ESAT ESAT Further Mathematics 1 — Questions and Answers
Question 1: What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
- 3/2 (Correct answer)
- 2
- 4/3
- 3
Correct 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.
Question 2: Which is the correct binomial expansion of (1+x)^(-2) for |x| < 1, up to the x² term?
- 1 - 2x + 3x² (Correct answer)
- 1 + 2x + 3x²
- 1 - 2x - 3x²
- 1 - x + x²
Correct 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.
Question 3: Find the value of d²y/dx² at x = 0 for y = sin(3x).
- 0 (Correct answer)
- -9
- 9
- 3
Correct answer: 0
dy/dx = 3cos(3x) and d²y/dx² = -9sin(3x); at x=0, sin(0) = 0, so d²y/dx² = 0.
Question 4: What is the determinant of the matrix with rows [2, 3] and [1, 4]?
- 5 (Correct answer)
- 8
- 11
- 2
Correct answer: 5
det = (2)(4) - (3)(1) = 8 - 3 = 5.
Question 5: If z = 3 + 4i, what is |z|, the modulus of z?
- 5 (Correct answer)
- 7
- 25
- √7
Correct answer: 5
|z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
Question 6: Evaluate the definite integral ∫₀^π sin(x) dx.
- 2 (Correct answer)
- 0
- 1
- π
Correct answer: 2
[-cos(x)]₀^π = -cos(π) + cos(0) = 1 + 1 = 2.
Question 7: What are the eigenvalues of the upper triangular matrix with rows [3, 1] and [0, 2]?
- 3 and 2 (Correct answer)
- 1 and 0
- 5 and 1
- 2 only (repeated)
Correct answer: 3 and 2
For any triangular matrix, the eigenvalues are exactly the diagonal entries: λ₁ = 3 and λ₂ = 2.
ESAT ESAT Further Mathematics 1
What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?