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CAT Quant

CAT algebra practice questions

Algebra in CAT is less about long manipulation and more about spotting a shortcut: an identity that turns a messy expression into a simple one, the discriminant that tells you about the roots without solving, or a substitution that removes a variable. Work through the questions below and notice which identity or property makes each one quick.

Key ideas

  • (a + b)^2 = a^2 + b^2 + 2ab, so a^2 + b^2 = (a + b)^2 - 2ab.
  • For ax^2 + bx + c = 0, the sum of roots is -b/a and the product is c/a.
  • Equal roots need the discriminant b^2 - 4ac = 0.
  • |x - a| = k gives x = a + k or x = a - k.
  • For inequalities, flip the sign when multiplying or dividing by a negative number.

Common traps

  • Dropping a root when dividing by a variable.
  • Forgetting both cases for a modulus equation.
  • Counting integer solutions without checking endpoints.

Practice questions with solutions

Try each question first, then open the solution. Aim for about 90 seconds per question.

  1. If x + y = 10 and xy = 21, what is the value of x^2 + y^2?

    • A 58
    • B 79
    • C 42
    • D 100
    Show answer and solution

    Answer: A. 58

    x^2 + y^2 = (x + y)^2 - 2xy = 100 - 42 = 58.

  2. If 2x + 3y = 12 and 3x - y = 7, what is x + y?

    • A 4
    • B 6
    • C 7
    • D 5
    Show answer and solution

    Answer: D. 5

    From the second equation y = 3x - 7. Then 2x + 9x - 21 = 12, so x = 3 and y = 2. x + y = 5.

  3. For what value of k does kx^2 - 12x + 9 = 0 have equal roots?

    • A 3
    • B 4
    • C 9
    • D 1
    Show answer and solution

    Answer: B. 4

    Discriminant = 144 - 36k = 0, so k = 4.

  4. What is the sum of all values of x that satisfy |2x - 3| = 7?

    • A 5
    • B -2
    • C 7
    • D 3
    Show answer and solution

    Answer: D. 3

    2x - 3 = 7 gives x = 5. 2x - 3 = -7 gives x = -2. The sum is 3.

  5. If f(x) = 3x^2 - 2x + 1, what is f(2)?

    • A 7
    • B 11
    • C 9
    • D 13
    Show answer and solution

    Answer: C. 9

    3 x 4 - 4 + 1 = 9.

  6. How many integers x satisfy |x - 2| < 4?

    • A 6
    • B 8
    • C 9
    • D 7
    Show answer and solution

    Answer: D. 7

    The inequality gives -2 < x < 6, so x = -1, 0, 1, 2, 3, 4, 5, which is 7 integers.

  7. How many integers x satisfy x^2 - 7x + 10 < 0?

    Type in the answer (no options).

    Show answer and solution

    Answer: 2

    Factorise as (x - 2)(x - 5) < 0, so 2 < x < 5. The integers are 3 and 4.

  8. If a/b = 3/4, what is the value of (a + b)/(b - a)?

    Type in the answer (no options).

    Show answer and solution

    Answer: 7

    Take a = 3 and b = 4. Then (3 + 4)/(4 - 3) = 7.

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