Rajan invested Rs. ‘x’ at 25% p.a. simple interest for 4 years and Rs. (x + 400) at 20% p.a. compound interest, compounded half yearly for 1 year. The sum of the interest received from the two investments is Rs. 3956. Find the amount received when the Rs. (x + 600) is invested at 12% p.a. simple interest for 2 years. Which of the following statements is true?
I. Rs. (x + 1512)
II. Rs. (3x/4 + 2312)
III. Rs. (x/2 + 3112)

ATQ, First investment: principal = x, rate = 25% p.a., time = 4 years Simple interest₁ = x × 25 × 4 / 100 = x × 100/100 = x Second investment: principal = x + 400, rate = 20% p.a., compounded half yearly Rate per half-year = 10%, n = 2 Amount factor = (11/10)² = 121/100 Interest factor = 121/100 − 1 = 21/100 Interest₂ = (x + 400) × 21/100 Sum of interests = 3956 So, x + (x + 400) × 21/100 = 3956 Multiply both sides by 100: 100x + 21(x + 400) = 395600 100x + 21x + 8400 = 395600 121x + 8400 = 395600 121x = 395600 − 8400 = 387200 x = 387200/121 = 3200 Third investment principal = x + 600 = 3200 + 600 = 3800 Rate = 12% p.a., time = 2 years Simple interest₃ = 3800 × 12 × 2 / 100 = 3800 × 24/100 = 3800 × 6/25 = 912 Amount = 3800 + 912 = 4712 Now check options: I. x + 1512 = 3200 + 1512 = 4712 → true II. 3x/4 + 2312 = 3 × 3200/4 + 2312 = 2400 + 2312 = 4712 → true III. x/2 + 3112 = 3200/2 + 3112 = 1600 + 3112 = 4712 → true Hence, all I, II and III are true.