NIMCET 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 Fundamental Theorem of Calculus for evaluating a definite integral.

   ∫_a^b f(x) dx = F(b) − F(a), where F is an antiderivative of f (F' = f).

2. In applications of derivatives, how do you classify a critical point using the second derivative test?

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

3. What does the sign of the first derivative f'(x) tell us about a function's behavior?

   f'(x) > 0 means the function is increasing; f'(x) < 0 means it is decreasing; f'(x) = 0 indicates a stationary/critical point.

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

   Order = the highest derivative present; degree = the power of that highest-order derivative after the equation is made polynomial (rational/radical-free) in derivatives.

5. State the standard form and integrating factor of a linear first-order differential equation.

   Form: dy/dx + P(x)y = Q(x). Integrating factor IF = e^(∫P dx); solution: y·IF = ∫Q·IF dx + C.

6. What distinguishes a scalar quantity from 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., displacement, velocity, force).

7. Define the dot (scalar) product of two vectors a and b and state its geometric meaning.

   a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃; it is a scalar and equals zero when the vectors are perpendicular.

8. Define the cross (vector) product of a and b, including its magnitude and direction.

   a×b has magnitude |a||b|sinθ and direction perpendicular to both (right-hand rule); a×b = 0 when vectors are parallel; |a×b| equals the area of the parallelogram they form.

9. What is the scalar triple product a·(b×c) and what does its value represent geometrically?

   It is a scalar equal to the determinant of the components; its absolute value is the volume of the parallelepiped formed by a, b, c. It is zero when the vectors are coplanar.

10. State the vector equation of a line passing through point a in the direction of vector b.

    r = a + λb, where λ is a scalar parameter.

11. State the vector equation of a plane through point a with normal vector n.

    (r − a)·n = 0, equivalently r·n = a·n = d.

12. Give the values of sin, cos, and tan for the standard angle 30°, 45°, and 60°.

    sin: 1/2, 1/√2, √3/2; cos: √3/2, 1/√2, 1/2; tan: 1/√3, 1, √3 for 30°, 45°, 60° respectively.

13. State the three Pythagorean trigonometric identities.

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

14. State the sine and cosine addition formulas: sin(A+B) and cos(A+B).

    sin(A+B) = sinA cosB + cosA sinB; cos(A+B) = cosA cosB − sinA sinB.

15. State the double angle formulas for sin 2θ and cos 2θ.

    sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

16. State the principal value ranges (domains of definition for the angle) of sin⁻¹x, cos⁻¹x, and tan⁻¹x.

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

17. State the formula relating slope to the angle a line makes with the x-axis, and the slope of a line through two points.

    Slope m = tanθ (θ is inclination); through (x₁,y₁) and (x₂,y₂): m = (y₂−y₁)/(x₂−x₁).

18. Give the conditions for two lines (slopes m₁, m₂) to be parallel and perpendicular.

    Parallel: m₁ = m₂; perpendicular: m₁·m₂ = −1.

19. State the distance of a point (x₁, y₁) from the line Ax + By + C = 0.

    d = |Ax₁ + By₁ + C| / √(A² + B²).

20. How is the type of conic determined from the eccentricity e?

    Circle: e = 0; ellipse: 0 < e < 1; parabola: e = 1; hyperbola: e > 1.

21. State the standard equation of a circle with centre (h, k) and radius r, and the general second-degree form.

    Standard: (x−h)² + (y−k)² = r². General: x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius √(g²+f²−c).

22. For the parabola y² = 4ax, give the vertex, focus, directrix, and length of latus rectum.

    Vertex (0,0); focus (a,0); directrix x = −a; length of latus rectum = 4a.

23. For the ellipse x²/a² + y²/b² = 1 (a > b), state the foci, eccentricity, and length of major/minor axes.

    Foci (±ae, 0) where b² = a²(1−e²); eccentricity e = √(1 − b²/a²); major axis = 2a, minor axis = 2b.

24. For the hyperbola x²/a² − y²/b² = 1, state the eccentricity, foci, and equations of the asymptotes.

    e = √(1 + b²/a²); foci (±ae, 0); asymptotes y = ±(b/a)x.

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