From the relation R = \( R_0A^{\dfrac{1}{3}}\), where R_{0} is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

General anesthesia is used for major operations to cure the patients and conduct pain free surgeries. Propofol is a commonly used anesthetic injected for major operations such as knee replacement or open heart surgery. It also acts as a sedative and an analgesic.

A patient is rushed to operation theatre for a 2-hour cardiac surgery. A person is anesthetized when its blood stream contains at least 3 mg of propofol per kg of body weight. The rate of change of propofol(x), in the body is proportional to the quantity of propofol present at that time. Based on the above information. Answer the following questions:

**(a)** Show that propofol given intravenously is eliminated exponentially from the patients’ blood stream

**(b)** What dose of propofol should be injected to induce unconsciousness in a 50 Kg adult for a two hours operation?

(Given \( (2)^{\dfrac{2}{5}}\)=1.1487 & assume half-life of propofol = 5 hours)

A ₹2000, 8% bond is redeemable at the end of 10 years at ₹105. Find the purchase price to yield 10% effective rate. [Given(1.1)^{−10 }=0.3855]

A couple wishes to purchase a house for ₹15,00,000 with a down payment of ₹4,00,000. If they can amortize the balance at an interest rate 9% per annum compounded monthly for 10 years, find the monthly installment (EMI). Also find the total interest paid. [Given \( (1.0075)^{−120}\)=0.4079 ]

A machine costing ₹50,000 is to be replaced at the end of 10 years, when it will have a salvage value of ₹5000. In order to provide money at that time for a machine costing the same amount, a sinking fund is set up. If equal payments are placed in the fund at the end of each quarter and the fund earns 8% compounded quarterly, then what should each payment be? [Given (1.02)^{40} =2.208 ]