Answer available — download the full PDF for complete details.

MTE-14 Solved Assignment January 2026 | MATHEMATICAL MODELLING | IGNOU BDP

Q1. null

(a)) State two real-world problems where you think that mathematical modelling is the only approach to find the solution of the problem. Give 4 essentials for each of the problems. Why do you think that there is no other scientific alternative for the treatment of these problems. (200 words)
(b)) Classify the following into linear and non-linear models, justifying your classification. i) Simple harmonic motion for small amplitude of oscillation. ii) Population growth model given by dN/dt = aN (B-kN), a, B, k are constants. iii) Equation for velocity v of a particle at any time t, moving with a constant acceleration a, and initial velocity u. iv) Equation describing dynamic stability of market equilibrium price given by p_t = [1 + k(a - A)p_t-1 + k (b − B)], a, b, A, B, k are constants and p_t is the price in period t. (300 words)

Key Points:

Mathematical modeling is essential for problems too complex, large, ethical-sensitive, or long-term for direct study.
Climate change prediction relies solely on mathematical models due to its vast scale and temporal scope.
Drug dosage optimization employs modeling for ethical, safety, and efficiency reasons in pharmacology.
Linear models have dependent variables and derivatives appearing only to the first power, without products.

Answer: Mathematical modeling serves as an indispensable tool for understanding and solving complex real-world problems that are otherwise intractable through direct experimentation or observation. It involves representing a system using mathematical concepts and language, allowing for analysis, prediction, and optimization. This approach is particularly crucial when systems are too large, too small, too dangerous, too slow, too fast, or too complex to study directly. Models can be broadly classified b...

Q2. null

(a)) 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. (200 words)
(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 sqrt( (a^3 - x^3) / x^3 ). Find the law of force. (150 words)
(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 (sqrt(2)/8) times the period of the planet's revolution. (150 words)

Key Points:

Discrete models: Variables change at distinct, separated points in time.
Continuous models: Variables change smoothly and continuously over time.
Newton's Second Law (F=ma) is fundamental for deriving force from motion.
Acceleration can be expressed as a = v(dv/dx) when velocity is a function of position.

Answer: Mathematical modeling involves classifying phenomena as either discrete or continuous, depending on how their variables change over time or space. This distinction influences the type of mathematical tools used, such as difference equations for discrete systems and differential equations for continuous systems. Furthermore, understanding fundamental laws of motion and gravitation is crucial for modeling physical systems, as demonstrated by analyzing forces and orbital mechanics. This response ad...

Q3. null

(a)) A particle moving in S.H.M has got the velocities 8cm/sec and 6cm/sec when it is at distance 3cm and 4cm respectively from the centre of its motion. Determine the period and the amplitude of motion. (200 words)
(b)) Consider the following system of equations dx/dt = y, dy/dt = -k² sin x. Find the nature of the critical point (0, 0) of the corresponding linear system. (150 words)
(c)) A string of length l is connected to a fixed point at one end and to a stick of mass m at the other. The stick is whirling in a circle at constant velocity v. Use dimensional analysis to find the equation of the force in the string. (150 words)

Key Points:

SHM velocity squared equals angular frequency squared times (amplitude squared minus displacement squared).
Amplitude and angular frequency in SHM are derivable from two velocity-displacement data points.
Period of SHM is calculated as T = 2π/ω, where ω is the angular frequency.
Linearizing a non-linear system's critical point involves computing its Jacobian matrix.

Answer: This response comprehensively addresses three distinct problems from Mathematical Modelling (MTE-14), covering Simple Harmonic Motion, stability analysis of critical points for non-linear systems, and dimensional analysis. Each sub-question is tackled using fundamental principles and relevant mathematical techniques, presenting step-by-step solutions as expected in the IGNOU MTE-14 course.

Q4. null

(a)) A parachutist, whose weight (actually mass) is 64 kg, drops from a helicopter 5000 m. 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 k1 = 16 kg/sec when the parachute is closed, and is k2 = 100 kg/sec when it is open. If the parachute does not open until 1 minute after the parachutist leaves the helicopter, after how many seconds will she hit the ground? (400 words)
(b)) A projectile is fixed with a constant speed v at two different angles of projection α and β such that it gives the same range. Show that cosec α = sec β. (100 words)

Key Points:

Newton's Second Law: `m(dv/dt) = mg - kv` models falling objects with linear air resistance.
Velocity solution for linear air resistance: `v(t) = (mg/k) * (1 - e^(-kt/m))` for `v(0)=0`.
Distance solution requires integrating the velocity function over time.
Projectile range formula: `R = (v² sin(2θ)) / g`.

Answer: The provided question involves mathematical modeling of physical phenomena: a parachutist's fall under gravity and air resistance, and projectile motion. Both sub-questions require applying fundamental physics principles and solving differential equations or using known kinematic formulas. Sub-question (a) demands a piecewise approach to model the fall, accounting for changing air resistance, while sub-question (b) tests the understanding of projectile range and trigonometric identities.

Q5. null

(a)) 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 0 ≤ x ≤ 2 and the initial concentration of phytoplankton as 30 moles/cm³, find the concentration distribution of phytoplankton in 0 ≤ x ≤ 2 at any time t. (350 words)
(b)) When an aeroplane ascends from take-off to an altitude of 10 km, by how much does the gravitational attraction acting on it decrease? (150 words)

Key Points:

Reaction-diffusion model describes concentration changes due to diffusion and net reaction (growth/mortality).
Phytoplankton dynamics PDE: ∂c/∂t = D∂²c/∂x² + (r-R)c, where D is diffusion, r growth, R mortality.
For uniform initial concentration and no-flux boundaries, the solution is c(x, t) = c₀ * e^((r-R)t).
Gravitational force formula: F = G * M * m / r², where r is distance from Earth's center.

Answer:

Q6. Consider the group of individuals born in a given year (t = 0) and let n(t) be the number of these individuals surviving t year later. Let x(t) be the number of members of this group who have not had smallpox by year t and are therefore still susceptible. Let β be the rate at which susceptibles contract smallpox and let v be the rate at which people who contract smallpox die from the disease. Finally, let μ(t) be the death rate from all causes other than smallpox. If dx/dt and dn/dt are, respectively the rates at which the number of susceptibles and entire population decline due to contraction from smallpox and also due to death from all causes then

(i)) Formulate the above problem by writing equations for dx/dt and dn/dt. (125 words)
(ii)) Taking z = x/n, show that z satisfies the initial value problem dz/dt = -βz (1-vz), z(0) = 1 (125 words)
(iii)) Find z(t) at any time t. (125 words)
(iv)) Bernoulli estimated that v = β = 1/8. Using these values, determine the proportion of 20 years old who have not had smallpox. (125 words)

Key Points:

dx/dt models susceptibles' decline due to smallpox contraction and other deaths.
dn/dt models total population decline due to smallpox deaths and other causes.
z = x/n represents the proportion of susceptibles in the surviving population.
The initial condition z(0)=1 signifies everyone starts as susceptible.

Answer: The problem involves modeling the dynamics of a cohort of individuals over time, specifically tracking the number of individuals susceptible to smallpox and the total surviving population. This epidemiological model considers rates of smallpox contraction, smallpox-induced death, and death from other causes. By formulating differential equations for these quantities, we can analyze the proportion of susceptible individuals within the surviving population.

Q7. A monopolist sets a price 'p' per unit and the quantity demanded 'q' is given by the following relation: q = 17 - p. 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. (100 words)
(ii)) For maximum profit, find the number 'x' of units produced. Also find the maximum profit. (100 words)
(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. (100 words)
(iv)) For maximum profit of entrant, find the number 'z' of units produced. (100 words)
(v)) Find the maximum profit of the entrant. Explain whether he should enter into the business or not. (50 words)
(vi)) Find the profit made by the monopolist after the entrant has entered into business. (50 words)

Key Points:

Monopolist's profit function is TR - TC, derived from inverse demand and cost functions.
Profit maximization occurs where the first derivative of the profit function with respect to quantity is zero.
An entrant's profit function considers the incumbent's fixed output, affecting market price.
Entrant's optimal output 'z' is found by maximizing their profit function.

Answer: This problem involves analyzing market structures, specifically moving from a monopoly to a duopoly scenario, a common topic in mathematical modelling of economics. We will derive profit functions, determine optimal output levels for maximum profit, and evaluate entry decisions based on profitability. The analysis uses fundamental economic principles of demand, cost, and profit maximization, which are crucial for understanding market dynamics.

Q8. null

(a)) Consider the cubic total cost function C = 0.06q³ - 0.8q² + 13q + 10. Assume that the price of q is 15 per unit. Find the output which yields maximum profit. (150 words)
(b)) Apply dominance to find the optimum strategies of A and B from the pay-off matrix given below Player B u v x 6(2) 7(3) Player A y 4(7) 9(3) z 7(1) 8(2) (150 words)
(c)) Suppose that the previous forecast was 2090 and the actual value of the variable of interest for the period was 1985 and the oldest value of interest was 1955. Using the moving average technique based upon the most recent four observations find new forecast for the next period. (200 words)

Key Points:

Profit maximization involves setting the first derivative of the profit function to zero.
Quadratic formula solves for critical points of a quadratic equation.
Second derivative test confirms maximum profit if its value is negative.
Dominance eliminates strategies where one strategy is always better than another for a player.

Answer: As an expert IGNOU tutor, I will address the given questions comprehensively, providing step-by-step solutions and clarifying any ambiguities in the problem statements by making explicit, reasonable assumptions where necessary. These problems draw upon key concepts in economic optimization, game theory, and forecasting from the MTE-14 Mathematical Modelling course. For part (a), the objective is to maximize profit by determining the optimal output level using calculus. Part (b) involves applyin...

Q9. null

(a)) Compare the phase diagrams of the systems: i) ẋ = y, ẏ = -x ii) ẋ = xy, ẏ = -x² by locating the equilibrium points and sketching the phase paths. (250 words)
(b)) Consider the epidemic model governed by the following equation dx/dt = -βx (n+1-x) with initial condition x = n at t=0. Here x(t) is the number of susceptibles at time t, β 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. (250 words)

Key Points:

System ẋ=y, ẏ=-x has a center at (0,0) with closed circular phase paths.
System ẋ=xy, ẏ=-x² has a line of equilibrium points along the y-axis (x=0).
Phase paths for ẋ=xy, ẏ=-x² are circular segments flowing downwards to the y-axis.
The rate of appearance of new cases in the epidemic model is -dx/dt = βx(N-x).

Answer:

Q10. For a given set of securities, all their portfolios lie on or within the boundary of the region shown in Fig. 1.

Key Points:

Fig. 1 depicts the feasible region of all possible portfolios on a risk-return plane.
The region's boundary is the minimum variance frontier, showing lowest risk for each return.
The upper part of the boundary is the efficient frontier, containing optimal risk-return portfolios.
Portfolios within the boundary are inefficient, offering suboptimal risk-return combinations.

Answer: In the context of mathematical modelling in finance, particularly Modern Portfolio Theory (MPT), the region described in Fig. 1 (assuming it depicts a risk-return plane) represents the **feasible set** of all possible portfolios constructed from a given set of securities. This region is typically convex, with expected return plotted on the y-axis and portfolio standard deviation (risk) on the x-axis. Any point *within* this boundary represents a portfolio whose combination of expected return an...

MTE-14 Solved Assignment — January 2026 / July 2026

MATHEMATICAL MODELLING | IGNOU BA CHELORS DEGREE PROGRAMME (BDP)

MTE-14—January 2026 / July 2026

MATHEMATICAL MODELLING

Questions & Answers

Download Study Materials

Related Courses

MTE-14 Assignment Questions — January 2026

Q1. null

Q2. null

Q3. null

Q4. null

Q5. null

Q7. A monopolist sets a price 'p' per unit and the quantity demanded 'q' is given by the following relation: q = 17 - p. Let there be a fixed cost of Rs.9 and a marginal cost of Rs.1 per unit.

Q8. null

Q9. null

Q10. For a given set of securities, all their portfolios lie on or within the boundary of the region shown in Fig. 1.

More MTE-14 Solved Materials