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ompare the phase diagrams of the systems:
i)
ii)
by locating the equilibrium points and sketching the phase paths.
b) Consider the epidemic model governed by the following equation
with initial condition at
. Here x(t) is the number of susceptibles at time t,
is the contact rate. The population is assumed to be closed and homogeneously mixing. Let the contact rate be 0.002 and the number of susceptibles be 5000 initially
i) Find the density of the population when the rate of appearance of new cases is maximum.
ii) Find the time (in weeks) at which the rate of appearance of new cases is maximum.
iii) Obtain the maximum rate of appearance of new cases.
यदि एक ग्रह अपनी कक्षा में, जिसे वृत्ताकार माना गया है, अचानक रूक जाता हो, तो दिखाइए कि वह उस कालावधि में सूर्य में गिर जाएगा जो कि ग्रह के परिक्रमण काल का गुना है। √2 8
Consider a one-dimensional growth c(x, t) of phytoplankton in a water mass. Formulate the model describing the dynamics of growth taking into account the following: D, its diffusion coefficient, r its rate of growth, R its mortality rate due to sinking. Fixing the area of interest as and the initial concentration of phytoplankton as
, find the concentration distribution of phytoplankton in
at any time t.
b) When an aeroplane ascends from take-off to an altitude of , by how much does the gravitational attraction acting on it decrease?
A parachutist, whose weight (actually mass) is , drops from a helicopter
above the ground. She falls towards the earth under the influence of gravity. Assume that the gravitational force is constant. Assume that the force due to air resistance is proportional to the velocity of the parachutist. The proportionality constant is
when the parachute is closed, and is
when it is open. If the parachute does not open until
after the parachutist leaves the helicopter, after how many seconds will she hit the ground?
b) A projectile is fixed with a constant speed v at two different angles of projection and
such that it gives the same range. Show that
Characterise the following as discrete or continuous giving reasons for your answers.
i) Effects of radiation treatment on a tumour when applied for short period of time but at regular intervals.
ii) Effects of chemotherapy drugs on a tumour when introduced into a patient for a given duration of time.
b) A particle of mass m moves on a straight line towards the centre of attraction, starting from rest at a distance a from the centre. Its velocity at a distance x from the centre varies as . Find the law of force.
c) If a planet was suddenly stopped in its orbit, supposed circular, show that it will fall into the sun in a time which is times the period of the planet’s revolution.
) निम्नलिखित त्रिघात कुल लागत फलन लीजिए। यहां यह मान लीजिए कि q की कीमत प्रति एकक 15 है। वह उत्पादन ज्ञात कीजिए जिस पर अधिकतम लाभ प्राप्त होता है।
ख) प्रमुखता लागू करके नीचे दिए गए भुगतान आव्यूह से A और B की इष्टतम युक्तियां ज्ञात कीजिए।
ग) मान लीजिए पिछला पूर्वानुमान 2090 था और पिछली अवधि में विशेष चर का वास्तविक मान 1985 था और इसका सबसे पुराना मान 1955 था। हाल ही में किए गए चार प्रेक्षणों पर आधारित गतिमान औसत तकनीक को लागू करके अगली अवधि के लिए किया गया नया पूर्वानुमान ज्ञात कीजिए।
A monopolist sets a price 'p' per unit and the quantity demanded 'q' is given by the following relation:
Let there be a fixed cost of Rs.9 and a marginal cost of Rs.1 per unit.
i) Write the profit function of monopolist.
ii) For maximum profit, find the number 'x' of units produced. Also find the maximum profit.
iii) A potential entrant enters into the business of the monopolist. He believes that the monopolist will go on making 'x' units. Write the profit function of the entrant.
iv) For maximum profit of entrant, find the number 'z' of units produced.
v) Find the maximum profit of the entrant. Explain whether he should enter into the business or not.
vi) Find the profit made by the monopolist after the entrant has entered into business.