ECAT 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. When does a square matrix A have an inverse (is non-singular)?

   A is invertible if and only if det(A) ≠ 0. If det(A) = 0, the matrix is singular and has no inverse.

2. What is the inverse of a 2×2 matrix [[a, b], [c, d]]?

   (1/(ad-bc)) × [[d, -b], [-c, a]], provided ad - bc ≠ 0.

3. State Cramer's Rule for solving a system of linear equations.

   For AX = B, each unknown x_i = det(A_i) / det(A), where A_i is A with its i-th column replaced by B. Requires det(A) ≠ 0.

4. What does it mean if the determinant of the coefficient matrix of a square linear system is zero?

   The system has either no solution (inconsistent) or infinitely many solutions; it does not have a unique solution.

5. What is the quadratic formula for solving ax^2 + bx + c = 0?

   x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a ≠ 0.

6. What is the discriminant of a quadratic equation and its symbol?

   The discriminant is D = b^2 - 4ac. It determines the nature of the roots of ax^2 + bx + c = 0.

7. For ax^2 + bx + c = 0, what is the sum and product of the roots in terms of a, b, c?

   Sum of roots = -b/a; Product of roots = c/a.

8. If the discriminant b^2 - 4ac > 0, what is the nature of the roots?

   The roots are real, distinct (unequal). If D is also a perfect square (with rational coefficients), they are rational; otherwise irrational.

9. If the discriminant b^2 - 4ac = 0, what is the nature of the roots?

   The roots are real and equal (a repeated/double root), each equal to -b/(2a).

10. If the discriminant b^2 - 4ac < 0, what is the nature of the roots?

    The roots are complex (imaginary) and occur as a conjugate pair.

11. How do you form a quadratic equation given that its roots are α and β?

    x^2 - (α + β)x + αβ = 0, i.e. x^2 - (sum of roots)x + (product of roots) = 0.

12. What substitution reduces the equation x^4 - 5x^2 + 4 = 0 to a quadratic?

    Let y = x^2. The equation becomes y^2 - 5y + 4 = 0, solve for y, then find x = ±sqrt(y).

13. When is a rational fraction called 'proper' for partial fraction decomposition?

    When the degree of the numerator is less than the degree of the denominator. Otherwise divide first to get a polynomial plus a proper fraction.

14. What is the partial fraction form for a repeated linear factor (x-a)^2 in the denominator?

    A/(x-a) + B/(x-a)^2 — one term for each power up to the highest.

15. What partial fraction form is used for an irreducible quadratic factor (x^2 + bx + c) in the denominator?

    A linear numerator over the quadratic: (Ax + B)/(x^2 + bx + c).

16. What is the nth term (general term) of an arithmetic sequence with first term a and common difference d?

    a_n = a + (n - 1)d.

17. What is the sum of the first n terms of an arithmetic series?

    S_n = (n/2)[2a + (n-1)d] = (n/2)(a + l), where l is the last term.

18. What is the arithmetic mean (A.M.) between two numbers a and b?

    A.M. = (a + b)/2.

19. What is the nth term of a geometric sequence with first term a and common ratio r?

    a_n = a × r^(n-1).

20. What is the sum to infinity of a geometric series, and when does it converge?

    S_infinity = a/(1 - r), valid only when |r| < 1 (the series converges).

21. What is the relationship between the Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H) of two positive numbers?

    G^2 = A × H, and A ≥ G ≥ H (equality when the numbers are equal).

22. What is the formula for the number of permutations of n distinct objects taken r at a time?

    P(n, r) = n! / (n - r)!

23. What is the formula for combinations of n distinct objects taken r at a time, and how does it relate to permutations?

    C(n, r) = n! / [r!(n - r)!] = P(n, r) / r!. Combinations ignore order; permutations count order.

24. How is the probability of an event E defined for equally likely outcomes, and what is its range?

    P(E) = (number of favorable outcomes) / (total number of possible outcomes). Its value lies between 0 and 1 inclusive, and P(E) + P(not E) = 1.

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