10,\infty )$
3. $[9,10)$
4. $(5,7)\cup (10,\infty )$

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 domain of the function $f\left(x\right)=\sqrt{{\log }_{x^2-1}x}$ is

1. $(0,\sqrt{2})$
2. $(0,\infty )$
3. $(1,\infty )$
4. $(\sqrt{2},\infty )$

Q. If exactly two integers lie between the roots of the equation $x^2+ax-1=0$, then possible integral value(s) of $a$ is (are)

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

Q. If the equation $2^{2x}+a\cdot 2^{x+1}+a+1=0$ has roots of opposite sign, then ${}^{\prime }{a}^{\prime }$ lies in the interval

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

Q. If $(x^2+x+2{)}^2-(a-3\left)\right(x^2+x+1\left)\right(x^2+x+2)+(a-4\left)\right(x^2+x+1{)}^2=0$ has at least one real root, then the complete set of values of $a$ is

1. $[\frac{3}{4},\infty )$
2. $(\frac{3}{4},\infty )$
3. $[5,\frac{19}{3})$
4. $(5,\frac{19}{3}]$

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. Let $a,b(b>a)$ are the roots of the quadratic equation $(k+1)x^2-(20k+14)x+91k+40=0;$ where $k>0$, then which among the following option(s) is/are correct for the roots

1. $a\in (4,7)$
2. $b\in (4,7)$
3. $a\in (7,10)$
4. $b\in (10,13)$

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}{x}^2+x-a=0\text{has integral roots}& \text{(P)}& 2\\ & \text{and}a\in N,\text{than}a\text{can be equal to}& & \\ \text{(B)}& \text{If the equation}a{x}^2+2bx+4c=16& \text{(Q)}& 12\\ & \text{has no real roots and}a+c>b+4,& & \\ & \text{then the integral value of}c\text{can be}& & \\ \text{(C)}& \text{If the equation}{x}^2+2bx+9b-14=0& \text{(R)}& 1\\ & \text{has only negative roots, then the integral}& & \\ & \text{values of}b\text{can be}& & \\ \text{(D)}& \text{If}n\text{is the number of solutions of}& \text{(S)}& 30\\ & \text{the equation}|x-|4-x||-2x=4,\text{then}& & \\ & \text{the value of}n\text{is}& & \end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(P), (Q), (S)}$
2. $\text{(A)}\to \text{(P), (Q), (R)}$
3. $\text{(B)}\to \text{(P), (Q)}$
4. $\text{(C)}\to \text{(R)}$

Q. The domain of ${{}^{9-x}P}_{x-2}$ is

1. $x\in \{2,3,4,5,6\}$
2. $x\in \{2,3,4,5\}$
3. $x\in \{3,4,5\}$
4. $x\in \{3,4,5,6\}$

Q. The number of integral values of $a$ for which $4^t-(a-4)2^t+\frac{9a}{4}<0,\forall t\in (1,2)$ is

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 $5,5r,5r^2$ are the side lengths of a triangle, then the possible value(s) of $r$ is/are

1. $\frac{3}{2}$
2. $\frac{3}{4}$
3. $\frac{5}{2}$
4. $\frac{7}{4}$

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. $4$