ESAT Further Mathematics Flashcards

What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?

Answer: 3/2

For an infinite geometric series with first term a=1 and ratio r=1/3, the sum is a/(1-r) = 1/(2/3) = 3/2.

Which is the correct binomial expansion of (1+x)^(-2) for |x| < 1, up to the x² term?

Answer: 1 - 2x + 3x²

Using the binomial series with n=-2: the coefficient of x is -2 and of x² is (-2)(-3)/2! = 3, giving 1 - 2x + 3x².

Find the value of d²y/dx² at x = 0 for y = sin(3x).

Answer: 0

dy/dx = 3cos(3x) and d²y/dx² = -9sin(3x); at x=0, sin(0) = 0, so d²y/dx² = 0.

What is the determinant of the matrix with rows [2, 3] and [1, 4]?

Answer: 5

det = (2)(4) - (3)(1) = 8 - 3 = 5.

If z = 3 + 4i, what is |z|, the modulus of z?

Answer: 5

|z| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Evaluate the definite integral ∫₀^π sin(x) dx.

Answer: 2

[-cos(x)]₀^π = -cos(π) + cos(0) = 1 + 1 = 2.

What are the eigenvalues of the upper triangular matrix with rows [3, 1] and [0, 2]?

Answer: 3 and 2

For any triangular matrix, the eigenvalues are exactly the diagonal entries: λ₁ = 3 and λ₂ = 2.