The machine as shown has 2 rods of length 1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :
(1) Speed which is $\frac { 3 } { 4 }$ th of that of the roller when the weight is 0.4 m above the ground
(2) Constant speed
(3) Decreasing speed
(4) Increasing speed

A man in a car at location Q on a straight highway is moving with speed v . He decides to reach a point $P$ in a field at a distance $d$ from highway (point $M$) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach $P$ is minimum? [Figure]
(1) $\frac { \mathrm { d } } { \sqrt { 3 } }$
(2) $\frac { d } { 2 }$
(3) $\frac { d } { \sqrt { 2 } }$
(4) d

The shortest distance between the point $\left( \frac { 3 } { 2 } , 0 \right)$ and the curve $y = \sqrt { x } , ( x > 0 )$, is
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { \sqrt { 5 } } { 2 }$

Let the function, $f : [-7, 0] \rightarrow R$ be continuous on $[-7, 0]$ and differentiable on $(-7, 0)$. If $f(-7) = -3$ and $f ^ { \prime } (x) \leq 2$ for all $x \in (-7, 0)$, then for all such functions $f$, $f(-1) + f(0)$ lies in the interval
(1) $(-\infty, 20]$
(2) $[-3, 11]$
(3) $(-\infty, 11]$
(4) $[-6, 20]$

A constant power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time $t$ is proportional to:-
(1) $t ^ { \frac { 2 } { 3 } }$
(2) $t ^ { \frac { 3 } { 2 } }$
(3) $t$
(4) $t ^ { \frac { 1 } { 2 } }$

The shortest distance between the line $x - y = 1$ and the curve $x ^ { 2 } = 2 y$ is:
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 2 \sqrt { 2 } }$
(4) 0

If the normal to the curve $y ( x ) = \int _ { 0 } ^ { x } \left( 2 t ^ { 2 } - 15 t + 10 \right) d t$ at a point $( a , b )$ is parallel to the line $x + 3 y = - 5 , a > 1$, then the value of $| a + 6 b |$ is equal to $\_\_\_\_$.

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is equal to (if the full question text was truncated, this is the standard formulation of this problem).

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is $\tan ^ { - 1 } \frac { 3 } { 4 }$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is $\_\_\_\_$ .

A body of mass 500 g moves along $x$-axis such that it's velocity varies with displacement $x$ according to the relation $v = 10 \sqrt { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the force acting on the body is:
(1) 125 N
(2) 25 N
(3) 166 N
(4) 5 N

Let $f(x) = (x+3)^2(x-2)^3$, $x \in [-4, 4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4, 4]$, then the value of $M - m$ is:
(1) 600
(2) 392
(3) 608
(4) 108

Q5. A body is moving unidirectionally under the influence of a constant power source. Its displacement in time $t$ is proportional to :
(1) $t$
(2) $t ^ { 3 / 2 }$
(3) $t ^ { 2 }$
(4) $t ^ { 2 / 3 }$

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81 \mathrm {~cm} ^ { 3 } / \mathrm { min }$ and the thickness of the ice-cream layer decreases at the rate of $\frac { 1 } { 4 \pi } \mathrm {~cm} / \mathrm { min }$. The surface area (in $\mathrm { cm } ^ { 2 }$) of the chocolate ball (without the ice-cream layer) is :
(1) $196 \pi$
(2) $256 \pi$
(3) $225 \pi$
(4) $128 \pi$

4.
For Oxford applicants in Mathematics / Mathematics \& Statistics / Mathematics \& Philosophy, OR those not applying to Oxford, ONLY.
Point $A$ is on the parabola $y = \frac { 1 } { 2 } x ^ { 2 }$ at ( $a , \frac { 1 } { 2 } a ^ { 2 }$ ) with $a > 0$. The line $L$ is normal to the parabola at $A$, and point $B$ lies on $L$ such that the distance $| A B |$ is a fixed positive number $d$, with $B$ above and to the left of $A$. [0pt] (i) [6 marks] Find the coordinates of $B$ in terms of $a$ and $d$. [0pt] (ii) [4 marks] Show that in order for $B$ to lie on the parabola, we must have
$$a ^ { 2 } d = 2 \left( 1 + a ^ { 2 } \right) ^ { 3 / 2 }$$
(iii) [2 marks] Let $t = a ^ { 2 }$ and express the equality ( $*$ ) in the form $d ^ { 2 / 3 } = f ( t )$ for some function $f$ which you should determine explicitly. [0pt] (iv) [3 marks] Find the minimum value of $f ( t )$. Hence show that the equality ( $*$ ) holds for some real value of $a$ if and only if $d$ is greater than or equal to some value, which you should identify.

A medicine is administered to a patient and t hours later the blood concentration of the active ingredient is given by $c ( t ) = t e ^ { - t / 2 }$ milligrams per milliliter. Determine the maximum value of $c ( t )$ and indicate at what moment this maximum value is reached. Knowing that the maximum safe concentration is $1 \mathrm { mg } / \mathrm { ml }$, indicate whether there is risk to the patient at any time.

The power generated by a battery is given by the expression $P(t) = 25te^{-t^{2}/4}$, where $t > 0$ is the operating time.\ a) (0.5 points) Calculate the value towards which the power generated by the battery tends if left operating indefinitely.\ b) (0.75 points) Determine the maximum power generated by the battery and the instant at which it is reached.\ c) (1.25 points) The total energy generated by the battery up to instant $t$, $E(t)$, is related to power by $E'(t) = P(t)$, with $E(0) = 0$. Calculate the energy produced by the battery between instant $t = 0$ and instant $t = 2$.

12. The cross-section of a triangular prism is an equilateral triangle with side $2 x \mathrm {~cm}$. The length of the prism is $d \mathrm {~cm}$.
Let the total surface area of the prism be $T \mathrm {~cm} ^ { 2 }$. Given that the volume of the prism is $T \mathrm { cm } ^ { 3 }$, which one of the following is an expression for $d$ in terms of $x$ ?
A $\frac { x } { 2 x - 3 }$
B $\frac { 3 x } { 3 x - 2 \sqrt { 3 } }$
C $\frac { 2 x } { x - 4 \sqrt { 3 } }$
D $\frac { 2 x } { x - 2 \sqrt { 3 } }$
E $\frac { 2 x } { x - \sqrt { 3 } }$

The function $f(x) = mx - 1 + \frac{1}{x}$ is given.
Accordingly, what is the smallest value of $m$ that satisfies the property $f(x) \geq 0$ for all $x > 0$?
A) $\frac{1}{2}$
B) $\frac{1}{3}$
C) $\frac{1}{4}$
D) $\frac{1}{5}$
E) $\frac{1}{6}$

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) $\frac { 11 } { 2 }$

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

An internet company can serve at most 1000 customers and can reach this number when it sets the monthly internet fee at 40 TL. The company has observed that after each 5 TL increase in the monthly internet fee, the number of customers decreases by 50.
At what monthly internet fee should this company set its rate to maximize the total revenue from the monthly internet fee?
A) 55 B) 60 C) 65 D) 70 E) 75