COMEDK UGET Mathematics - Algebra and Trigonometry Flashcards

24 sample cards from the Mathematics - Algebra and Trigonometry deck

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

1. State the relationship between ⁿPᵣ and ⁿCᵣ.

   ⁿPᵣ = ⁿCᵣ × r! (permutations equal combinations times the arrangements of r items).

2. What is the number of permutations of n objects where p are alike of one kind and q alike of another?

   n! / (p! q!).

3. State the Binomial Theorem for (a + b)ⁿ.

   (a + b)ⁿ = Σ_{r=0}^{n} ⁿCᵣ a^{n−r} b^r.

4. What is the general (r+1)th term in the expansion of (a + b)ⁿ?

   T_{r+1} = ⁿCᵣ a^{n−r} b^r.

5. How many terms are there in the expansion of (a + b)ⁿ?

   n + 1 terms.

6. What is the nth term of an arithmetic progression (AP)?

   aₙ = a + (n − 1)d, where a is the first term and d the common difference.

7. What is the sum of the first n terms of an AP?

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

8. What is the nth term and the sum of n terms of a geometric progression (GP)?

   nth term: aₙ = arⁿ⁻¹; Sum Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1.

9. What is the sum of an infinite GP when |r| < 1?

   S∞ = a / (1 − r).

10. What is the arithmetic mean (AM) and geometric mean (GM) of two numbers a and b?

    AM = (a + b)/2; GM = √(ab). Also AM ≥ GM.

11. What is the condition for two matrices to be conformable for multiplication?

    The number of columns of the first matrix must equal the number of rows of the second.

12. Define a symmetric and a skew-symmetric matrix.

    Symmetric: A' = A (a_ij = a_ji); Skew-symmetric: A' = −A (a_ij = −a_ji, so diagonal entries are 0).

13. What is an identity matrix and a scalar matrix?

    Identity matrix: diagonal matrix with all diagonal entries 1. Scalar matrix: diagonal matrix with all diagonal entries equal.

14. How do you compute the determinant of a 2×2 matrix [[a,b],[c,d]]?

    det = ad − bc.

15. What is the relationship det(AB) and det(A)·det(B)?

    det(AB) = det(A) × det(B).

16. What is the formula for the inverse of a matrix A using its adjoint?

    A⁻¹ = (1/det A) · adj(A), provided det A ≠ 0.

17. When does a system of linear equations have a unique solution (Cramer's rule / matrix method)?

    When the coefficient determinant Δ ≠ 0 (matrix is non-singular/invertible).

18. In Cramer's rule for two equations, how is x found?

    x = Δ₁/Δ, where Δ is the coefficient determinant and Δ₁ replaces the x-column with the constants.

19. What is the formula for the area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) using a determinant?

    Area = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|.

20. State the basic Pythagorean trigonometric identities.

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

21. State the sum formulas for sin(A+B) and cos(A+B).

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

22. What is the general solution of sin θ = sin α?

    θ = nπ + (−1)ⁿ α, where n ∈ Z.

23. What are the principal value ranges of sin⁻¹x and cos⁻¹x?

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

24. State the Sine Rule and the Cosine Rule for a triangle.

    Sine Rule: a/sinA = b/sinB = c/sinC = 2R. Cosine Rule: a² = b² + c² − 2bc·cosA.

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Mathematics - Algebra and Trigonometry flashcards FAQ