Conditions for a system of linear equations to have no solutions

Q. A motorboat, whose speed is $15$ km/hr in still water goes $30$ km downstream and comes back in a total of $4$ hours $30$ minutes. The speed of the stream is

1. $4$ km/hr
   
2. $6$ km/hr
   
3. $10$ km/hr
   
4. $5$ km/hr
   

Q.

For which values of $a$ and $b$ does the following pair of linear equations have an infinite number of solutions? $2x+3y=7,\left(a-b\right)x+\left(a+b\right)y=3a+b-2$.

Q.

If two equations of line represent the equation of same line, then their coefficients should be equal.

1. True

2. False

Q. Liam's bookstore sold 40 notebooks and 20 newspapers for a total of $\$130$. A day later, the bookstore sold 8 notebooks and 4 newspapers at the same prices, for a total of $\$28$. How much does a notebook and a newspaper cost at Liam's bookstore?

1. Notebook costs $\$1.50$ and newspaper is $\$2.50$
2. There is not enough information to determine the exact costs
3. The given situation describes an impossible situation
4. Notebook is $\$2.50$ and newspaper is $\$1.50$

Q. $-5+bx=2x+3(x+2)$
In the equation shown above, $b$ is constant. For what value of $b$ does the equation have no solution?

1. 3
2. 5
3. 4
4. 6

Q.

Write an expression having only two terms that is equivalent to $2x-3$.

Q. y = 0.25 x + 12 ;
y = m(x + 3)
In the above system of equations, for what value of constant m, will the system of equation have no solution?

1. 1
2. 0.75
3. 0.25
4. 0

Q. 22s = -11t + 16 ;
44(t + 2s) = 12
Consider the system of equations above. How many solutions (s, t) does this system have ?

1. 0
2. Exactly 1
3. Exactly 2
4. Infinitely Many

Q.

If 10 is a root of quadratic equation $11x^2-kx-120=0$, then the value of $k$ is _________.

Q. $5x-2y=6$, $15x-6y=c$
Which of the following choices of $c$ will result in a system of linear equation with no solution?

1. $c$ can be any number other than $-18$
2. $c$ can be any number
3. $c$ can be any number other than 18
4. $c=18$

Q. -6x + 4y = 2;
3x – 2y = 4
Consider the system of equations above. If (x, y) is a solution of the equations above, then how many possible values are there for (x, y) ?

1. No Solution
2. One solution
3. Two solutions
4. Infinitely many solutions

Q. $-6x+8y=2$ , $3x-4y=-3$
Consider the system of equations above. How many $(x,y)$ solutions does this system have?

1. Exactly one solution
2. No solutions
3. Exactly two solutions
4. Infinitely many solutions

Q. Find the shortest distance between the lines whose vector equations are :
$\overrightarrow{r}=(1-p)\hat{i}+(p-2)\hat{j}+(3-2p)\hat{k}$
$\overrightarrow{r}=(q+1)\hat{i}+(2q-1)\hat{j}-(2q-1)\hat{k}$

Q.

In the following, determine the set of values of $p$ for which the quadratic equation has real roots:

$p{x}^2+4x+1=0$