If the system of equations $x + y + z = 5 , x + 2 y + 3 z = 9 , x + 3 y + \alpha z = \beta$ has infinitely many solutions, then $\beta - \alpha$ equals
(1) 8
(2) 21
(3) 5
(4) 18

The value of $k \in R$, for which the following system of linear equations $3 x - y + 4 z = 3$ $x + 2 y - 3 z = - 2$ $6 x + 5 y + k z = - 3$ has infinitely many solutions, is:
(1) 3
(2) - 5
(3) 5
(4) - 3

Let $[ \lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x + y + z = 4,3 x + 2 y + 5 z = 3,9 x + 4 y + ( 28 + [ \lambda ] ) z = [ \lambda ]$ has a solution is: (1) $R$ (2) $( - \infty , - 9 ) \cup [ - 8 , \infty )$ (3) $( - \infty , - 9 ) \cup ( - 9 , \infty )$ (4) $[ - 9 , - 8 )$

The following system of linear equations $2 x + 3 y + 2 z = 9$ $3 x + 2 y + 2 z = 9$ $x - y + 4 z = 8$
(1) has infinitely many solutions
(2) has a unique solution
(3) has a solution ( $\alpha , \beta , \gamma$ ) satisfying $\alpha + \beta ^ { 2 } + \gamma ^ { 3 } = 12$
(4) does not have any solution

If the following system of linear equations $2 x + y + z = 5$ $x - y + z = 3$ $x + y + a z = b$ has no solution, then :
(1) $a = - \frac { 1 } { 3 } , b \neq \frac { 7 } { 3 }$
(2) $a \neq \frac { 1 } { 3 } , b = \frac { 7 } { 3 }$
(3) $a \neq - \frac { 1 } { 3 } , b = \frac { 7 } { 3 }$
(4) $a = \frac { 1 } { 3 } , b \neq \frac { 7 } { 3 }$

If the system of equations $$\begin{aligned} & k x + y + 2 z = 1 \\ & 3 x - y - 2 z = 2 \\ & - 2 x - 2 y - 4 z = 3 \end{aligned}$$ has infinitely many solutions, then $k$ is equal to

If the system of linear equations $8 x + y + 4 z = - 2$ $x + y + z = 0$ $\lambda x - 3 y = \mu$ has infinitely many solutions, then the distance of the point $\left( \lambda , \mu , - \frac { 1 } { 2 } \right)$ from the plane $8 x + y + 4 z + 2 = 0$ is:
(1) $3 \sqrt { 5 }$
(2) 4
(3) $\frac { 26 } { 9 }$
(4) $\frac { 10 } { 3 }$

The ordered pair $( a , b )$, for which the system of linear equations $3 x - 2 y + z = b$ $5 x - 8 y + 9 z = 3$ $2 x + y + a z = - 1$ has no solution, is
(1) $\left( 3 , \frac { 1 } { 3 } \right)$
(2) $\left( - 3 , \frac { 1 } { 3 } \right)$
(3) $\left( - 3 , - \frac { 1 } { 3 } \right)$
(4) $\left( 3 , - \frac { 1 } { 3 } \right)$

For the system of linear equations $x + y + z = 6$ $\alpha x + \beta y + 7z = 3$ $x + 2y + 3z = 14$ which of the following is NOT true?
(1) If $\alpha = \beta = 7$, then the system has no solution
(2) If $\alpha = \beta$ and $\alpha \neq 7$ then the system has a unique solution.
(3) There is a unique point $(\alpha, \beta)$ on the line $x + 2y + 18 = 0$ for which the system has infinitely many solutions
(4) For every point $(\alpha, \beta) \neq (7,7)$ on the line $x - 2y + 7 = 0$, the system has infinitely many solutions.

For the system of linear equations $$2x - y + 3z = 5$$ $$3x + 2y - z = 7$$ $$4x + 5y + \alpha z = \beta$$ which of the following is NOT correct?
(1) The system has infinitely many solutions for $\alpha = -5$ and $\beta = 9$
(2) The system has infinitely many solutions for $\alpha = -6$ and $\beta = 9$
(3) The system is inconsistent for $\alpha = -5$ and $\beta = 8$
(4) The system has a unique solution for $\alpha \neq -5$ and $\beta = 8$

If the system of equations $$\begin{aligned} & 2 x + y - z = 5 \\ & 2 x - 5 y + \lambda z = \mu \\ & x + 2 y - 5 z = 7 \end{aligned}$$ has infinitely many solutions, then $( \lambda + \mu ) ^ { 2 } + ( \lambda - \mu ) ^ { 2 }$ is equal to
(1) 904
(2) 916
(3) 912
(4) 920

Let the system of linear equations $- x + 2 y - 9 z = 7$ $- x + 3 y + 7 z = 9$ $- 2 x + y + 5 z = 8$ $- 3 x + y + 13 z = \lambda$ has a unique solution $x = \alpha , y = \beta , z = \gamma$. Then the distance of the point $( \alpha , \beta , \gamma )$ from the plane $2 x - 2 y + z = \lambda$ is
(1) 11
(2) 7
(3) 9
(4) 13

Let $S _ { 1 }$ and $S _ { 2 }$ be respectively the sets of all $a \in R - \{ 0 \}$ for which the system of linear equations $a x + 2 a y - 3 a z = 1$ $( 2 a + 1 ) x + ( 2 a + 3 ) y + ( a + 1 ) z = 2$ $( 3 a + 5 ) x + ( a + 5 ) y + ( a + 2 ) z = 3$ has unique solution and infinitely many solutions. Then
(1) $\mathrm { n } \left( S _ { 1 } \right) = 2$ and $S _ { 2 }$ is an infinite set
(2) $S _ { 1 }$ is an infinite set and $n \left( S _ { 2 } \right) = 2$
(3) $S _ { 1 } = \phi$ and $S _ { 2 } = \mathbb { R } - \{ 0 \}$
(4) $S _ { 1 } = \mathbb { R } - \{ 0 \}$ and $S _ { 2 } = \phi$

Consider the following system of questions
$$\begin{aligned} & \alpha x + 2 y + z = 1 \\ & 2 \alpha x + 3 y + z = 1 \\ & 3 x + \alpha y + 2 z = \beta \end{aligned}$$
For some $\alpha , \beta \in \mathbb { R }$. Then which of the following is NOT correct.
(1) It has no solution if $\alpha = - 1$ and $\beta \neq 2$
(2) It has no solution for $\alpha = - 1$ and for all $\beta \in \mathbb { R }$
(3) It has no solution for $\alpha = 3$ and for all $\beta \neq 2$
(4) It has a solution for all $\alpha \neq - 1$ and $\beta = 2$

For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations $x - y + z = 5$ $2x + 2y + \alpha z = 8$ $3x - y + 4z = \beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of
(1) $x^{2} - 10x + 16 = 0$
(2) $x^{2} + 18x + 56 = 0$
(3) $x^{2} - 18x + 56 = 0$
(4) $x^{2} + 14x + 24 = 0$

Consider the system of linear equations $x + y + z = 5$, $x + 2y + \lambda^2 z = 9$ and $x + 3y + \lambda z = \mu$, where $\lambda, \mu \in R$. Then, which of the following statement is NOT correct?
(1) System has infinite number of solution if $\lambda = 1$
(2) System is inconsistent if $\lambda = 1$ and $\mu \neq 13$ and $\mu = 13$
(3) System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$
(4) System is consistent if $\lambda \neq 1$ and $\mu = 13$

The values of $m , n$, for which the system of equations \begin{align*} x + y + z &= 4, 2x + 5y + 5z &= 17, x + 2y + \mathrm{m}z &= \mathrm{n} \end{align*} has infinitely many solutions, satisfy the equation:
(1) $m ^ { 2 } + n ^ { 2 } - m n = 39$
(2) $m ^ { 2 } + n ^ { 2 } - m - n = 46$
(3) $m ^ { 2 } + n ^ { 2 } + m + n = 64$
(4) $m ^ { 2 } + n ^ { 2 } + m n = 68$

If the system of equations $x + 4 y - z = \lambda , 7 x + 9 y + \mu z = - 3,5 x + y + 2 z = - 1$ has infinitely many solutions, then $( 2 \mu + 3 \lambda )$ is equal to : (1) 3 (2) - 3 (3) - 2 (4) 2

If the system of equations $$2 x + 7 y + \lambda z = 3$$ $$3 x + 2 y + 5 z = 4$$ $$x + \mu y + 32 z = - 1$$ has infinitely many solutions, then $( \lambda - \mu )$ is equal to $\_\_\_\_$

Consider the matrices : $A = \left[ \begin{array} { c c } 2 & - 5 \\ 3 & m \end{array} \right] , B = \left[ \begin{array} { l } 20 \\ m \end{array} \right]$ and $X = \left[ \begin{array} { l } x \\ y \end{array} \right]$. Let the set of all $m$, for which the system of equations $A X = B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval ( $a , b$ ). Then $8 \int _ { a } ^ { b } | A | d m$ is equal to $\_\_\_\_$

Let $\alpha , \beta ( \alpha \neq \beta )$ be the values of m , for which the equations $x + y + z = 1 ; x + 2 y + 4 z = \mathrm { m }$ and $x + 4 y + 10 z = m ^ { 2 }$ have infinitely many solutions. Then the value of $\sum _ { n = 1 } ^ { 10 } \left( n ^ { \alpha } + n ^ { \beta } \right)$ is equal to :
(1) 3080
(2) 560
(3) 3410
(4) 440

Let $x$, $y$ and $z$ satisfy the following two equations:
$$x+y-z=0 \tag{1}$$ $$2x-y+1=0 \tag{2}$$
We are to find the values of $a$, $b$ and $c$ such that the equation
$$ax^2+by^2+cz^2=1 \tag{3}$$
holds for all $x$, $y$ and $z$ satisfying (1) and (2).
First, given (1) and (2), we may express $y$ and $z$ in terms of $x$ as
$$y = \mathbf{A}x+\mathbf{B}, \quad z = \mathbf{C}x+\mathbf{D}. \tag{4}$$
This shows that the values of both $y$ and $z$ depend on the value of $x$.
Next, when (4) is substituted into (3) and the left side is arranged in descending order of powers of $x$, we obtain
$$(a+\mathbf{E}b+\mathbf{F}c)x^2+(\mathbf{G}b+\mathbf{H}c)x+b+c=1.$$
Since this equation holds for any $x$, it holds also when $x=0$, $x=1$ and $x=-1$ are substituted into it, from which we obtain
$$\left\{\begin{aligned} b+c &= 1 \\ a+9b+\mathbf{IJ}c &= 1 \\ a+b+\mathbf{K}c &= 1 \end{aligned}\right.$$
When we regard these as simultaneous equations and solve them for $a$, $b$ and $c$, we have
$$a = \mathbf{L}, \quad b = \mathbf{M}, \quad c = \mathbf{NO}.$$

A company has only three members: a manager, a secretary, and a salesperson. If only the secretary receives a 10\% salary increase, the company's total salary expenditure increases by 3\%; if only the salesperson receives a 20\% salary increase, the company's total salary expenditure increases by 4\%. If only the manager's salary is reduced by 15\%, then the company's total salary expenditure will decrease by (12).(13)\%.

Let $c$ be a real number such that the system of linear equations $\left\{ \begin{array}{c} x - y + z = 0 \\ 2x + cy + 3z = 1 \\ 3x - 3y + cz = 0 \end{array} \right.$ has no solution. Select the value of $c$.
(1) $-3$
(2) $-2$
(3) $0$
(4) $2$
(5) $3$

A newly opened juice shop offers three types of beverages: juice, milk tea, and coffee. The sales volume (in cups) and total revenue (in yuan) for each type of beverage over the first 3 days are shown in the table below. For example, on the first day, the sales of juice, milk tea, and coffee were 60 cups, 80 cups, and 50 cups respectively, with total revenue of 12,900 yuan.
It is known that the price of each type of beverage is the same every day. Then the price per cup of coffee is

	Juice (cups)	Milk Tea (cups)	Coffee (cups)	Total Revenue (yuan)
Day 1	60	80	50	12900
Day 2	30	40	30	6850
Day 3	50	70	40	10800