Assertion-Reason Questions
Q1: Assertion (A): ₹100000 amounts to ₹121000 in 1 year and to ₹146410 in 2 years, compounded annually.
Reason (R): When the interest is compounded annually, then, we have \(A=P \left( 1 + \frac{R}{100} \right)^n\)
i. Verify the Reason (R) Statement
Step 1: Examine the formula provided in the Reason.
The standard mathematical formula to find the total amount under annual compound interest is \(A = P\left(1+\frac{R}{100}\right)^n\).
Therefore, the statement given in Reason (R) is true.
ii. Verify the Assertion (A) Statement
Step 1: Determine the underlying interest rate from the first year values.
Principal \( (P) = \text{₹}100000 \)
Amount after 1 year \( (A_1) = \text{₹}121000 \)
\( 121000 = 100000 \times \left(1 + \frac{R}{100}\right)^1 \)
\( \frac{121000}{100000} = 1 + \frac{R}{100} \)
\( 1.21 = 1 + \frac{R}{100} \)
\( \frac{R}{100} = 0.21 \Rightarrow R = 21\% \text{ per annum} \)
Step 2: Calculate the expected amount after 2 years using this calculated rate.
Time period \( (n) = 2 \text{ years} \)
\( A_2 = 100000 \times \left(1 + \frac{21}{100}\right)^2 \)
\( A_2 = 100000 \times \left(1.21\right)^2 \)
\( A_2 = 100000 \times 1.4641 \)
\( A_2 = \text{₹}146410 \)
The calculated value exactly matches the value given in the Assertion statement.
Therefore, the statement given in Assertion (A) is true.
Answer:c. Both A and R are true
Q2: Assertion (A): The present population of a town decreases at the rate of 5% p.a. due to migration. Its present population is 21600 and 2 years ago it was 24000.
Reason (R): Let there be an increase in population at R% p.a. Then population after n years is given by \(P\left(1-\frac{R}{100}\right)^n\), where P is the present population.
i. Verify the Reason (R) Statement
Step 1: Analyze the logic and formula given in Reason (R).
The statement mentions an **increase** in population, but the formula provided uses a minus sign \( \left(1-\frac{R}{100}\right) \), which is for a decrease.
For an increase in population, the correct formula is \( P\left(1+\frac{R}{100}\right)^n \).
Therefore, the statement given in Reason (R) is false.
ii. Verify the Assertion (A) Statement
Step 1: Test if a population of 24000 decreasing at 5% p.a. results in 21600 after 2 years.
Let population 2 years ago be \( P_{\text{ago}} = 24000 \).
Rate of decrease \( (R) = 5\% \)
Time period \( (n) = 2 \text{ years} \)
Step 2: Calculate the expected present population value.
Formula: \( \text{Present Population} = P_{\text{ago}} \times \left(1 – \frac{R}{100}\right)^n \)
\( \text{Present Population} = 24000 \times \left(1 – \frac{5}{100}\right)^2 \)
\( \text{Present Population} = 24000 \times \left(\frac{19}{20}\right)^2 \)
\( \text{Present Population} = 24000 \times \frac{361}{400} \)
\( \text{Present Population} = 60 \times 361 \)
\( \text{Present Population} = 21660 \)
Step 3: Compare the computed value with the value given in the text statement.
The calculated population is 21660, whereas the Assertion statement claims it is 21600.
Therefore, the statement given in Assertion (A) is also false.
Answer:d. Both A and R are false.
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