ICAR-IASRI MSc and PhD Differential Equations, Finite Differences & Numerical Analysis: practice questions

ICAR-IASRI MSc and PhD (Agricultural Statistics, Computer Application, Bioinformatics) · India · ICAR-Indian Agricultural Statistics Research Institute with the IARI Graduate School

Original practice questions written to the published syllabus · not past papers

10worked here, free

34written to this exam in this subject

11syllabus units

What this subject covers

Ordinary differential equations, interpolation, numerical differentiation and integration and root-finding: what AIEEA-PG Group 07 names explicitly, and what growth models, life tables and iterative estimation (Newton-Raphson for maximum likelihood) rely on.

In the exams: AIEEA-PG Group 07 Units II-III (heavy: differential equations, finite differences, interpolation, numerical integration, Newton-Raphson named); AICE-JRF Computer Application Unit 10 numerical analysis (heavy)

Ordinary differential equations of first order: exact, linear and Bernoulli equations
Higher-order linear equations: complementary function and particular integral
Calculus of finite differences and difference operators
Interpolation: Newton forward and backward, Lagrange and divided differences
Numerical differentiation
Numerical integration: trapezoidal and Simpson's rules
Difference equations
Solution of non-linear equations: bisection and Newton-Raphson
Solution of linear systems: Gaussian elimination and iterative methods
Numerical solution of ODEs: Euler and Runge-Kutta methods
Coordinate geometry of conics

Where the 34 questions fall

How our questions for ICAR-IASRI MSc and PhD spread across the units of Differential Equations, Finite Differences & Numerical Analysis. Difficulty: 12 easy, 20 medium, 2 hard.

Calculus of finite differences and difference operators	4
Interpolation: Newton forward and backward, Lagrange and divided differences	4
Numerical differentiation	4
Ordinary differential equations of first order: exact, linear and Bernoulli equations	3
Numerical integration: trapezoidal and Simpson's rules	3
Solution of non-linear equations: bisection and Newton-Raphson	3
Solution of linear systems: Gaussian elimination and iterative methods	3
Numerical solution of ODEs: Euler and Runge-Kutta methods	3
Coordinate geometry of conics	3
Higher-order linear equations: complementary function and particular integral	2
Difference equations	2

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Ordinary differential equations of first order: exact, linear and Bernoulli equations · easy

A population grows as dN/dt = 0.07N. Its doubling time is about:

14.3 time units
9.9 time units
0.07 time units
70 time units

Answer and working

B. 9.9 time units

Doubling time = ln 2/r = 0.693/0.07 ≈ 9.9. 14.3 is 1/r, the mean time constant; 70 forgets to divide by the rate in per cent.

Further reading: Boyce, DiPrima and Meade, Elementary Differential Equations (2017); Burden, Faires and Burden, Numerical Analysis (2016); Hildebrand, Introduction to Numerical Analysis (1987); Thornley and France, Mathematical Models in Agriculture (2007)

2 · Ordinary differential equations of first order: exact, linear and Bernoulli equations · easy

The monomolecular (Mitscherlich) model dN/dt = r(K − N), used for fertiliser response, has:

An inflection at K/2
An inflection at K/e
No inflection point
Exponential growth without limit

Answer and working

C. No inflection point

Growth rate is greatest at the start and declines steadily; logistic and Gompertz are sigmoid.

3 · Higher-order linear equations: complementary function and particular integral · medium

A particular integral of y″ + 4y = sin x is:

(sin x)/3
(sin x)/5
−(sin x)/3
(sin x)/4

Answer and working

A. (sin x)/3

Replace D² by −1: f = −1 + 4 = 3.

4 · Interpolation: Newton forward and backward, Lagrange and divided differences · medium

In Newton's forward formula, the variable p equals:

(x − xₙ)/h
x − x₀
h/(x − x₀)
(x − x₀)/h

Answer and working

D. (x − x₀)/h

(x − xₙ)/h is used in the backward formula.

5 · Numerical differentiation · easy

The central difference approximation of f′(x) has truncation error of order:

h
h³
h⁴
h²

Answer and working

D. h²

[f(x + h) − f(x − h)]/(2h) − f′(x) = h²f‴/6 + …

6 · Difference equations · medium

Characteristic root m = 3 repeated twice gives the solution:

A3ⁿ + B3ⁿ
A3ⁿ + Bn²
A(3n)ⁿ
(A + Bn)3ⁿ

Answer and working

D. (A + Bn)3ⁿ

A repeated root needs the extra factor n.

7 · Solution of linear systems: Gaussian elimination and iterative methods · easy

Partial pivoting in Gaussian elimination chooses as pivot:

The smallest non-zero entry in the whole matrix
The diagonal entry always
The largest entry in absolute value in the column
The entry in the last row

Answer and working

C. The largest entry in absolute value in the column

Large pivots limit growth of rounding errors.

8 · Solution of linear systems: Gaussian elimination and iterative methods · medium

For the same system, one Gauss–Seidel iteration from (0, 0) gives:

(1.1, 0.99)
(1.1, 1.1)
(1.0, 1.0)
(0.99, 1.1)

Answer and working

A. (1.1, 0.99)

x = 1.1, then y = (11 − 1.1)/10 = 0.99 using the new x.

9 · Coordinate geometry of conics · medium

The parabola y² = 12x has focus:

(12, 0)
(0, 3)
(6, 0)
(3, 0)

Answer and working

D. (3, 0)

Compare with y² = 4ax: 4a = 12 gives a = 3, focus (a, 0).

10 · Coordinate geometry of conics · medium

A fitted quadratic response surface in two fertilisers has an indefinite quadratic part. Its stationary point is:

A maximum
A saddle point
A minimum
Undefined

Answer and working

B. A saddle point

Indefinite quadratic forms give hyperbolic contours around a saddle.

Ten here, 34 in the app. The Laisha app has every practice question we have written to ICAR-IASRI MSc and PhD in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.

Free for ten days, then one subscription of US$10 a month. We would rather say so here than have you find it later.

Practise the full set in Laisha →