If $f(x)=x^2+b x+c$ and $f(1+k)=f(1-k) \forall k \in R$, for two real numbers $b$ and $c$ then

If $\alpha, \beta$ are the roots of the equation $x^2+3 x+k=0$ and $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are the roots of the equation $4 x^2+p x+18=0$, then $k$ satisfies the equation

If $f(x)$ is a second degree polynomial such that $f(x) \geq 0 \forall x \in R, f(-3)=0$ and $f(0)=18$, then $f(3)=$

If one of the roots of the equation $6 x^3-25 x^2+2 x+8=0$ is an integer and $\alpha>0, \beta<0$ are the other two roots, then $\frac{4}{\alpha}+\frac{1}{\beta}=$

If $\alpha, \beta, \gamma, \delta$ and $\varepsilon$ are the roots of the equation $x^5+x^4-13 x^3-13 x^2+36 x+36=0$ and $\alpha<\beta<\gamma<\delta<\varepsilon$ then $\frac{\varepsilon}{\alpha}+\frac{\delta}{\beta}+\frac{1}{\gamma}=$

If $\tan \theta$ and $\cot \theta$ are two distinct roots of the equation $a x^2+b x+c=0, a \neq 0, b \neq 0$, then

Sum of all the roots of the equation $||2 x-3|-4|=2$ is

If the quotient and remainder obtained when the expression $3 x^5-6 x^4+2 x^3+4 x^2-5 x+8$ is divided by the expression $x^2-2 x+3$ are $a x^3+b x^2+c x+d$ and $p x+q$ respectively, then $a b+c d=$

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $12 x^4-56 x^3+89 x^2-56 x+12=0$ such that $\alpha \beta=\gamma \delta=1$ and $\frac{\alpha+\beta}{\gamma+\delta}>1$, then $\frac{\alpha+\beta}{\gamma+\delta}=$

If the equations $x^2+p x+2=0$ and $x^2+x+2 p=0$ have a common root, then the sum of the roots of the equation $x^2+2 p x+8=0$ is

If both roots of the equation $x^2-5 a x+6 a=0$ exceed 1 , then the range of ' $a$ ' is

If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4-4 x^3+3 x^2+2 x-2=0$ such that $\alpha$ and $\beta$ are integers and $\gamma, \delta$ are irrational numbers, then $\alpha+2 \beta+\gamma^2+\delta^2=$

The equation having the multiple root of the equation $x^4+4 x^3-16 x-16=0$ as its roots is

If the equation $x^2-3 a x+a^2-2 a-k=0$ has different real roots for every rational number $a$, then $k$ lies in the interval

The number of all common roots of the equation $x^4-10 x^3+37 x^2-60 x+36=0$ and the transformed equation of it obtained by increasing any two distinct roots of it by 1 , keeping the other two roots fixed, is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-P x^2+Q x-R=0$ and $(\alpha-2)^2,(\beta-2)^2,(\gamma-2)^2$ are the roots of the equation $x^3-5 x^2+4 x=0$, then the possible least value of $P+Q+R$ is

The number of integral values of ' $a$ ' for which the quadratic equation $a x^2+a x+5=0$ cannot have real roots is

If the roots of the equation $32 x^3-48 x^2+22 x-3=0$ are in arithmetic progression, then the square of the common difference of the roots is

With respect to the roots of the equation $3 x^{3}+b x^{2}+b x+3=0$, match the items of List I with those fo List II

$\alpha, \beta, \gamma, 2$ and $\varepsilon$ are the roots of the equation

$ \begin{aligned} & \alpha, \beta, \gamma+4 x^4-13 x^3-52 x^2+36 x+144=0 . \text { If } \\ & \alpha<\beta<\gamma<2<\varepsilon \text {, then } \alpha+2 \beta+3 \gamma+5 \varepsilon= \end{aligned} $

If the quadratic equation $3 x^2+(2 k+1) x-5 k=0$ has real and equal roots, then the value of $k$ such that

$\frac{1}{2}$ < $k$ < 0 is

If $\alpha, \beta$ are the roots of $x^2+a x+2=0$ and $1 / \alpha, 1 / \beta$ are the roots of $x^2-b x+c=0$, then

$ \left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)= $

The sum of all the real values of $x$ satisfying the equation $\left(x^2-7 x+11\right)^{x^2-6 x-7}=1$ is

If $x^2+2 p x-2 p+8>0$ for all real values of $x$, then the set of all possible values of $p$ is

If $R-(\alpha, \beta)$ is the range of $\frac{x+3}{(x-1)(x+2)}$, then the sum of the intercepts of the line $\alpha x+\beta y+1=0$ on the coordinate axes is

The roots of the equation $x^4+x^3-4 x^2+x+1=0$ are diminished by $h$ so that, the transformed equation does not contain $x^2$ term. If the values of such $h$ are $\alpha$ and $\beta$, then $12(\alpha-\beta)^2=$

If $\sin 2 \theta$ and $\cos 2 \theta$ are solutions of $x^2+a x-c=0$, then

Let the equations $a x^2-7 x+c=0$ and $a x^2+5 x-c=0$ have a common root and $a c \neq 0$. If 3 is a root of $a x^2-7 x+c=0$ other than the common root, then the common root of the given equations is

The set of all values of $x$ for which the inequalities $x^2-7 x+10 \geq 0$ and $2 x+3-x^2>0$ hold simultaneously is

If $x^2+3 x-2 k=0$ and $x^2-2 x-7 k=0$ have a non-zero common root, then the positive root of the equation $k x^2+(k+2) x-(k+1)=0$ is

The values of $\frac{x^2-2 x+1}{x^2+x-1}$ do not lie in the interval

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$

If the sum of two particular roots of the equation $x^4-4 x^3-7 x^2+22 x+24=0$ is equal to the sum of the remaining two roots, then the sum of the cubes of all the roots of this equation is

The set of all values of $x$ which satisfy both the inequations $x^2-1 \leq 0$ and $x^2-x-2 \geq 0$ simultaneously is

If $\alpha$ and $\beta$ are the roots of the equation $x^2+2 x+2=0$, then $\alpha^{15}+\beta^{15}=$

If the equation whose roots are $P$ times the roots of the equation $x^4-2 a x^3+4 b x^2+8 a x+16=0$ is a reciprocal equation, then $|P|=$

Statement I The set of solutions of $|x|^2-4|x|+3<0$ is the interval $(-3,3)$

Statement II If $x<3$ or $x>5$, then $x^2-8 x+15>0$

Which of the above statements is (are) true?

If $6 x-x^2+12$ attains its extreme value $\beta$ at $x=\alpha$, then $\beta=$

Let $a$ be a common root of the equations $x^3-2 x-25 \lambda=0,3 x^3-8 x-\frac{175}{3} \lambda=0$ and $\lambda>0$. Then, $\lambda=$

If the sum of two roots of the equation $x^3-7 p x^2+5 q x-6 r=0$ is zero, then

If $\alpha$ and $\beta$ are the irrational roots of the equation $3 p^2 x^3+p x^2+q x+3=0$ when $p=1$ and $q=-7$, then $|\alpha-\beta|=$

The roots of a cubic equation $f(x)=0$ are diminished by $\frac{-3}{2}$ so, as to remove the term containing $x^2$ and the transformed equation is $8 x^3-54 x-78=0$. Then, the equation $f(x)=0$ is

If $\alpha$ and $\beta$ are the roots of a quadratic equation $x^2+b x+c=0$ such that $\alpha^2+\beta^2=5$ and $\alpha^3+\beta^3=9$, then $b+c=$

The set of all real values of the expression $\frac{x^2-x+2}{x^2+x-2} \forall x \in R-\{-2,1\}$ is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-9 x^2+23 x-15=0$, then $\alpha^3+\beta^3+\gamma^3=$

If $\alpha, \beta$ and $2 \beta$ are the real roots of the equation $x^3-9 x^2+k=0$ and $k \in R-\{0\}$, then $14 \beta=$

The sum of all distinct roots of the equation $x^5-3 x^4+5 x^3-5 x^2+3 x-1=0$ is

$\left(x^4+1\right)=\frac{1}{a}(x+1)^4$ is a reciprocal equation

Let $f(x)=A x^2+B x, g(x)=L x^2+M x+N$. Given that $f(2)-g(2)=1, f(3)-g(3)=4, f(4)-g(4)=9$. Then, a root of $f(x)-g(x)=0$ is

If $f(x)=\frac{2 x-3}{(x-2)(x-3)}$ is a real valued function, then the value that $f(x)$ does not take is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-3 x^2+2 x-4=0$, then $\Sigma \alpha^2 \beta^2=$

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ such that $\alpha+\beta=0$, then $\alpha^2+2 \beta^2+3 \gamma^2=$

If $m$ and $M$ are respectively, the smallest and greatest rational roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$, then $M-m=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2 \sqrt{3} x+4=0$, then $\alpha^6+\beta^6=$

When $b=17$, it is found that the roots of the equation $x^2+b x+c=0$ are -2 and -15 . If $\alpha$ and $\beta$ are the roots of the same equation when $b=13$, then $|\alpha-\beta|=$

Let $x$ be a real number. Malch the following:

$ \text { The correct match is : } $

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-2 x-4=0$, then $\alpha^3+\beta^3+\gamma^3=$

If the roots of $x^5-a x^4+b x^3-c x^2+d x-1=0$ are all positive such that their arithmetic mean and geometric mean are equal, then $a+b+c+d=$

The number of non-real roots of the equation $x^{10}-3 x^8+5 x^6-5 x^4+3 x^2-1=0$ is

If the quadratic equations $x^2-7 x+3 c=0$ and $x^2+x-5 c=0$ have a common root, then for non-zero real value of $c$ the sign of the expression $x^2-3 x+c$ is

II. Let $f(x)=\frac{6 x^2-18 x+21}{6 x^2-18 x+17}$. If $m$ is the maximum value of $f(x)$ and $f(x)>n \forall x \in R$. Then, $14 m-7 n=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+r=0$ and $\alpha^3+\beta^3+\gamma^3=5$, then $r=$

18. If $\frac{5}{2}$ is the sum of two roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$ then the sum of all non-real roots of the equation is

If $1-\sqrt{2}$ and $2+i$ are the roots of the equation $x^4+b x^3+c x^2+d x+e=0$ where $b, c, d, e$ are rational numbers, then the roots of the equation $b x^2+c x+d=0$ are

Let the transformed equation of $2 x^4-8 x^3+3 x^2-1=0$ so that the term containing the cubic power of $x$ is absent be $2 x^4+b x^2+c x+d=0$. Then, $b=$

If $\tan 15^{\circ}$ and $\tan 30^{\circ}$ are the roots of equation $x^2+p x+q=0$, then $p q=$

If the extreme value of $3 x-2 x^2+1$ is $k$, then the set of all real values of $x$ for which $k x^2+2 x+1>0$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-5 x^2-2 x+24=0$, then $\frac{\beta \gamma}{\alpha}+\frac{\gamma \alpha}{\beta}+\frac{\alpha \beta}{\gamma}=$

Let $p(x)$ be a quadratic polynomial with real coefficients. If $p(x)=0$ has only purely imaginary roots, then the zeroes of the polynomial $p(p(x))$ are

If $\alpha, \beta, \gamma$ are the roots of the equation $4 x^3+12 x^2-7 x+165=0$ and $\alpha+5, \beta+5, \gamma+5$ are the roots of the equation $a x^3+b x^2+c x+d=0$ then the product of the roots of the second equation is

For $n>2$ and $n \in \mathbf{N}$, the product of the roots of $(x-n)\left(\left(x^2-2 n x\right)^2+\left(2 n^2-5\right)\left(x^2-2 n x\right)\right. \left.+\left(n^4-5 n^2+4\right)\right)=0$ is divisible by

If $\alpha, \beta$ are the roots of $a x^2+b x+c=0$ then $\left(\frac{\alpha}{a \beta+b}\right)^3-\left(\frac{\beta}{a \alpha+b}\right)^3=$

The maximum value of $\left\{x \in \mathbf{R} / \sqrt{x+2}>\sqrt{8-x^2}\right\}=$

If $x$ is real, then the maximum and minimum values of $\frac{x^2+14 x+9}{x^2+2 x+3}$ are respectively

When $\mathbf{R}$ is the set of all real numbers,

$ \left\{x \in \mathbf{R}: \frac{\sqrt{12-x-x^2}}{x+10} \leq \frac{\sqrt{12-x-x^2}}{2 x+9}\right\}= $

If $\alpha$ and $\beta$ are two complex roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$, then $\alpha+\beta=$

If $\alpha$ and $\beta$ are the real roots of the equation $\sqrt{\frac{5 x}{x-2}}+\sqrt{\frac{x-2}{5 x}}=\frac{29}{10}$ and $\alpha>\beta$, then $\sqrt{\alpha^2-11^4 \beta^2}=$

The minimum value of $\frac{9 \cdot 3^{2 x}+6 \cdot 3^x+4}{9 \cdot 3^{2 x}-6 \cdot 3^x+4}$ is

$p$ is non-zero real number. If the equation whose roots are the squares of the roots of the equation $x^3-p x^2+p x-1=0$ is identical with the given equation, then $p=$

Let $S$ be the set of all possible integral values of $\lambda$ in the interval $(-3,7)$ for which the roots of the quadratic equation $\lambda x^2+13 x+7=0$ are all rational numbers. Then the sum of the elements in $S$ is

$\alpha$ is the maximum value of $1-2 x-5 x^2$ and $\beta$ is the minimum value of $x^2-2 x+r$. If $5 \alpha x^2+\beta x+6>0$ for all real values $x$, then the interval in which $r$ lies is

For the equation $x^4+x^3-4 x^2+x-1=0$ the ratio of the sum of the squares of all the roots to the product of the distinct roots is

If $\alpha_1, \beta_1, \gamma_1, \delta_1$ are the roots of the equation $a x^4+b x^3+c x^2+d x+e=0$ and $\alpha_2, \beta_2, \gamma_2, \delta_2$ are the roots of the equation $e x^4+d x^3+c x^2+b x+a=0$ such that $0<\alpha_1<\beta_1<\gamma_1<\delta_1, 0<\alpha_2<\beta_2<\gamma_2<\delta_2$, $\alpha_1-\delta_2=2=\beta_1-\gamma_2 ; \gamma_1-\beta_2=\delta_1-\alpha_2=4$, then $a+b+c+d+e=$