UPSC NDA Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the condition for a function f(x) to be continuous at x = a.

   f is continuous at a if: (1) f(a) is defined, (2) lim_{x→a} f(x) exists, and (3) lim_{x→a} f(x) = f(a).

2. Give the limit definition of the derivative of f(x).

   f'(x) = lim_{h→0} [f(x + h) − f(x)] / h, the instantaneous rate of change / slope of the tangent.

3. State the product rule and quotient rule of differentiation.

   Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

4. How is the derivative used to determine whether a function is increasing or decreasing?

   If f'(x) > 0 on an interval, f is increasing there; if f'(x) < 0, f is decreasing there; f'(x) = 0 gives critical (stationary) points.

5. State the second-derivative test for local maxima and minima.

   At a critical point where f'(x) = 0: if f''(x) < 0 it is a local maximum; if f''(x) > 0 it is a local minimum; if f''(x) = 0 the test is inconclusive.

6. What does it mean that integration is the inverse of differentiation, and give the power rule for integration.

   ∫f'(x) dx = f(x) + C; differentiating an antiderivative returns the original function. Power rule: ∫xⁿ dx = x^{n+1}/(n+1) + C, n ≠ −1.

7. State the Fundamental Theorem of Calculus for a definite integral.

   ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f (F' = f).

8. How do you find the area between a curve y = f(x) and the x-axis from x = a to x = b?

   Area = ∫ₐᵇ |f(x)| dx. Where the curve dips below the axis the integral is taken as the absolute value (or split at the roots) so areas add positively.

9. What is the order and degree of a differential equation?

   Order = the highest derivative present. Degree = the power of the highest-order derivative after the equation is made polynomial (free of radicals/fractions in derivatives).

10. How is a differential equation formed from a family of curves with n arbitrary constants?

    Differentiate the equation n times (n = number of arbitrary constants), then eliminate the constants among the resulting equations to obtain a DE of order n.

11. What is the difference between the general solution and a particular solution of a differential equation?

    The general solution contains arbitrary constants (the full family of solutions); a particular solution is obtained by assigning specific values to those constants using given initial/boundary conditions.

12. What is the difference between a scalar and a vector quantity? Give one example of each.

    A scalar has only magnitude (e.g. mass, temperature, speed); a vector has both magnitude and direction (e.g. velocity, force, displacement).

13. State the triangle (or parallelogram) law for the addition of two vectors.

    If two vectors are represented as two sides of a triangle taken in order, their sum (resultant) is the third side taken in the opposite order. Vector addition is commutative and associative.

14. What is the effect of scalar multiplication k·⃗a on a vector?

    It scales the magnitude to |k||⃗a|; the direction is unchanged if k > 0 and reversed if k < 0. If k = 0 the result is the zero vector.

15. Define the dot (scalar) product of two vectors and state when they are perpendicular.

    ⃗a · ⃗b = |⃗a||⃗b|cosθ = a₁b₁ + a₂b₂ + a₃b₃, a scalar. The vectors are perpendicular when ⃗a · ⃗b = 0.

16. Define the cross (vector) product, including its magnitude and direction.

    ⃗a × ⃗b has magnitude |⃗a||⃗b|sinθ and direction perpendicular to both (right-hand rule). It is a vector; ⃗a × ⃗b = 0 when the vectors are parallel. |⃗a × ⃗b| equals the area of the parallelogram formed by them.

17. Give two geometric applications of the dot and cross products.

    Dot product: projection of one vector on another and the angle between vectors (cosθ = ⃗a·⃗b/(|⃗a||⃗b|)). Cross product: area of a triangle (½|⃗a×⃗b|) and a normal direction to a plane.

18. Name the three measures of central tendency and define the mode.

    Mean, median, and mode. The mode is the value that occurs most frequently in a data set; the median is the middle value of ordered data; the mean is the arithmetic average.

19. Name the common measures of dispersion and give the formula for standard deviation.

    Range, mean deviation, variance, and standard deviation. Standard deviation σ = √[Σ(xᵢ − x̄)² / n]; variance = σ².

20. What does the correlation coefficient r measure, and what is its range?

    r measures the strength and direction of a linear relationship between two variables. It ranges from −1 to +1: +1 perfect positive, −1 perfect negative, 0 no linear correlation.

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

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

22. What is a random variable, and how do discrete and continuous random variables differ?

    A random variable assigns a numerical value to each outcome of a random experiment. A discrete random variable takes countable distinct values (e.g. number of heads); a continuous one takes any value in an interval (e.g. height).

23. State the binomial distribution probability formula and its mean and variance.

    P(X = r) = ⁿCr · pʳ · q^{n−r}, where q = 1 − p, for n independent trials. Mean = np; variance = npq.

24. State the key properties of the normal distribution curve.

    It is symmetric and bell-shaped about its mean, with mean = median = mode. The total area under the curve is 1, and about 68%, 95%, and 99.7% of data lie within 1, 2, and 3 standard deviations of the mean (empirical rule).

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