Q61. If 2 and 6 are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, then the quadratic equation, whose roots are $\frac { 1 } { 2 a + b }$ and $\frac { 1 } { 6 a + b }$, is :
(1) $2 x ^ { 2 } + 11 x + 12 = 0$
(2) $x ^ { 2 } + 8 x + 12 = 0$
(3) $4 x ^ { 2 } + 14 x + 12 = 0$
(4) $x ^ { 2 } + 10 x + 16 = 0$

Q61. Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } + 2 \sqrt { 2 } x - 1 = 0$. The quadratic equation, whose roots are $\alpha ^ { 4 } + \beta ^ { 4 }$ and $\frac { 1 } { 10 } \left( \alpha ^ { 6 } + \beta ^ { 6 } \right)$, is :
(1) $x ^ { 2 } - 190 x + 9466 = 0$
(2) $x ^ { 2 } - 180 x + 9506 = 0$
(3) $x ^ { 2 } - 195 x + 9506 = 0$
(4) $x ^ { 2 } - 195 x + 9466 = 0$

2. For ALL APPLICANTS.

(i) Show, with working, that
$$x ^ { 3 } - ( 1 + \cos \theta + \sin \theta ) x ^ { 2 } + ( \cos \theta \sin \theta + \cos \theta + \sin \theta ) x - \sin \theta \cos \theta ,$$
equals
$$( x - 1 ) \left( x ^ { 2 } - ( \cos \theta + \sin \theta ) x + \cos \theta \sin \theta \right)$$
Deduce that the cubic in (1) has roots
$$1 , \quad \cos \theta , \quad \sin \theta$$
(ii) Give the roots when $\theta = \frac { \pi } { 3 }$.
(iii) Find all values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ such that two of the three roots are equal.
(iv)What is the greatest possible difference between two of the roots, and for what values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ does this greatest difference occur?
Show that for each such $\theta$ the cubic (1) is the same.

2. (i) Show, with working, that
$$x ^ { 3 } - ( 1 + \cos \theta + \sin \theta ) x ^ { 2 } + ( \cos \theta \sin \theta + \cos \theta + \sin \theta ) x - \sin \theta \cos \theta$$
equals
$$( x - 1 ) \left( x ^ { 2 } - ( \cos \theta + \sin \theta ) x + \cos \theta \sin \theta \right)$$
Deduce that the cubic in (1) has roots
$$1 , \quad \cos \theta , \quad \sin \theta$$
(ii) Give the roots when $\theta = \frac { \pi } { 3 }$.
(iii) Find all values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ such that two of the three roots are equal.
(iv) What is the greatest possible difference between two of the roots, and for what values of $\theta$ in the range $0 \leqslant \theta < 2 \pi$ does this greatest difference occur?
Show that for each such $\theta$ the cubic (1) is the same.

2. For ALL APPLICANTS.

Let $a$ and $b$ be real numbers. Consider the cubic equation
$$x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 } = 0$$
(i) Show that if $x = 1$ is a solution of ( $*$ ) then
$$1 - \sqrt { 2 } \leqslant b \leqslant 1 + \sqrt { 2 }$$
(ii) Show that there is no value of $b$ for which $x = 1$ is a repeated root of ( $*$ ).
(iii) Given that $x = 1$ is a solution, find the value of $b$ for which $( * )$ has a repeated root. For this value of $b$, does the cubic
$$y = x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 }$$
have a maximum or minimum at its repeated root?

Let the polynomial function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ are all rational numbers. Select the correct options.
(1) The graph of $y = f ( x )$ and the parabola $y = x ^ { 2 } + 100$ may have no intersection points
(2) If $f ( 0 ) f ( 1 ) < 0 < f ( 0 ) f ( 2 )$, then the equation $f ( x ) = 0$ must have three distinct real roots
(3) If $1 + 3 i$ is a complex root of the equation $f ( x ) = 0$, then the equation $f ( x ) = 0$ has a rational root
(4) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form an arithmetic sequence in order
(5) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form a geometric sequence in order

19. The positive real numbers $a , b$, and $c$ are such that the equation
$$x ^ { 3 } + a x ^ { 2 } = b x + c$$
has three real roots, one positive and two negative.
Which one of the following correctly describes the real roots of the equation
$$x ^ { 3 } + c = a x ^ { 2 } + b x ?$$
A It has three real roots, one positive and two negative.
B It has three real roots, two positive and one negative.
C It has three real roots, but their signs differ depending on $a , b$, and $c$.
D It has exactly one real root, which is positive.
E It has exactly one real root, which is negative.
F It has exactly one real root, whose sign differs depending on $a , b$, and $c$.
G The number of real roots can be one or three, but the number of roots differs depending on $a , b$, and $c$.

The leading coefficient is 1, and the fourth-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with real coefficients has roots $-i$ and $2i$. What is $\mathbf { P } ( \mathbf { 0 } )$?
A) 2
B) 4
C) 6
D) 7
E) 8

Let $a$ and $b$ be integers,
$$P(x) = x^{3} + ax^{2} + bx - 2$$
It is known that the polynomial has exactly one real root.
If $\mathbf{P}(1) = 0$, what is the smallest value that the integer $a$ can take?
A) $-6$ B) $-5$ C) $-4$ D) $-3$ E) $-2$