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Algebra, sequences and functions

Written by Hassan Raza (Mathematics)

ECAT · Mathematics · 9 questions

Name: ____________________   Date: ____________

  1. What are the roots of x² - 5x + 6 = 0?

    • A 2 and 3
    • B 1 and 6
    • C -1 and 6
    • D -2 and -3

  2. What is the sum of the roots of 2x² - 7x + 3 = 0?

    • A 7/2
    • B -7/2
    • C 3/2
    • D -3/2

  3. For which values of k does x² + kx + 9 = 0 have equal roots?

    • A k = 3 or -3
    • B k = 3 only
    • C k = 9 only
    • D k = 6 or -6

  4. What is the 10th term of the arithmetic progression 3, 7, 11, ...?

    • A 39
    • B 43
    • C 40
    • D 36

  5. What is the sum of the first 10 terms of the geometric progression 1, 2, 4, 8, ...?

    • A 2047
    • B 512
    • C 1024
    • D 1023

  6. What is the coefficient of x² in the expansion of (1 + x)⁵?

    • A 5
    • B 15
    • C 10
    • D 20

  7. If f(x) = 2x + 3, then the inverse function f⁻¹(x) is:

    • A 2x - 3
    • B x / 2 - 3
    • C (x - 3) / 2
    • D (x + 3) / 2

  8. What is the domain of f(x) = sqrt(x - 4)?

    • A x > 4
    • B All real numbers
    • C x >= 4
    • D x <= 4

  9. What is the sum to infinity of the series 1 + 1/2 + 1/4 + 1/8 + ...?

    • A 2
    • B 4
    • C It has no sum
    • D 1

Answer key

1. A, 2. A, 3. D, 4. A, 5. D, 6. C, 7. C, 8. C, 9. A

  1. Factorise as (x - 2)(x - 3) = 0, so x = 2 or x = 3.

  2. For ax² + bx + c = 0 the sum of the roots is -b/a = 7/2.

  3. Equal roots need discriminant zero: k² - 36 = 0, so k = 6 or k = -6.

  4. The nth term is a + (n - 1)d = 3 + 9 x 4 = 39.

  5. S = a(r^n - 1) / (r - 1) = (2¹⁰ - 1) / 1 = 1023.

  6. The coefficient is 5C2 = 10.

  7. Let y = 2x + 3. Then x = (y - 3) / 2, so f⁻¹(x) = (x - 3) / 2.

  8. The expression under the square root must be at least zero, so x - 4 >= 0 and x >= 4.

  9. S = a / (1 - r) = 1 / (1 - 1/2) = 2.