The number of values of $b$ for which there is an isosceles triangle with sides of lengths $b+5$,$3b-2$,and $6-b$ is

Consider the curves given by the following quadratic functions:

$f_1(x) = 5 x^2 + 2 x + 1$	$f_2(x) = 5 x^2 + 6 x + 1$
$f_3(x) = x^2 - 7 x + 6$	$f_4(x) = 64 x^2 + 48 x + 9$

If $A_1, A_2, A_3$ and $A_4$ denote the lengths of the intercepts on the $X$-axis made by the above curves respectively,then which of the following is true?

If $a, b, c, d$ are real numbers such that $a < b < c < d$,then the roots of the equation $(x-a)(x-c)+2(x-b)(x-d)=0$ are

If $\alpha, \beta, \gamma$ are the roots of $x^3-3x^2-4x+12=0$,then $\sum(\alpha+\beta)^2$ is equal to

Let $P(x) = x^3 - ax^2 + bx + c$ where $a, b, c \in \mathbb{R}$ have integral roots such that $P(6) = 3$. Then $a$ cannot be equal to:

The quadratic equation whose roots are $\sin^2 18^{\circ}$ and $\cos^2 36^{\circ}$ is: