Exercise 4 — Candidates who have chosen the specialization option

In the mountains, a hiker made reservations in two types of accommodations: Accommodation A and Accommodation B. One night in accommodation A costs $24 €$ and one night in accommodation B costs $45 €$. He remembers that the total cost of his reservation is $438 €$. We wish to find the numbers $x$ and $y$ of nights spent respectively in accommodation $A$ and accommodation $B$.

1. a. Show that the numbers $x$ and $y$ are respectively less than or equal to 18 and 9. b. Copy and complete lines (1), (2) and (3) of the following algorithm so that it displays the possible pairs ( $x ; y$ ).
   

   Input: Processing:	\begin{tabular}{l} $x$ and $y$ are numbers
   For $x$ varying from $0$ to $\ldots$ (1)
   For $y$ varying from $0$ to $\ldots$ (2)
   If $\ldots$ (3) Display $x$ and $y$ End If End For End For

   \hline \end{tabular}
   
2. Justify that the total cost of the reservation is a multiple of 3.
   
3. a. Justify that the equation $8 x + 15 y = 1$ admits at least one solution in integers. b. Determine such a solution. c. Solve the equation (E): $8 x + 15 y = 146$ where $x$ and $y$ are integers.
   
4. The hiker remembers having spent at most 13 nights in accommodation A. Show then that he can find the exact number of nights spent in accommodation A and that of nights spent in accommodation B. Calculate these numbers.

Question 155
A solução do sistema de equações $$\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}$$ é o par ordenado $(x, y)$ igual a
(A) $(1, 5)$ (B) $(2, 3)$ (C) $(3, 1)$ (D) $(4, -1)$ (E) $(5, -3)$

A solução do sistema de equações $$\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}$$ é
(A) $x = 1$ e $y = 5$ (B) $x = 2$ e $y = 3$ (C) $x = 3$ e $y = 1$ (D) $x = 4$ e $y = -1$ (E) $x = 5$ e $y = -3$

The supply and demand curves of a product represent, respectively, the quantities that sellers and consumers are willing to trade as a function of the product's price. In some cases, these curves can be represented by straight lines. Suppose that the quantities of supply and demand of a product are, respectively, represented by the equations: $$Q_{0} = -20 + 4P$$ $$Q_{D} = 46 - 2P$$ where $Q_{o}$ is the quantity supplied, $Q_{D}$ is the quantity demanded, and P is the price of the product.
From these supply and demand equations, economists find the market equilibrium price, that is, when $\mathrm{Q}_{\circ}$ and $\mathrm{Q}_{\mathrm{D}}$ are equal.
For the situation described, what is the value of the equilibrium price?

QUESTION 163
The solution of the system $$\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$$ is
(A) $x = 1, y = 4$
(B) $x = 2, y = 3$
(C) $x = 3, y = 2$
(D) $x = 4, y = 1$
(E) $x = 5, y = 0$

The figure shows three lines in the Cartesian plane, with $P, Q$ and $R$ being the intersection points between the lines, and $A, B$ and $C$ being the intersection points of these lines with the $x$-axis.
This figure is the graphical representation of a linear system of three equations and two unknowns that
(A) has three distinct real solutions, represented by points $P, Q$ and $R$, since they indicate where the lines intersect.
(B) has three distinct real solutions, represented by points $A, B$ and $C$, since they indicate where the lines intersect the $x$-axis.
(C) has infinitely many real solutions, since the lines intersect at more than one point.
(D) has no real solution, since there is no point that belongs simultaneously to all three lines.
(E) has a unique real solution, since the lines have points where they intersect.

A rectangular plot of land with sides whose measurements, in meters, are $x$ and $y$ will be fenced for the construction of an amusement park. One side of the plot is located on the banks of a river. Observe the figure.
To fence the entire plot, the owner will spend R\$ 7500.00. The fence material costs R\$ 4.00 per meter for the sides of the plot parallel to the river, and R\$ 2.00 per meter for the other sides.
Under these conditions, the dimensions of the plot and the total cost of the material can be related by the equation
(A) $4(2x + y) = 7500$
(B) $4(x + 2y) = 7500$
(C) $2(x + y) = 7500$
(D) $2(4x + y) = 7500$
(E) $2(2x + y) = 7500$

The sum of three consecutive even integers is 78. What is the largest of these integers?
(A) 24
(B) 26
(C) 28
(D) 30
(E) 32

A rectangle has a perimeter of 36 cm and its length is twice its width. What is the area, in square centimeters, of this rectangle?
(A) 48
(B) 72
(C) 96
(D) 108
(E) 144

The number of solutions of the system $$\begin{cases} x + y = 5 \\ 2x + 2y = 10 \end{cases}$$ is:
(A) 0
(B) 1
(C) 2
(D) 3
(E) Infinite

Given three distinct positive constants $a , b , c$ we want to solve the simultaneous equations
$$\begin{aligned} a x + b y & = \sqrt { 2 } \\ b x + c y & = \sqrt { 3 } \end{aligned}$$
(a) There exists a combination of values for $a , b , c$ such that the above system has infinitely many solutions $( x , y )$.
(b) There exists a combination of values for $a , b , c$ such that the above system has exactly one solution $( x , y )$.
(c) Suppose that for a combination of values for $a , b , c$, the above system has NO solution. Then $2 b < a + c$.
(d) Suppose $2 b < a + c$. Then the above system has NO solution.

Solve the system of equations:
$$\begin{aligned} & x _ { 1 } + 2 x _ { 2 } - 3 x _ { 3 } = 5 \\ & 2 x _ { 1 } - x _ { 2 } + 4 x _ { 3 } = 7 \\ & 3 x _ { 1 } + 4 x _ { 2 } + x _ { 3 } = - 2 \end{aligned}$$

Solve the system of equations:
$$\begin{aligned} & x _ { 1 } + 2 x _ { 2 } - 3 x _ { 3 } = 5 \\ & 2 x _ { 1 } - x _ { 2 } + 4 x _ { 3 } = 7 \\ & 3 x _ { 1 } + 4 x _ { 2 } + x _ { 3 } = - 2 \end{aligned}$$

140- Three planes are given by the following matrix equation: $$\begin{bmatrix} 2 & -1 & 1 \\ 1 & 3 & -1 \\ 1 & -11 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ 2 \end{bmatrix}$$ What is the relationship of the two planes with respect to each other?
(1) Parallel (2) Coincident (3) Perpendicular (4) Lacking a common line of intersection

140- In solving the system of equations $\begin{cases} x + y - z = 7 \\ 4x - y + 5z = 3 \\ 5x + y + z = 17 \end{cases}$ using Gaussian elimination, the matrix $\begin{bmatrix} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \end{bmatrix}$ is obtained. What is $b$?
(1) $2$ (2) $3$ (3) $4$ (4) $5$

140. Three planes with the matrix equation $\begin{bmatrix} 1 & 3 & -1 \\ 3 & -2 & 3 \\ 5 & 4 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 7 \\ 3 \\ 9 \end{bmatrix}$ are given. The common intersection of these two planes is which of the following?
(1) Parallel (2) Coincident (3) Divergent (4) Pass through one point

Consider the system of equations $x + y = 2$, $ax + y = b$. Find conditions on $a$ and $b$ under which

1. [(i)] the system has exactly one solution;
2. [(ii)] the system has no solution;
3. [(iii)] the system has more than one solution.

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(A) a circle
(B) a circle and a pair of straight lines
(C) a rectangular hyperbola
(D) a pair of straight lines

For a real number $\theta$, consider the following simultaneous equations:
$$\begin{aligned} & \cos ( \theta ) x - \sin ( \theta ) y = 1 \\ & \sin ( \theta ) x + \cos ( \theta ) y = 2 \end{aligned}$$
The number of solutions of these equations in $x$ and $y$ is
(A) 0
(B) 1
(C) infinite for some values of $\theta$
(D) finite only when $\theta = \frac { m \pi } { n }$ for integers $m$, and $n \neq 0$.

8. If $P ( 1,2 ) , Q ( 4,6 ) , R ( 5,7 )$ and $S ( a , b )$ are the vertices of a parallelogram $P Q R S$, then :
(A) $\mathrm { a } = 2 , \mathrm {~b} = 4$
(B) $\mathrm { a } = 3 , \mathrm {~b} = 4$
(C) $\mathrm { a } = 2 , \mathrm {~b} = 3$
(D) $\mathrm { a } = 3 , \mathrm {~b} = 5$ ... Powered By IITians

13. The number of integer values of $m$, for which the $x$-coordinate of the point of intersection of the lines $3 x + 4 y = 9$ and $y = m x + 1$ is also an integer, is :
(A) 2
(B) 0
(C) 4
(D) 1

7. The number of values of k for which the system of equations $( \mathrm { k } + 1 ) \mathrm { x } + 8 \mathrm { y } = 4 \mathrm { k }$ $\mathrm { kx } + ( \mathrm { k } + 3 ) \mathrm { y } = 3 \mathrm { k } - 1$ has infinitely many solution is
(A) 0
(B) 1
(C) 2
(D) Infinite

Consider the system of equations $$\begin{aligned} & x - 2 y + 3 z = - 1 \\ & - x + y - 2 z = k \\ & x - 3 y + 4 z = 1 . \end{aligned}$$ STATEMENT-1 : The system of equations has no solution for $k \neq 3$. and STATEMENT-2 : The determinant $\left| \begin{array} { c c c } 1 & 3 & - 1 \\ - 1 & - 2 & k \\ 1 & 4 & 1 \end{array} \right| \neq 0$, for $k \neq 3$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let $a , \lambda , \mu \in \mathbb { R }$. Consider the system of linear equations
$$\begin{aligned} & a x + 2 y = \lambda \\ & 3 x - 2 y = \mu \end{aligned}$$
Which of the following statement(s) is(are) correct?
(A) If $a = - 3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
(B) If $a \neq - 3$, then the system has a unique solution for all values of $\lambda$ and $\mu$
(C) If $\lambda + \mu = 0$, then the system has infinitely many solutions for $a = - 3$
(D) If $\lambda + \mu \neq 0$, then the system has no solution for $a = - 3$

For a real number $\alpha$, if the system $$\left[\begin{array}{ccc}1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right] = \left[\begin{array}{r}1 \\ -1 \\ 1\end{array}\right]$$ of linear equations, has infinitely many solutions, then $1 + \alpha + \alpha^2 =$