NDA (National Defence Academy) Exam Mathematics (Paper I) Flashcards

24 sample cards from the Mathematics (Paper I) deck

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

1. State the three Pythagorean trigonometric identities.

   sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

2. State the compound angle formulas for sin(A±B) and cos(A±B).

   sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB.

3. State the general solutions of sinθ = 0, cosθ = 0, and tanθ = 0.

   sinθ=0 → θ = nπ; cosθ=0 → θ = (2n+1)π/2; tanθ=0 → θ = nπ, where n is any integer.

4. State the general solution of sinθ = sinα and cosθ = cosα.

   sinθ = sinα → θ = nπ + (−1)ⁿα; cosθ = cosα → θ = 2nπ ± α, where n is any integer.

5. State the principal value ranges of sin⁻¹x, cos⁻¹x, and tan⁻¹x.

   sin⁻¹x ∈ [−π/2, π/2]; cos⁻¹x ∈ [0, π]; tan⁻¹x ∈ (−π/2, π/2).

6. State the sine rule and cosine rule for a triangle.

   Sine rule: a/sinA = b/sinB = c/sinC = 2R. Cosine rule: a² = b² + c² − 2bc·cosA (and cyclic).

7. Give two formulas for the area of a triangle in terms of its sides/angles.

   Area = ½·ab·sinC (two sides and included angle); and Heron's formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2.

8. State the distance formula and section formula in 2D coordinate geometry.

   Distance between (x₁,y₁) and (x₂,y₂) = √[(x₂−x₁)² + (y₂−y₁)²]. Section formula (ratio m:n internally): ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

9. State the various forms of the equation of a straight line: slope-intercept, point-slope, and the angle between two lines.

   Slope-intercept: y = mx + c; point-slope: y − y₁ = m(x − x₁). Angle θ between lines of slopes m₁,m₂: tanθ = |(m₁−m₂)/(1+m₁m₂)|; parallel if m₁=m₂, perpendicular if m₁m₂=−1.

10. Give the standard equations and eccentricities of the parabola, ellipse, and hyperbola.

    Parabola: y²=4ax, e=1. Ellipse: x²/a²+y²/b²=1, e<1 (e=√(1−b²/a²)). Hyperbola: x²/a²−y²/b²=1, e>1 (e=√(1+b²/a²)). Circle is a special ellipse with e=0.

11. State the distance formula between two points in three-dimensional space.

    Distance between (x₁,y₁,z₁) and (x₂,y₂,z₂) = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].

12. What are direction cosines of a line, and what relation do they satisfy?

    Direction cosines (l,m,n) are the cosines of the angles the line makes with the x, y, z axes; they satisfy l² + m² + n² = 1.

13. State the equation of a plane in normal form and the angle between a line and a plane.

    Plane: lx + my + nz = p (l,m,n direction cosines of normal, p = perpendicular distance from origin). The angle φ between a line and a plane satisfies sinφ = |line direction · plane normal| / (|d||n|).

14. What are the necessary conditions for a function f(x) to be continuous at x = a?

    f(a) is defined, lim_{x→a} f(x) exists (left limit = right limit), and lim_{x→a} f(x) = f(a).

15. State the standard limit results: lim_{x→0} (sin x)/x and lim_{x→0} (e^x − 1)/x.

    lim_{x→0} (sin x)/x = 1 and lim_{x→0} (e^x − 1)/x = 1 (with x in radians).

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

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

17. State the chain rule and give d/dx of sin x, ln x, and aˣ.

    Chain rule: dy/dx = (dy/du)(du/dx). d/dx(sin x)=cos x; d/dx(ln x)=1/x; d/dx(aˣ)=aˣ ln a.

18. How are increasing/decreasing functions and points of local maxima/minima determined using derivatives?

    f'(x)>0 → increasing, f'(x)<0 → decreasing. At a critical point (f'(x)=0): if f''(x)<0 it is a local maximum, if f''(x)>0 it is a local minimum (second-derivative test).

19. Give the basic integration results for ∫xⁿ dx, ∫(1/x) dx, and ∫eˣ dx.

    ∫xⁿ dx = x^{n+1}/(n+1) + C (n≠−1); ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C.

20. State the Fundamental Theorem of Calculus for definite integrals.

    If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

21. How do you find the order and degree of a differential equation, and how is a variable-separable equation solved?

    Order = highest order derivative present; degree = power of the highest-order derivative (when free of radicals/fractions). Variable-separable: write as f(y)dy = g(x)dx, then integrate both sides.

22. State the scalar (dot) product and vector (cross) product of two vectors in terms of the angle between them.

    a·b = |a||b|cosθ (scalar; zero if perpendicular). a×b = |a||b|sinθ·n̂ (vector perpendicular to both; magnitude = area of parallelogram; zero if parallel).

23. In statistics, define mean, median, and mode, and state the empirical relation between them.

    Mean = sum/number of observations; median = middle value when ordered; mode = most frequent value. Empirical relation: Mode = 3·Median − 2·Mean.

24. State the addition theorem of probability and the formula for conditional probability.

    P(A∪B) = P(A)+P(B) − P(A∩B) (for mutually exclusive events, = P(A)+P(B)). Conditional: P(A|B) = P(A∩B)/P(B), P(B)≠0; for independent events P(A∩B)=P(A)·P(B).

What this deck covers

Mathematics (Paper I) flashcards FAQ