A curve $C$ has equation $y = f ( x )$ where
$$f ( x ) = p ^ { 3 } - 6 p ^ { 2 } x + 3 p x ^ { 2 } - x ^ { 3 }$$
and $p$ is real.
The gradient of the normal to the curve $C$ at the point where $x = - 1$ is $M$.
What is the greatest possible value of $M$ as $p$ varies?
A $- \frac { 3 } { 2 }$
B $- \frac { 2 } { 3 }$
C $- \frac { 1 } { 2 }$
D $\frac { 1 } { 4 }$
E $\frac { 2 } { 3 }$
F $\frac { 3 } { 2 }$

The curve $C$ has equation $y = x ^ { 2 } + b x + 2$, where $b \geq 0$.
Find the value of $b$ that minimises the distance between the origin and the stationary point of the curve $C$.
A $\quad b = 0$
B $b = 1$
C $b = 2$
D $b = \frac { \sqrt { 6 } } { 2 }$
E $\quad b = \sqrt { 2 }$
F $\quad b = \sqrt { 6 }$

Find the shortest distance between the curve $y = x^2 + 4$ and the line $y = 2x - 2$.

The following shape has two lines of reflectional symmetry.
$M N O P$ is a square of perimeter 40 cm .
The vertices of rectangle $R S T U$ lie on the edge of square $M N O P$.
$M R$ has length $x \mathrm {~cm}$.
What is the largest possible value of $x$ such that $R S T U$ has area $20 \mathrm {~cm} ^ { 2 }$ ?

Consider an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in the $xy$-plane. Here, $a$ and $b$ are constants satisfying $a > b > 0$. Answer the following questions.

1. Find the equation of the tangent line at a point $(X, Y)$ on the ellipse in the first quadrant.
2. The tangent line obtained in Question I. 1 intersects the $x$- and $y$-axes. Find the coordinates $(X, Y)$ at the tangent point that minimizes the length of the segment connecting the two intersects and obtain the minimum length of the segment.
3. Consider a region bounded by the segment obtained in Question I. 2 and the $x$- and $y$-axes, and let $C_{1}$ be a cone formed by rotating the region around the $x$-axis. Next, let $C_{2}$ be a cone with the maximum volume while having the same surface area (including a base area) as the cone $C_{1}$. Find $\frac{S_{2}}{S_{1}}$, where $S_{1}$ and $S_{2}$ are the base areas of the cones $C_{1}$ and $C_{2}$, respectively.

A workplace consisting of a corridor, kitchen, and study room has the model shown above as rectangle ABCD, and the perimeter of this rectangle is 72 meters.
For the kitchen in this workplace to have the largest area, what should $x$ be in meters?
A) 1
B) 2
C) 3
D) 4
E) 5

A line $d$ with negative slope passing through the point $(1,2)$ forms a triangular region with the coordinate axes. What is the minimum area of this triangular region in square units?
A) 2
B) 3
C) 4
D) $\frac { 9 } { 2 }$
E) $\frac { 7 } { 2 }$

For $x > 0$; if the point $(a, b)$ on the graph of the curve $y = 6 - x^2$ is closest to the point $(0, 1)$, what is b?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 9 } { 2 }$
E) $\frac { 11 } { 2 }$

A tour company charges 140 TL per person for a tour it will organize. If the number of registered participants exceeds 80, a refund of 50 kuruş will be made to all participants for each person above 80. The capacity is limited to 200 people.
For example, if 100 people participate in the tour, everyone receives a 10 TL refund and the per-person fee is 130 TL.
Accordingly, how many people should participate in the tour so that the company's revenue from participants is maximum?
A) 160
B) 165
C) 175
D) 180
E) 185

In the rectangular coordinate plane, the graph of the curve $y = e ^ { \left( - x ^ { 2 } \right) }$ is given.
In this plane, a rectangle with one side on the x-axis and two vertices on the curve is drawn with the maximum possible area.
What is the area of this rectangle in square units?
A) $\sqrt { \mathrm { e } }$
B) $\sqrt { 2 e }$
C) $\frac { \sqrt { e } } { 2 }$
D) $\sqrt { \frac { 2 } { \mathrm { e } } }$
E) $2 \sqrt { e }$

In the rectangular coordinate plane, rectangles are drawn such that two vertices lie on the x-axis and the other two vertices lie on the parabola $y = 27 - x ^ { 2 }$, and the rectangles lie between this parabola and the x-axis.
Accordingly, what is the perimeter of the rectangle with the largest area?
A) 40
B) 42
C) 44
D) 46
E) 48

A crystal in the shape of a cube with one edge of length $x$ units has a production cost of 5 TL per unit cube based on volume, and a selling price of 20 TL per unit square based on surface area.
Accordingly, for what value of x in units will the profit from selling this crystal be maximum?\ A) 16\ B) 18\ C) 20\ D) 22\ E) 24