CFTRI M.Sc. (Food Technology) Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in CFTRI M.Sc. (Food Technology), not a summary of it.

1. 1

   Theory of Quadratic Equations

   3 topics

   • Binomial Theorem
   • Uses of Natural and Common Logarithms
   • Exponential Series
     • Differentiation
     • Successive Differentiation
     • Maxima Minima
2. 2

   Differential Equations

   2 topics

   • First Order
   • Linear
3. 3

   Integration and Integral Equations

   overview

   Examined as a single unit within Mathematics — no further topic split in the official outline.

4. 4

   Trigonometry

   1 topic

   • Ratios and Their Relations
5. 5

   Matrices, Vectors, Determinants

   overview

   Examined as a single unit within Mathematics — no further topic split in the official outline.

6. 6

   Statistics

   2 topics

   • Sampling
   • Probability

Mathematics flashcards for CFTRI M.Sc. (Food Technology)

21 of 51 cards from the Mathematics deck — real questions with worked answers.

1. State the Binomial Theorem for a positive integer exponent n, expanding (a + b)^n.

   (a + b)^n = Σ (from k=0 to n) C(n,k) a^(n-k) b^k, where C(n,k) = n!/(k!(n-k)!).

2. What is the general (r+1)th term in the expansion of (a + b)^n?

   T(r+1) = C(n,r) a^(n-r) b^r.

3. How many terms are there in the expansion of (a + b)^n for a positive integer n?

   There are n + 1 terms.

4. What is the binomial coefficient C(n,r) and how does it relate to C(n, n-r)?

   C(n,r) = n!/(r!(n-r)!); it equals C(n, n-r) (symmetry property).

5. State Pascal's rule for binomial coefficients.

   C(n,r) + C(n, r-1) = C(n+1, r).

6. What is the sum of all binomial coefficients in the expansion of (1 + x)^n when x = 1?

   The sum equals 2^n: C(n,0)+C(n,1)+...+C(n,n) = 2^n.

7. Write the binomial series for (1 + x)^n when n is not a positive integer (|x| < 1).

   (1+x)^n = 1 + nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ... , valid for |x| < 1.

8. In the expansion of (1 + x)^n, what is the middle term when n is even?

   There is one middle term: the (n/2 + 1)th term, C(n, n/2) x^(n/2).

9. How do you find the term independent of x (constant term) in a binomial expansion?

   Write the general term, set the total power of x equal to zero, solve for r, then substitute r back into the general term.

10. Define the natural logarithm and state its base.

    The natural logarithm, ln x or log_e x, is the logarithm to the base e ≈ 2.71828.

11. Define the common logarithm and state its base.

    The common logarithm, log x or log_10 x, is the logarithm to the base 10.

12. State the change-of-base formula relating natural and common logarithms.

    log_b x = ln x / ln b = log x / log b; specifically ln x = log x / log e ≈ 2.302585 × log x.

13. What is the product rule of logarithms?

    log(MN) = log M + log N.

14. What is the quotient rule of logarithms?

    log(M/N) = log M - log N.

15. What is the power rule of logarithms?

    log(M^p) = p · log M.

16. What are the characteristic and mantissa of a common logarithm?

    The characteristic is the integer part of log10 of a number (indicating order of magnitude); the mantissa is the non-negative decimal/fractional part.

17. What is the value of log_a 1 and log_a a for any valid base a?

    log_a 1 = 0 and log_a a = 1.

18. Give a practical use of natural logarithms in growth/decay problems.

    Natural logs solve continuous growth/decay: from A = A0 e^(kt), t = (1/k) ln(A/A0), e.g. in microbial growth, radioactive decay, or first-order kinetics.

19. Why are common (base-10) logarithms convenient for quantities like pH?

    Base-10 logs compress wide ranges into manageable scales; e.g. pH = -log10[H+], so each unit represents a tenfold change in concentration.

20. Write the exponential series expansion of e^x.

    e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... = Σ (n=0 to ∞) x^n/n!.

21. What is the value of the constant e expressed as a series (set x = 1)?

    e = 1 + 1/1! + 1/2! + 1/3! + ... ≈ 2.71828.

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