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 $\_\_\_\_$
If the system of equations $$x + 2y - 3z = 2$$ $$2x + \lambda y + 5z = 5$$ $$14x + 3y + \mu z = 33$$ has infinitely many solutions, then $\lambda + \mu$ is equal to : (1) 13 (2) 10 (3) 12 (4) 11
If the system of equations $$2x - y + z = 4$$ $$5x + \lambda y + 3z = 12$$ $$100x - 47y + \mu z = 212$$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) 57 (2) 59 (3) 55 (4) 56
If the system of linear equations : $$x + y + 2z = 6$$ $$2x + 3y + \mathrm { a } z = \mathrm { a } + 1$$ $$- x - 3 y + \mathrm { b } z = 2 \mathrm {~b}$$ where $a , b \in \mathbf { R }$, has infinitely many solutions, then $7a + 3b$ is equal to : (1) 16 (2) 12 (3) 22 (4) 9
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
Q69. $$11 x + y + \lambda z = - 5$$ If the system of equations $2 x + 3 y + 5 z = 3$ has infinitely many solutions, then $\lambda ^ { 4 } - \mu$ is equal to : $$8 x - 19 y - 39 z = \mu$$ (1) 51 (2) 45 (3) 47 (4) 49
Q69. $3 x + 5 y + \lambda z = 3$ Let $\lambda , \mu \in \mathbf { R }$. If the system of equations $7 x + 11 y - 9 z = 2$ has infinitely many solutions, then $\mu + 2 \lambda$ is $97 x + 155 y - 189 z = \mu$ equal to : (1) 24 (2) 25 (3) 22 (4) 27
Q70. 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
Q85. $$2 x + 7 y + \lambda z = 3$$ If the system of equations $3 x + 2 y + 5 z = 4$ has infinitely many solutions, then $( \lambda - \mu )$ is equal $$x + \mu y + 32 z = - 1$$ to $\_\_\_\_$
$x - n y + z = 6$ $\mathbf { x } - ( \mathbf { n } - \mathbf { 2 } ) \mathbf { y } + ( \mathbf { n } + \mathbf { 1 } ) \mathbf { z } = \mathbf { 8 }$ $( \mathrm { n } - 1 ) \mathrm { y } + \mathrm { z } = 1$ Let $\mathbf { n } \boldsymbol { = }$ number on the dies when rolled randomly then $\mathbf { P }$ (that system equation has unique solution) $= \left( \frac { \mathrm { k } } { 6 } \right)$ then sum of value of k and all possible value of n is (A) 22 (B) 24 (C) 20 (D) 21
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 $$a = \mathbf{L}, \quad b = \mathbf{M}, \quad c = \mathbf{NO}.$$
Given the following system of equations $\left\{ \begin{array} { l } 2 x + a y + z = a \\ x - 4 y + ( a + 1 ) z = 1 \\ 4 y - a z = 0 \end{array} \right.$ a) (2 points) Discuss depending on the values of the real parameter $a$. b) ( 0.5 points) Solve the system for $a = 1$. c) ( 0.5 points) Solve the system for $a = 2$.
There are three alloys $\mathrm { A } , \mathrm { B }$ and C that contain, among other metals, gold and silver in the proportions indicated in the attached table.
Gold (\%)
Silver (\%)
A
100
0
B
75
15
C
60
22
It is desired to obtain an ingot of 25 grams, with a proportion of $72 \%$ gold and a proportion of $16 \%$ silver, taking $x$ grams of $\mathrm { A } , y$ grams of B and $z$ grams of C . Determine the quantities $x$, $y , z$.
Given the system of equations $$\left\{ \begin{array} { l }
x + m y = 1 \\
- 2 x - ( m + 1 ) y + z = - 1 \\
x + ( 2 m - 1 ) y + ( m + 2 ) z = 2 + 2 m
\end{array} \right.$$ it is requested: a) (2 points) Discuss the system as a function of the parameter $m$. b) ( 0.5 points) Solve the system in the case $m = 0$.
Given the system of equations $\left\{ \begin{array} { l } k x + ( k + 1 ) y + z = 0 \\ - x + k y - z = 0 \\ ( k - 1 ) x - y = - ( k + 1 ) \end{array} \right.$ it is requested: a) (2 points) Discuss the system according to the values of the real parameter k. b) ( 0.5 points) Solve the system for $\mathrm { k } = - 1$.
Consider the following system of equations depending on the real parameter a: $$\left. \begin{array} { l }
x + a y + z = a + 1 \\
- a x + y - z = 2 a \\
- y + z = a
\end{array} \right\}$$ It is requested:\ a) (2 points) Discuss the system according to the different values of a.\ b) (0.5 points) Solve the system for $\mathrm { a } = 0$.
Three friends, Sara, Cristina and Jimena, have a total of 15000 followers on a social network. If Jimena lost $25 \%$ of her followers she would still have three times as many followers as Sara. Furthermore, half of Sara's followers plus one-fifth of Cristina's followers equal one-quarter of Jimena's followers. Calculate how many followers each of the three friends has.
a) ( 0.75 points) Find a single system of two linear equations in the variables $x$ and $y$, which has as solutions $\{ x = 1 , y = 2 \}$ and $\{ x = 0 , y = 0 \}$. b) (1 point) Find a system of two linear equations in the variables $x , y$ and z whose solutions are, as a function of the parameter $\lambda \in \mathbb { R }$ : $$\left\{ \begin{array} { l }
x = \lambda \\
y = \lambda - 2 \\
z = \lambda - 1
\end{array} \right.$$ c) ( 0.75 points) Find a system of three linear equations with two unknowns, x and y, that has only the solution $x = 1$ and $y = 2$.
Three brothers want to divide equally a total of 540 shares valued at 1560 euros, which correspond to three companies A, B and C. Knowing that the current stock market value of share A is three times that of B and half that of C, that the number of shares of C is half that of B, and that the current stock market value of share B is 1 euro, find the number of each type of share that corresponds to each brother.
Consider the following system of equations depending on the real parameter a: $$\left. \begin{array} { l }
a x - 2 y + ( a - 1 ) z = 4 \\
- 2 x + 3 y - 6 z = 2 \\
- a x + y - 6 z = 6
\end{array} \right\}$$ a) (2 points) Discuss the system according to the different values of $a$.\ b) ( 0.5 points) Solve the system for $a = 1$.
Given the following system of linear equations dependent on the real parameter $m$ : $$\left. \begin{array} { l }
x - 2 m y + z = 1 \\
m x + 2 y - z = - 1 \\
x - y + z = 1
\end{array} \right\}$$ a) (2 points) Discuss the system as a function of the values of $m$. b) ( 0.5 points) Solve the system for the value $m = \frac { 1 } { 2 }$
Three cousins, Pablo, Alejandro and Alicia, are going to share a prize of 9450 euros in direct proportion to their ages. The sum of the ages of Pablo and Alejandro exceeds by three years twice the age of Alicia. Furthermore, the age of the three cousins together is 45 years. Knowing that in the distribution of the prize Pablo receives 420 euros more than Alicia, calculate the ages of the three cousins and the money each one receives from the prize.
On a library shelf there are essays, novels and biographies. Three out of every sixteen books on the shelf are essays. The biographies together with one third of the essays exceed the novels by two. If we removed half of the essays and one fifth of the novels, one hundred five books would remain. Calculate the number of books of each type on the shelf.
At a construction site, to transport the earth extracted for the construction of a building's foundations, three different types of trucks are used: A, B, and C. Type A trucks have a capacity of 14 tonnes, type B trucks have 24 tonnes, and type C trucks have 28 tonnes. One more type A truck would be needed to equal the number of remaining trucks. 10\% of the capacity of all type B trucks equals one-seventh of the capacity of the largest tonnage trucks. Today, with each truck making a single trip at maximum capacity, 302 tonnes of earth have been extracted from the site. How much earth has been transported today by trucks of each type?