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?

In the calibration of a new traffic light, the times are adjusted so that, in each complete cycle (green-yellow-red), the yellow light remains on for 5 seconds, and the time in which the green light remains on is equal to $\frac{2}{3}$ of the time in which the red light stays on. The green light is on, in each cycle, for $X$ seconds and each cycle lasts $Y$ seconds.
Which expression represents the relationship between $X$ and $Y$?
(A) $5X - 3Y + 15 = 0$ (B) $5X - 2Y + 10 = 0$ (C) $3X - 3Y + 15 = 0$ (D) $3X - 2Y + 15 = 0$ (E) $3X - 2Y + 10 = 0$

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.

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.

For two sets $A = \{ a + 2,6 \} , B = \{ 3 , b - 1 \}$, when $A = B$, what is the value of $a + b$? (Note: $a , b$ are real numbers.) [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

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

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.

Let $a$ be a real number. The number of distinct solutions $(x, y)$ of the system of equations $(x - a)^2 + y^2 = 1$ and $x^2 = y^2$, can only be
(A) $0, 1, 2, 3, 4$ or 5
(B) 0, 1 or 3
(C) $0, 1, 2$ or 4
(D) $0, 2, 3$, or 4

Let $a$ be a real number. The number of distinct solutions $(x, y)$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be
(A) $0, 1, 2, 3, 4$ or 5
(B) 0, 1 or 3
(C) $0, 1, 2$ or 4
(D) $0, 2, 3$, or 4

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 $n \neq 0$.

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 $1 + \alpha + \alpha^2 =$

Let $S$ be the set of all column matrices $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right]$ such that $b _ { 1 } , b _ { 2 } , b _ { 3 } \in \mathbb { R }$ and the system of equations (in real variables)
$$\begin{aligned} - x + 2 y + 5 z & = b _ { 1 } \\ 2 x - 4 y + 3 z & = b _ { 2 } \\ x - 2 y + 2 z & = b _ { 3 } \end{aligned}$$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right] \in S$ ?
(A) $x + 2 y + 3 z = b _ { 1 } , 4 y + 5 z = b _ { 2 }$ and $x + 2 y + 6 z = b _ { 3 }$
(B) $x + y + 3 z = b _ { 1 } , 5 x + 2 y + 6 z = b _ { 2 }$ and $- 2 x - y - 3 z = b _ { 3 }$
(C) $- x + 2 y - 5 z = b _ { 1 } , 2 x - 4 y + 10 z = b _ { 2 }$ and $x - 2 y + 5 z = b _ { 3 }$
(D) $x + 2 y + 5 z = b _ { 1 } , 2 x + 3 z = b _ { 2 }$ and $x + 4 y - 5 z = b _ { 3 }$

Let $p , q , r$ be nonzero real numbers that are, respectively, the $10 ^ { \text {th } } , 100 ^ { \text {th } }$ and $1000 ^ { \text {th } }$ terms of a harmonic progression. Consider the system of linear equations
$$\begin{gathered} x + y + z = 1 \\ 10 x + 100 y + 1000 z = 0 \\ q r x + p r y + p q z = 0 \end{gathered}$$
List-I (I) If $\frac { q } { r } = 10$, then the system of linear equations has (II) If $\frac { p } { r } \neq 100$, then the system of linear equations has (III) If $\frac { p } { q } \neq 10$, then the system of linear equations has (IV) If $\frac { p } { q } = 10$, then the system of linear equations has
List-II (P) $x = 0 , \quad y = \frac { 10 } { 9 } , z = - \frac { 1 } { 9 }$ as a solution (Q) $x = \frac { 10 } { 9 } , \quad y = - \frac { 1 } { 9 } , z = 0$ as a solution (R) infinitely many solutions (S) no solution (T) at least one solution
The correct option is:
(A) (I) → (T); (II) → (R); (III) → (S); (IV) → (T)
(B) (I) → (Q); (II) → (S); (III) → (S); (IV) → (R)
(C) (I) → (Q); (II) → (R); (III) → (P); (IV) → (R)
(D) (I) → (T); (II) → (S); (III) → (P); (IV) → (T)

The number of values of $k$, for which the system of equations: $(k+1)x + 8y = 4k$ $kx + (k+3)y = 3k - 1$ has no solution, is:
(1) 2
(2) 3
(3) Infinite
(4) 1

If $a , b , c$ are non-zero real numbers and if the system of equations $$( a - 1 ) x = y + z$$ $$( b - 1 ) y = x + z$$ $$( c - 1 ) z = x + y$$ has a non-trivial solution, then $ab + bc + ca$ equals:
(1) $-1$
(2) $a + b + c$
(3) $abc$
(4) 1

If $S$ is the set of distinct values of $b$ for which the following system of linear equations
$$\begin{aligned} & x + y + z = 1 \\ & x + ay + z = 1 \\ & ax + by + z = 0 \end{aligned}$$
has no solution, then $S$ is:
(1) An empty set
(2) An infinite set
(3) A finite set containing two or more elements
(4) A singleton

The number of real values of $\lambda$ for which the system of linear equations, $2 x + 4 y - \lambda z = 0, 4 x + \lambda y + 2 z = 0$ and $\lambda x + 2 y + 2 z = 0$, has infinitely many solutions, is:
(1) 3
(2) 1
(3) 2
(4) 0

Let $S$ be the set of all real values of $k$ for which the system of linear equations $x + y + z = 2$ $2 x + y - z = 3$ $3 x + 2 y + k z = 4$ has a unique solution. Then, $S$ is :
(1) equal to $R - \{ 0 \}$
(2) an empty set
(3) equal to $R$
(4) equal to $\{ 0 \}$

Let $S$ be the set of all real values of $k$ for which the system of linear equations
$$\begin{aligned} & x + y + z = 2 \\ & 2 x + y - z = 3 \\ & 3 x + 2 y + k z = 4 \end{aligned}$$
has a unique solution. Then $S$ is
(1) an empty set
(2) equal to $\mathrm { R } - \{ 0 \}$
(3) equal to $\{ 0 \}$
(4) equal to $R$

If the system of linear equations $x - 4y + 7z = g$; $3y - 5z = h$; $-2x + 5y - 9z = k$ is consistent, then:
(1) $g + h + 2k = 0$
(2) $g + 2h + k = 0$
(3) $2g + h + k = 0$
(4) $g + h + k = 0$