ICAR AIEEA-PG Differential Equations, Finite Differences & Numerical Analysis: practice questions

ICAR AIEEA-PG - Group 07 Statistical Sciences (Master's admission and PG scholarship) · India · National Testing Agency for ICAR

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

10worked here, free

59written 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 59 questions fall

How our questions for ICAR AIEEA-PG spread across the units of Differential Equations, Finite Differences & Numerical Analysis. Difficulty: 17 easy, 39 medium, 3 hard.

Ordinary differential equations of first order: exact, linear and Bernoulli equations	7
Calculus of finite differences and difference operators	7
Higher-order linear equations: complementary function and particular integral	6
Interpolation: Newton forward and backward, Lagrange and divided differences	6
Numerical differentiation	6
Numerical integration: trapezoidal and Simpson's rules	5
Solution of linear systems: Gaussian elimination and iterative methods	5
Coordinate geometry of conics	5
Difference equations	4
Solution of non-linear equations: bisection and Newton-Raphson	4
Numerical solution of ODEs: Euler and Runge-Kutta methods	4

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 · medium

The equation (2xy + 3)dx + (x² + 4y)dy = 0 is exact because:

M and N are both polynomials
∂M/∂y = ∂N/∂x = 2x
∂M/∂x = ∂N/∂y
M = N at the origin

Answer and working

B. ∂M/∂y = ∂N/∂x = 2x

M = 2xy + 3 gives ∂M/∂y = 2x; N = x² + 4y gives ∂N/∂x = 2x.

3 · Ordinary differential equations of first order: exact, linear and Bernoulli equations · hard

With K = 1,000, N₀ = 100 and r = 0.5, the logistic solution N(t) = K/(1 + Ae^(−rt)) gives N(2) of about:

Answer and working

C. 232

A = (1,000 − 100)/100 = 9; N(2) = 1,000/(1 + 9e^(−1)) = 1,000/4.311 ≈ 232. 272 uses unchecked exponential growth 100e¹.

4 · 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.

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 · Numerical differentiation · medium

The standard approximation of f″(x) from three equally spaced values is:

[f(x + h) − f(x − h)]/(2h)
[f(x + h) − f(x)]/h²
[f(x + h) + 2f(x) + f(x − h)]/h²
[f(x + h) − 2f(x) + f(x − h)]/h²

Answer and working

D. [f(x + h) − 2f(x) + f(x − h)]/h²

It is the second central difference divided by h².

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 eccentricity of the ellipse x²/25 + y²/9 = 1 is:

0.8
0.6
1.25
0.36

Answer and working

A. 0.8

e = √(1 − 9/25) = 4/5.

10 · Coordinate geometry of conics · medium

The eccentricity of the hyperbola x²/16 − y²/9 = 1 is:

0.8
1.25
1.33
0.75

Answer and working

B. 1.25

e = √(1 + 9/16) = 5/4.

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