Location of Roots

Q.

Write the number of quadratic equations, with real roots, which do not change by squaring their roots.

Q.

If $\alpha$and $\beta$ are the roots of the equation$x^2-2x+4=0$, then the value ${\alpha }^n+{\beta }^n$of will be

1. $i{\left(2\right)}^{n+1}\sin (n\pi /3)$

2. ${\left(2\right)}^{n+1}\cos (n\pi /3)$

3. $i{\left(2\right)}^{n-1}\sin (n\pi /3)$

4. ${\left(2\right)}^{n-1}\cos (n\pi /3)$

Q.

The roots of the equation $x^4-2x^3+x=380$ are

1. $5,-4,\raisebox{1ex}{$\left[1\pm 5\sqrt{-3}\right]$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

2. $-5,4,\raisebox{1ex}{$\left[-1\pm 5\sqrt{-3}\right]$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

3. $5,4,\raisebox{1ex}{$\left[-1\pm 5\sqrt{-3}\right]$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

4. $-5,-4,\raisebox{1ex}{$\left[1\pm 5\sqrt{3}\right]$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

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

Q. If the roots of the equation $(m-2)x^2-(8-2m)x-(8-3m)=0$ are real and opposite in sign, then the number of integral value(s) of $m$ is

1. $0$
2. $1$
3. $2$
4. more than $2$

Q. Consider the equation $x^2+a|x|+1=0$, where $a$ is a parameter.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{No real roots}& \text{(P)}& a<-2\\ \text{(B)}& \text{Two real roots}& \text{(Q)}& \varphi \\ \text{(C)}& \text{Three real roots}& \text{(R)}& a=-2\\ \text{(D)}& \text{Four distinct real roots}& \text{(S)}& a\ge 0\\ & & \text{(T)}& a=2\end{array}$

Which of the following is the only CORRECT combination?

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

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. 1)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
i) $f\left(x\right)=x^3,xϵ[-2,2]$

2)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
ii) $f\left(x\right)=\sin x+\cos x,xϵ[0,\pi ]$

3)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
iii) $f\left(x\right)=4x-\frac{1}{2}{x}^2,xϵ[-2,\frac{9}{2}]$

4)Find the absolute maximum value and the absolute minimum value of the function in the given interval:
iv) $f\left(x\right)=(x-1{)}^2+3,xϵ[-3,1]$

Q. '$af\left(k\right)<0$' is the necessary and sufficient condition for a particular real number $k$ to lie between the roots of a quadratic equation $f\left(x\right)=0$, where $f\left(x\right)=a{x}^2+bx+c$. If $f\left(k_1\right)f\left(k_2\right)<0$, then exactly one of the roots will lie between $k_1$ and $k_2$.

If $a(a+b+c)<0<c(a+b+c)$, then

1. one root is less than $0$, the other is greater than $1$
2. exactly one of the roots lies in $(0,1)$
3. at least one of the roots lies in $(0,1)$
4. both the roots lie in $(0,1)$

Q.

The roots of the equation ${\left(a+\sqrt{b}\right)}^{x^2-15}+{\left(a-\sqrt{b}\right)}^{x^2-15}=2a$, where $a^2-b=1$ are

1. $\pm 2,\pm 3$

2. $\pm 4,\pm \sqrt{14}$

3. $\pm 4,\pm \sqrt{5}$

4. $\pm 5,\pm \sqrt{20}$

Q. Complete set of values of $k$ for which the equation $4^x-(k+2)2^x+2k=0,$ has exactly one positive root is

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

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 number of integral value(s) of $a$ for which the equation $2^a{x}^2-4^{a}x-2^a-1=0$ has exactly one root between $1$ and $2$ is

Q. Find the value of $k$ so that the function $f$ is continuous at the indicated point.
$f\left(x\right)=\{\begin{array}{cc}kx+1,& \text{if}x\le \pi \\ \cos x,& \text{if}x>\pi \end{array}$ at $x=\pi$.

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 $f\left(x\right)=(4+y^2)x^2+2xy+1$ and $g\left(y\right)$ be the minimum value of $f\left(x\right)$. If $y\in R,$ then the maximum value of $g\left(y\right)$ is

1. $1$
2. $4$
3. $16$
4. $\frac{1}{4}$

Q. The range of $p$ for which the number $6$ lies between the roots of $x^2+2(p-3)x+9=0$ is

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

Q.

If $\alpha$ is one real root of quadratic equation $x^2-4x+1=0$ then $2^{nd}$ root $\beta$ is $(\alpha <\beta )$

1. 

2. -

3. -4 +

4. 

Q. Consider the triangle $ABC$ with usual notations and the following equations
$a^2{\sin }^{2}B+2a\sin B+|\sin \theta |=0$ ... (i)
$b^2{\sin }^{2}A+2b\sin A+|\sin \theta |=0$ ... (ii)
The number of values of $\theta$ in $[0,10\pi ]$ is

1. $10$
2. $4$
3. $5$
4. $15$

Q. Which of the following cannot be true for the given quadratic equation $a{x}^2+bx+c=0;c\ne 0$ if ratio of roots of this equation is equal to it's reciprocal?

1. $b\ne 0$ and $c$ is positive when $a>0$.
2. $b=0$ and $c$ is negative when $a>0$.
3. $b\ne 0$ and $c$ is negative when $a<0$.
4. $b=0$ and $c$ is negative when $a<0$.

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. Consider the equation $(m^2+1)x^2-3x+(m^2+1{)}^2=0.$ Let $p$ be the least value of product of roots and $q$ be the greatest value of sum of roots of the equation. Then the sum of an infinitely decreasing G.P. whose first term is equal to $p+2$ and the common ratio is $\frac{2}{q},$ is

1. $6$
2. $14$
3. $8$
4. $9$

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 the quadratic equations $a{x}^2+2cx+b=0$ and $a{x}^2+2bx+c=0,(b\ne c)$ have a common root, then the value of $a+4b+4c$ is

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

Q. If $x,y$ and $z$ are all positive, then the minimum value of $f(x,y,z)=x^3+12\left(\frac{yz}{x}\right)+16{\left(\frac{1}{yz}\right)}^{3/2}$ is

1. $14$
2. $24$
3. $42$
4. $3$

Q. The range of $p$ for which the number $6$ lies between the roots of $x^2+2(p-3)x+9=0$ is

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

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. If the roots of the equation, $x^2+6x+b=0$ are real and distinct and they differ by at most $8$, then $b$ lies in the interval

1. $[-7,9)$
2. $[-7,16)$
3. $(-7,9]$
4. $(7,25)$

Q. The domain of the function $y={\sin }^{-1}(-x^2)$ is

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