To solve the question, we will analyze each statement one by one and determine if they are true (T) or false (F).
### Step 1: Analyze Statement (i)
**Statement (i):** If the area of a rectangle PQRS is `50 cm²` and PQ : QR is `2 : 1`, then PQ = 10 cm and QR = 5 cm.
1. Let PQ = 2x and QR = x (since PQ : QR = 2 : 1).
2. The area of the rectangle is given by the formula: Area = Length × Breadth.
3. Therefore, Area = PQ × QR = (2x) × (x) = 2x².
4. We know the area is 50 cm², so we set up the equation: 2x² = 50.
5. Dividing both sides by 2 gives us: x² = 25.
6. Taking the square root of both sides gives us: x = 5 cm.
7. Therefore, PQ = 2x = 2(5) = 10 cm and QR = x = 5 cm.
**Conclusion:** The statement is **True (T)**.
### Step 2: Analyze Statement (ii)
**Statement (ii):** Sonali has a total of 120 red and black pens. If 30% of them are black, then there are 40 red pens and 80 black pens.
1. Total pens = 120.
2. If 30% are black, then the number of black pens = 30% of 120 = (30/100) × 120 = 36.
3. Therefore, the number of red pens = Total pens - Black pens = 120 - 36 = 84.
**Conclusion:** The statement is **False (F)** because it claims there are 40 red pens, but there are actually 84.
### Step 3: Analyze Statement (iii)
**Statement (iii):** If `1/x : 1/y : 1/z = 4:5:6`, then `x : y : z = 6 : 5 : 4`.
1. From the ratio `1/x : 1/y : 1/z = 4:5:6`, we can express it as:
- 1/x = 4k
- 1/y = 5k
- 1/z = 6k
2. Taking the reciprocal gives us:
- x = 1/(4k)
- y = 1/(5k)
- z = 1/(6k)
3. Now, we need to find the ratio x : y : z:
- x : y : z = (1/(4k)) : (1/(5k)) : (1/(6k))
- This simplifies to 6 : 5 : 4.
**Conclusion:** The statement is **True (T)**.
### Step 4: Analyze Statement (iv)
**Statement (iv):** If the speed of three cars is in the ratio `3:4:5`, then the ratio of time taken by these cars to travel the same distance is `20 : 15 : 12`.
1. Let the speeds of the cars be 3x, 4x, and 5x.
2. Time taken is given by the formula: Time = Distance / Speed.
3. Since the distance is the same for all cars, the ratio of time taken will be inversely proportional to their speeds.
4. Therefore, the ratio of time taken = 1/(3x) : 1/(4x) : 1/(5x) = 5 : 4 : 3.
5. This does not match the given ratio of 20 : 15 : 12.
**Conclusion:** The statement is **False (F)**.
### Final Summary of Statements:
- (i) True (T)
- (ii) False (F)
- (iii) True (T)
- (iv) False (F)
### Correct Option:
The correct option is **T, F, T, F**.
