State T for true and F for false an select the correct option.
(i) If Sunil bought a laptop for Rs. 44000 including a tax of 10%, then the price of the laptop before tax was added was Rs. 40000.
(ii) x % of y is equal to y % of x.
(iii) The compound interest on Rs. 12000 for `1 1/2` years at 10% per annum compounded half yearly is Rs. 13891.50.
(iv) If a retailer sells an alarm clock for Rs. 350 and gains `1/6` of its cost price then the cost price is Rs. 250.

Options

A. `{:(i" "ii" "iii" "iv),(T" "T" "F" "F):}`
B. `{:(i" "ii" "iii" "iv),(F" "F" "T" "T):}`
C. `{:(i" "ii" "iii" "iv),(T" "F" "T" "F):}`
D. `{:(i" "ii" "iii" "iv),(T" "T" "T" "F):}`

Answer

To solve the question, we will evaluate each statement one by one and determine whether it is true (T) or false (F).

### Statement (i):
**If Sunil bought a laptop for Rs. 44000 including a tax of 10%, then the price of the laptop before tax was added was Rs. 40000.**

1. Let the price of the laptop before tax be \( P \).
2. The tax is 10% of \( P \), which can be expressed as \( 0.1P \).
3. Therefore, the total price including tax is:
\[ P + 0.1P = 1.1P \]
4. We know that the total price including tax is Rs. 44000:
\[ 1.1P = 44000 \]
5. To find \( P \), we divide both sides by 1.1:
\[ P = \frac{44000}{1.1} = 40000 \]
6. Thus, the statement is **True (T)**.

### Statement (ii):
**x % of y is equal to y % of x.**

1. Let's express \( x \% \) of \( y \):
\[ \frac{x}{100} \times y = \frac{xy}{100} \]
2. Now, let's express \( y \% \) of \( x \):
\[ \frac{y}{100} \times x = \frac{yx}{100} \]
3. Since \( \frac{xy}{100} = \frac{yx}{100} \), the statement is **True (T)**.

### Statement (iii):
**The compound interest on Rs. 12000 for 1 1/2 years at 10% per annum compounded half yearly is Rs. 13891.50.**

1. The principal \( P = 12000 \).
2. The rate \( r = 10\% \) per annum, compounded half yearly means the rate per half year is:
\[ \frac{10}{2} = 5\% \]
3. The time \( t = 1.5 \) years means there are \( 3 \) half-year periods.
4. The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where \( n \) is the number of compounding periods.
5. Substituting the values:
\[ A = 12000 \left(1 + \frac{5}{100}\right)^3 = 12000 \left(1.05\right)^3 \]
6. Calculating \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
7. Now calculate \( A \):
\[ A = 12000 \times 1.157625 \approx 13891.50 \]
8. The compound interest \( CI \) is given by:
\[ CI = A - P = 13891.50 - 12000 = 1891.50 \]
9. Thus, the statement is **False (F)**.

### Statement (iv):
**If a retailer sells an alarm clock for Rs. 350 and gains 1/6 of its cost price then the cost price is Rs. 250.**

1. Let the cost price be \( CP \).
2. The selling price \( SP = 350 \).
3. The gain is \( \frac{1}{6} \) of the cost price:
\[ Gain = SP - CP = \frac{1}{6} CP \]
4. Rearranging gives us:
\[ 350 - CP = \frac{1}{6} CP \]
5. Multiplying through by 6 to eliminate the fraction:
\[ 6(350 - CP) = CP \]
6. Expanding gives:
\[ 2100 - 6CP = CP \]
7. Adding \( 6CP \) to both sides:
\[ 2100 = 7CP \]
8. Solving for \( CP \):
\[ CP = \frac{2100}{7} = 300 \]
9. Thus, the statement is **False (F)**.

### Final Answers:
- (i) T
- (ii) T
- (iii) F
- (iv) F