How many real solutions are there to the equation $$2 \cos ^ { 4 } \theta - 5 \cos ^ { 2 } \theta + 3 = 0$$ in the interval $0 \leq \theta \leq 2 \pi$ ?
Find the complete set of values of $p$ for which the equation $$x ^ { 2 } - 2 p x + y ^ { 2 } - 6 y - p ^ { 2 } + 8 p + 9 = 0$$ describes a circle in the $x y$-plane.
These sectors of circles are similar. The arc length of the smaller sector is 6 . The difference between the areas of the sectors is 21 . Find the positive difference between the perimeters of the sectors.
The terms $x _ { n }$ of a sequence follow the rule $$x _ { n + 1 } = \frac { x _ { n } + p } { x _ { n } + q }$$ where $p$ and $q$ are real numbers. Given that $x _ { 1 } = 3 , x _ { 2 } = 5$, and $x _ { 3 } = 7$, find the value of $x _ { 4 }$
A geometric sequence has first term $a$ and common ratio $r$, where $a$ and $r$ are positive integers and $r$ is greater than 1. The sum of the first $n$ terms of this sequence is denoted by $S _ { n }$ It is given that the terms of the sequence satisfy $$S _ { 30 } - S _ { 20 } = k S _ { 10 }$$ for some positive integer $k$. What is the smallest possible value of $k$ ?
This question is about pairs of functions f and g that satisfy $$\begin{aligned}
f ( x ) - g ( x ) & = 2 \sin x \\
f ( x ) g ( x ) & = \cos ^ { 2 } x
\end{aligned}$$ for all real numbers $x$. Across all solutions for $\mathrm { f } ( x )$, what is the minimum value that $\mathrm { f } ( x )$ attains for any $x$ ?
A sequence of translations is applied to the graph of $y = x ^ { 3 }$ Which of the following graphs could be the result of this sequence of translations? $$\begin{array} { l l }
\text { I } & y = x ^ { 3 } - 3 x ^ { 2 } + 9 x - 27 \\
\text { II } & y = x ^ { 3 } - 9 x ^ { 2 } + 27 x - 3 \\
\text { III } & y = 27 x ^ { 3 } - 9 x ^ { 2 } + x - 3
\end{array}$$
A family of quadratic curves is given by $$y _ { k } = 2 \left( x - \frac { k } { 2 } \right) ^ { 2 } + \frac { k ^ { 2 } } { 2 } + 4 k + 3$$ where $k$ is any real number and $y _ { k }$ is a function of $x$. All these curves are sketched, and the point with the lowest $y$-coordinate among all the curves $y _ { k }$ is $( a , b )$. Find the value of $a + b$
Given that $$\left( a ^ { 3 } + \frac { 2 } { b ^ { 3 } } \right) \left( \frac { 2 } { a ^ { 3 } } - b ^ { 3 } \right) = \sqrt { 2 }$$ where $a$ and $b$ are real numbers, what is the least value of $a b$ ?
A circle has centre $O$ and radius 6 . $P , Q$ and $R$ are points on the circumference with angle $P O Q \geq \frac { \pi } { 2 }$ The area of the triangle $P O Q$ is $9 \sqrt { 3 }$ What is the greatest possible area of triangle $P R Q$ ?
A rectangle is drawn in the region enclosed by the curves $p$ and $q$, where $$\begin{aligned}
& p ( x ) = 8 - 2 x ^ { 2 } \\
& q ( x ) = x ^ { 2 } - 2
\end{aligned}$$ such that the sides of the rectangle are parallel to the $x$ - and $y$-axes. What is the maximum possible area of the rectangle?
The solutions to $7 x ^ { 4 } - 6 x ^ { 2 } + 1 = 0$ are $\pm \cos \theta$ and $\pm \cos \beta$. Which one of the following equations has solutions $\pm \sin \theta$ and $\pm \sin \beta$ ?
It is given that $$\begin{aligned}
& \mathrm { f } ( x ) = x ^ { 2 } ( x - 1 ) ^ { 2 } ( x - 2 ) \\
& \mathrm { g } ( x ) = - p ( x - q ) ^ { 2 } ( x - r ) ^ { 2 }
\end{aligned}$$ where $p , q$ and $r$ are positive and $q < r$ Find the set of values of $q$ and $r$ that guarantees the greatest number of distinct real solutions of the equation $\mathrm { f } ( x ) = \mathrm { g } ( x )$ for all $p$.
Circle $C _ { 1 }$ is defined as $x ^ { 2 } + y ^ { 2 } = 25$ A second circle $C _ { 2 }$ has radius 4 and centre $( a , b )$ where $$- 2 \leq a \leq 2 \text { and } - 3 \leq b \leq 3$$ If the centre of $C _ { 2 }$ is equally likely to be located anywhere within the given range, what is the probability that $C _ { 2 }$ intersects $C _ { 1 }$ ?
$n$ is the number of points of intersection of the graphs $$y = \left| x ^ { 2 } - a ^ { 2 } \right| \text { and } y = a ^ { 2 } | x - 1 |$$ where $a$ is a real number. What is the smallest value of $n$ that is not possible?