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CFTRI M.Sc. (Food Technology) Mathematics Syllabus

The Mathematics syllabus for CFTRI M.Sc. (Food Technology) in full: 6 chapters, 8 topics and 3 sub-topics, every one listed below, plus 51 Mathematics flashcards with answers.

6Chapters
8Topics
3Sub-topics
51Flashcards

Mathematics syllabus: all 6 chapters

Expand any chapter to see its topics and sub-topics.

  1. Theory of Quadratic Equations

     3 topics
    • Binomial Theorem
    • Uses of Natural and Common Logarithms
    • Exponential Series
      • Differentiation
      • Successive Differentiation
      • Maxima Minima

  2. Differential Equations

     2 topics
    • First Order
    • Linear

  3. Integration and Integral Equations

     overview

    No separate topics listed for this chapter.

  4. Trigonometry

     1 topic
    • Ratios and Their Relations

  5. Matrices, Vectors, Determinants

     overview

    No separate topics listed for this chapter.

  6. Statistics

     2 topics
    • Sampling
    • Probability

CFTRI M.Sc. (Food Technology) Mathematics flashcards — 51 questions with 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!.

Show the other 31 Mathematics flashcards

  1. 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.

  2. What is the radius of convergence of the exponential series for e^x?

    The series converges for all real (and complex) x; the radius of convergence is infinite.

  3. Write the series expansion of e^(-x).

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

  4. Using the exponential series, what is the coefficient of x^n in e^x?

    The coefficient of x^n is 1/n!.

  5. State Euler's formula linking the exponential and trigonometric functions.

    e^(ix) = cos x + i sin x.

  6. What general form defines a first-order differential equation?

    An equation involving the first derivative only: dy/dx = f(x, y), with no higher derivatives.

  7. What is a separable first-order differential equation and how is it solved?

    One that can be written as g(y) dy = h(x) dx; solved by integrating both sides separately.

  8. Solve the first-order equation dy/dx = ky.

    y = C e^(kx), where C is an arbitrary constant (exponential growth/decay).

  9. What distinguishes the order from the degree of a differential equation?

    Order is the highest derivative present; degree is the power of that highest derivative (after the equation is made polynomial in derivatives).

  10. What is the standard form of a first-order linear differential equation?

    dy/dx + P(x) y = Q(x).

  11. What is the integrating factor for dy/dx + P(x)y = Q(x)?

    The integrating factor is IF = e^(∫P(x) dx).

  12. State the general solution of a first-order linear differential equation using the integrating factor.

    y · (IF) = ∫ Q(x) · (IF) dx + C, where IF = e^(∫P dx).

  13. What is the standard form of a linear equation in two variables, and what does its graph look like?

    ax + by + c = 0 (a, b not both zero); its graph is a straight line.

  14. What is the slope-intercept form of a straight line?

    y = mx + c, where m is the slope and c is the y-intercept.

  15. For a system of two linear equations, what does it mean to have a unique solution, no solution, or infinitely many solutions?

    Unique solution: lines intersect (a1/a2 ≠ b1/b2). No solution: parallel lines (a1/a2 = b1/b2 ≠ c1/c2). Infinitely many: coincident lines (a1/a2 = b1/b2 = c1/c2).

  16. Define a ratio and how it is written.

    A ratio compares two quantities of the same kind by division, written a:b or a/b (b ≠ 0).

  17. Define a proportion and state the cross-multiplication property.

    A proportion states two ratios are equal, a:b = c:d; then ad = bc (product of extremes = product of means).

  18. What is the duplicate ratio of a:b?

    The duplicate ratio of a:b is a^2 : b^2.

  19. What is the sub-duplicate ratio of a:b?

    The sub-duplicate ratio of a:b is √a : √b.

  20. What does the componendo and dividendo rule state for a/b = c/d?

    If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d).

  21. Define the mean proportional (geometric mean) between two quantities a and b.

    The mean proportional x satisfies a:x = x:b, so x = √(ab).

  22. In a:b = c:d, identify the means and the extremes.

    The extremes are a and d (the outer terms); the means are b and c (the inner terms).

  23. Define sampling in statistics.

    Sampling is the process of selecting a subset (sample) from a population to estimate characteristics of the whole population.

  24. What is the difference between a population and a sample?

    A population is the entire set of items/individuals under study; a sample is a subset selected from that population for examination.

  25. Define simple random sampling.

    A sampling method in which every member of the population has an equal and independent chance of being selected.

  26. Distinguish stratified sampling from cluster sampling.

    Stratified: divide the population into homogeneous strata and sample from each. Cluster: divide into heterogeneous clusters and randomly select whole clusters to sample.

  27. What is the difference between a parameter and a statistic?

    A parameter is a numerical characteristic of a population (e.g. μ, σ); a statistic is a numerical characteristic computed from a sample (e.g. x̄, s).

  28. What is sampling error?

    The difference between a sample statistic and the true population parameter, arising because only a portion of the population is observed.

  29. State the classical (theoretical) definition of the probability of an event.

    P(E) = (number of favourable outcomes) / (total number of equally likely outcomes), with 0 ≤ P(E) ≤ 1.

  30. State the addition rule of probability for two events A and B.

    P(A ∪ B) = P(A) + P(B) - P(A ∩ B); for mutually exclusive events, P(A ∪ B) = P(A) + P(B).

  31. State the multiplication rule and the condition for independent events.

    P(A ∩ B) = P(A) · P(B|A); if A and B are independent, P(A ∩ B) = P(A) · P(B). Conditional: P(B|A) = P(A ∩ B)/P(A).

CFTRI M.Sc. (Food Technology) Mathematics FAQ

What is in the CFTRI M.Sc. (Food Technology) Mathematics syllabus?

CFTRI M.Sc. (Food Technology) Mathematics is split into 6 chapters: Theory of Quadratic Equations, Differential Equations, Integration and Integral Equations, Trigonometry, Matrices, Vectors, Determinants and Statistics. Together they contain 8 topics and 3 sub-topics.

How many chapters are in CFTRI M.Sc. (Food Technology) Mathematics?

6 chapters, containing 8 topics and 3 sub-topics. The largest chapter is Theory of Quadratic Equations, with 3 topics.

Are there flashcards for CFTRI M.Sc. (Food Technology) Mathematics?

Yes — a 51-card Mathematics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.