Financial mathematics is concerned with solving practical financial problems involving borrowing, lending, investments, insurance, mortgages, annuities, perpetuities, and sinking funds. The principles of compound interest, present value, and future value are fundamental to these activities. Among the given options, banking, insurance, and property dealing are the industries that directly depend on financial mathematics for pricing, valuation, investment decisions, loan repayment schedules, and risk assessment. Therefore, Option B correctly identifies the primary fields where financial mathematics is extensively applied.

The correct matching is obtained by understanding the role of each component. Use of money corresponds to money being exchanged in return for periodic payments. Interest plays a vital role in almost all financial activities, including investments, loans, and insurance. Finance companies are established to earn profits by providing financial services and lending money. Thus, the correct correspondence is 1 → b 2 → c 3 → a Hence, Option C is correct.

In lending transactions, the borrower uses another person's money and agrees to make periodic repayments while paying interest on the borrowed amount. Statement (I) is correct because loans generally involve periodic instalments or interest payments. Statement (III) is also correct because charging interest is the fundamental principle of lending. Statement (II) is incorrect because infinite geometric boundaries are mathematical concepts unrelated to the basic lending process. Therefore, only Statements I and III are correct.

Periodic payments are a fundamental feature of financial mathematics and occur in several financial instruments, including annuities, perpetuities, and sinking funds. A perpetuity consists of equal periodic payments that continue forever, whereas a sinking fund involves regular periodic deposits made to accumulate a specified future amount. Therefore, the statement that periodic payments apply only to sinking funds and never to perpetuities is incorrect. Hence, Option A is the incorrect statement.

The present value of a perpetuity is expressed as R/(1 + i) + R/(1 + i)² + R/(1 + i)³ + ⋯ Each term is obtained from the previous one by multiplying by 1/(1 + i) which is a constant ratio. Since 0 < 1/(1 + i) < 1, the series is an infinite geometric progression, and its sum is P = R/i Therefore, Option B is correct.

A perpetuity is an infinite sequence of equal periodic cash flows. By definition, it has no maturity date and continues indefinitely. Hence, the probability that the sequence terminates before reaching infinity is Any positive probability would imply that the perpetuity eventually stops, contradicting its definition. Therefore, Option C is correct.

The payment vector is P⃗ = ⟨R, R, R, …⟩ while the discount factor vector is v⃗ = ⟨(1 + i)⁻¹, (1 + i)⁻², (1 + i)⁻³, …⟩ The dot product is P⃗ · v⃗ = R(1 + i)⁻¹ + R(1 + i)⁻² + R(1 + i)⁻³ + ⋯ which is exactly the infinite geometric series representing the present value of an end-period perpetuity. Its value is P = R / i Therefore, Option D is correct.

A beginning-period perpetuity, also called an annuity due, makes its first payment at the beginning of the first period. Thus, First payment: t=0, Second payment: t=1, Third payment: t=2, and so on. This immediate payment distinguishes an annuity due from an ordinary perpetuity, where the first payment occurs at (t=1). Hence, Option A is correct.

The present value of an end-period perpetuity is R/(1 + i) + R/(1 + i)² + R/(1 + i)³ + ⋯ This is an infinite geometric progression whose common ratio is 1/(1 + i) where 0 < 1/(1 + i) < 1 Using the sum of an infinite geometric series, P = R / i Therefore, Option B is correct.

For an end-period perpetuity, the present value is P = R / i Rearranging the formula to determine the interest rate, Pi = R which gives i = R / P Substituting R = 450, P = 20,000, we obtain i = 450 / 20,000 = 0.0225 = 2.25% Thus, the Assertion is correct. The Reason is also correct because it states the standard perpetuity formula from which the assertion is directly derived. Hence, the reason correctly explains the assertion, making Option C the correct answer.

A payment every 6 months means one payment is made in each half-year. Since one year consists of 12 months, 12 / 6 = 2 payments occur annually. Thus, over any one-year interval, the moving average of the number of payments is 2 Hence, Option D is correct.

To solve a beginning-period perpetuity problem, the correct order is: Step 1: Identify the known variables: R = 2500, i = 0.03 Step 2: Write the appropriate formula: P = R + (R / i) Step 3: Substitute the given values: P = 2500 + (2500 / 0.03) = 2500 + 83333.33 = 85833.33 Thus, the correct sequence is III → II → I Hence, Option A is correct.

The passage clearly states that sinking fund problems are solved using the formulas for either: Ordinary annuity, when payments are made at the end of each period, or Annuity due, when payments are made at the beginning of each period. These formulas determine the periodic deposit required to accumulate a specified future amount. Therefore, Option B correctly reflects the statement given in the passage.

The passage clearly states that both sinking funds and savings accounts involve setting aside money for future use. However, the distinguishing feature of a sinking fund is that it is established for a specific purpose and within a definite time period. Examples include accumulating funds for replacing machinery, repaying a loan, or financing a future capital expenditure. In contrast, a savings account may be used for any future requirement and generally has no fixed objective or time horizon. Therefore, Option C correctly identifies the defining distinction.

The amount that must be accumulated through the sinking fund is the net replacement cost. Using the standard relationship, Required Accumulation = Replacement Cost − Salvage Value Substituting the given values, = 52,000 − 4,000 = 48,000 Therefore, the sinking fund should accumulate ₹48,000 Hence, Option D is correct.

In the sinking fund formula, A = R(Sₙ|ᵢ) where (A) = accumulated (future) amount, (R) = periodic deposit, (n) = number of instalments, (i) = interest rate per period. The objective of a sinking fund is to accumulate the required lump sum (A) by making equal periodic deposits. Therefore, Option A correctly defines the variable (A).

A savings account is a general-purpose financial instrument used to save money for any future requirement, such as education, emergencies, travel, or household expenses. It does not require a fixed objective or predetermined maturity date. In contrast, a sinking fund is established for a clearly defined purpose, such as repaying debt or replacing machinery, within a specified time period. Therefore, the financial instrument that is generally established for any purpose it may serve is the savings account. Hence, Option B is correct.

A sinking fund is created by making regular periodic deposits to accumulate a specified amount for a future obligation. In this case, the company deposits money annually to repay a debt of ₹1,00,000 after four years. The periodic deposits earn interest and accumulate until the required amount is available on the due date. Therefore, the company is using a sinking fund. Hence, Option C is correct.

The objective is to accumulate a fixed future amount through regular deposits, which is the purpose of a sinking fund. Since the deposits are made at the beginning of each year, the payment pattern is an annuity due. Therefore, the situation is an annuity due sinking fund, and the appropriate formulas for an annuity due are applied. Hence, Option D is correct.

When replacing an old machine, the amount received from selling or disposing of the existing machine is called its salvage value. This recovered amount contributes towards purchasing the new machine. Therefore, the sinking fund needs to accumulate only Replacement Cost Salvage Value Subtracting the salvage value prevents unnecessary accumulation of funds and ensures that only the net replacement cost is financed. Hence, Option A is correct.