21. State ' $$ T $$ ' for true or ' F ' for false. I. If $$ \frac{p}{q} $$ is a rational number, then $$ p
eq 0 $$. II. If $$ \frac{b}{a} $$ is multiplicative inverse of $$ \frac{a}{b} $$, then $$ a
eq 0 $$. III. For any rational number $$ x, x+(-1)=-x $$. IV. $$ \frac{x+y}{2} $$ is a rational number which lies between $$ x $$ and $$ y $$. V. If $$ \frac{p}{q}=\frac{3}{5} $$ and $$ \frac{x}{y}=\frac{2}{3} $$, then $$ \frac{p y+q x}{3 q y}=\frac{19}{45} $$ Codes $$ \begin{array}{ll} ext { I II III IV V } & ext { (a) F T F T T' }\end{array} $$ $$ \begin{array}{ll} ext { (c) F F F F F } & ext { (b) T F T F F F }\end{array} $$

Question

21. State ' $$ T $$ ' for true or ' F ' for false. I. If $$ \frac{p}{q} $$ is a rational number, then $$ p 
eq 0 $$. II. If $$ \frac{b}{a} $$ is multiplicative inverse of $$ \frac{a}{b} $$, then $$ a 
eq 0 $$. III. For any rational number $$ x, x+(-1)=-x $$. IV. $$ \frac{x+y}{2} $$ is a rational number which lies between $$ x $$ and $$ y $$. V. If $$ \frac{p}{q}=\frac{3}{5} $$ and $$ \frac{x}{y}=\frac{2}{3} $$, then $$ \frac{p y+q x}{3 q y}=\frac{19}{45} $$ Codes $$ \begin{array}{ll}	ext { I II III IV V } & 	ext { (a) F T F T T' }\end{array} $$ $$ \begin{array}{ll}	ext { (c) F F F F F } & 	ext { (b) T F T F F F }\end{array} $$

UpStudy AI · Accepted Answer

Let's evaluate each statement one by one to determine whether they are true (T) or false (F).

### I. If $$ \frac{p}{q} $$ is a rational number, then $$ p \neq 0 $$.
- **Evaluation**: A rational number can be expressed as $$ \frac{p}{q} $$ where $$ p $$ and $$ q $$ are integers and $$ q \neq 0 $$. However, $$ p $$ can be zero (e.g., $$ \frac{0}{1} = 0 $$). Therefore, this statement is **False**.
- **Result**: F

### II. If $$ \frac{b}{a} $$ is the multiplicative inverse of $$ \frac{a}{b} $$, then $$ a \neq 0 $$.
- **Evaluation**: The multiplicative inverse of $$ \frac{a}{b} $$ is $$ \frac{b}{a} $$ if $$ a \neq 0 $$ and $$ b \neq 0 $$. If $$ a = 0 $$, then $$ \frac{b}{a} $$ is undefined. Therefore, this statement is **True**.
- **Result**: T

### III. For any rational number $$ x, x + (-1) = -x $$.
- **Evaluation**: This statement simplifies to $$ x - 1 = -x $$. Rearranging gives $$ 2x = 1 $$, which is not true for all rational numbers $$ x $$. For example, if $$ x = 2 $$, then $$ 2 + (-1) = 1 $$ which is not equal to $$-2$$. Therefore, this statement is **False**.
- **Result**: F

### IV. $$ \frac{x+y}{2} $$ is a rational number which lies between $$ x $$ and $$ y $$.
- **Evaluation**: The average $$ \frac{x+y}{2} $$ is indeed a rational number if both $$ x $$ and $$ y $$ are rational. Additionally, it lies between $$ x $$ and $$ y $$ if $$ x $$ and $$ y $$ are distinct. Therefore, this statement is **True**.
- **Result**: T

### V. If $$ \frac{p}{q} = \frac{3}{5} $$ and $$ \frac{x}{y} = \frac{2}{3} $$, then $$ \frac{p y + q x}{3 q y} = \frac{19}{45} $$.
- **Evaluation**: Let's substitute $$ p = 3k $$, $$ q = 5k $$, $$ x = 2m $$, and $$ y = 3m $$ for some integers $$ k $$ and $$ m $$.
  $$
  p y + q x = 3k(3m) + 5k(2m) = 9km + 10km = 19km
  $$
  $$
  3 q y = 3(5k)(3m) = 45km
  $$
  Therefore,
  $$
  \frac{p y + q x}{3 q y} = \frac{19km}{45km} = \frac{19}{45}
  $$
  This statement is **True**.
- **Result**: T

### Summary of Results:
- I: F
- II: T
- III: F
- IV: T
- V: T

### Final Answer:
The correct code is **(a) F T F T T**.
The correct code is **(a) F T F T T**.

State ’ ’ for true or ’ F ’ for false.
I. If is a rational number, then .
II. If is multiplicative inverse of , then .
III. For any rational number .
IV. is a rational number which lies between and .
V. If and , then
Codes

Upstudy AI Solution

Answer

Solution

I. If is a rational number, then .

II. If is the multiplicative inverse of , then .

III. For any rational number .

IV. is a rational number which lies between and .

V. If and , then .

Summary of Results:

Final Answer:

The Deep Dive

Related Questions

Latest Algebra Questions

State ’ ’ for true or ’ F ’ for false. I. If is a rational number, then . II. If is multiplicative inverse of , then . III. For any rational number . IV. is a rational number which lies between and . V. If and , then Codes