Free Compass Math Test Practice Questions
Solutions with Explanations - Sample 3

Solutions with detailed explanations to compass math test practice questions in sample 3.

  1. If number x is increased by one 1/4 of itself, which of the following expressions represents the new number?

    A) x + 1/4
    B) x + 0.25
    C) x + 0.25x
    D) 4 x - 0.25
    E) x/4

    Solution

    x is the number and one quarter of itself is

    (1/4) x

    x increased by a quarter of itself is written as follws

    x + (1/4) x = x + 0.25 x , since 1/4 = 0.25

    Answer C

  2. (5 + 1/5)x = 6, find x.

    Solution

    Given equation has denominator equal to 5, we therefore need to multiply all terms of the equation by 5 in order to eliminate the denominator

    5 [ (5 + 1/5)x ] = 5 [ 6 ]

    Simplify and group like terms

    25 x + x = 30

    x = 30 / 26 = (26 + 4) / 26

    = 1 + 4/26 = 1 + 2/13

    = 1(2/3) mixed number

  3. If f(x) = 3x + 2, then f(2a + b) =

    Solution

    Substitute x in f(x) by 2a + b.

    f(2a + b) = 3(2a + b) + 2 = 6a + 3b + 2

  4. If f(x) = 3x + 2 and g(x) = x^2 – 2, then f(2) – g(3) =

    Solution

    Substitute x by 2 in f(x) to find f(2) as follows

    f(2) = 3(2) + 2 = 6 + 2 = 8

    Substitute x by 3 in g(x) to find g(3) as follows

    g(3) = (3)2 - 2 = 7

    Finally

    f(2) – g(3) = 8 - 7 = 1

  5. For all x, (x + 3)(-x + 3)=

    Solution

    Expand the given expression as follows

    (x + 3)(-x + 3) = - x 2 + 3x - 3x + 9

    = 9 - x 2

  6. What is the tenth term of the geometric series if the first, second and third terms are 0.5, 1.0, 2.0, ... ?

    Solution

    The n th term of a geometric series is written as

    an = a0 r n-1 , where a0 is the first term of the series and r is the common ratio

    r is found by dividing two successive terms of the series

    r = second term of series / first term of series = 1.0 / 0.5 = 2 or r = 2.0 / 1.0 = 2

    Hence

    a10 = 0.5 * 2 10-1 = 256

  7. If (2x)(2-4x) = 1/8, then x = ?

    Solution

    Rearrange the right and hand side terms so that the exponential expressions have the same base

    (2x)(2-4x) = 1/(23)

    2x - 4x = 2 -3

    now that the bases of the two sides are equal, their exponent must also be equal

    - 3x = - 3 , x = 1

  8. The solution of the equation 2x - 3 = 5x – 2 falls between two consecutive integers

    A) 0 and 1
    B) -1 and 0
    C) -2 and -1
    D) 1 and 2
    E) -3 and -2

    Solution

    Solve the given equation

    2x - 3 = 5x – 2

    2x - 5x = - 2 + 3

    - 3x = 1

    x = - 1/3

    -1/3 is between -1 and 0. Answer B

  9. What is the tenth term of the geometric sequence 1, -1/2, 1/4, -1/8 ...

    Solution

    The n th term of a geometric series is written as

    an = a0 r n-1 , where a0 is the first term of the series and r is the common ratio

    r is found by dividing two successive terms of the series

    r = second term of series / first term of series = (-1/2) / 1 = -1/2

    Hence

    a10 = 1 * (-1/2)10-1 = -1/512

  10. If f(x) = 3x3 + 2x2 + 3 and g(x) = x2 + 2, then f(x)/g(x) = ?

    Solution

    Use long division of polynomials to divide f(x) by g(x)

      3x + 2 (quotient)

    (divisor) x 2 + 2

    		 3x 3 + 2x 2 + 3     (dividend)

    3x 3 + 6x

      2x 2 - 6x + 3

    2x 2 + 4
     

    - 6x - 1 (remainder)


    and the fact that

    Dividend / divisor = quotient + remainder / divisor

    to write that

    f(x)/g(x) = (3x3 + 2x2 + 3) / (x2 + 2)

    = 3x + 2 - (6x + 1) / (x2 + 2)

  11. If i is the imaginary unit such that sqrt(-1) = i, then (7i)2 = ?

    Solution

    (7i)2 = 72(i)2

    = 49*(-1) = -49

  12. Which of these is a complete factorization of f(x) = 3x3 + x2 + (3x + 1)(2x - 3)?

    Solution

    Factor the term 3x3 + x2 as follows

    3x3 + x2 = x2(3x + 1)

    Substitute in f(x) and write

    f(x) = x2(3x + 1) + (3x + 1)(2x - 3)

    = (3x + 1)(x2 + 2x - 3) , factor 3x + 1 out

    We now factor the quadratic term x2 + 2x - 3 to complete factoring.

    f(x) = (3x + 1)(x - 1)(x + 3)

  13. Find k if 8! = 6!k.

    Solution

    Use the fact that

    8! = 8*7*6!

    to rewrite the equation as follows

    8*7*6! = 6! k

    Solve for k

    k = 56

  14. Find f(-2) if f(x) = 2x2 + kx + 2 and f(1) = 3.

    Solution

    We first use f(1) = 3 to find k

    f(1) = 2(1)2 + k(1) + 2 = 3

    Solve above equation for k

    k = -1

    Substitute k by - 1 in f(x) and write

    f(x) = 2x2 - x + 2

    We now evaluate f(-2)

    f(-2) = 2(-2)2 - (-2) + 2 = 12

  15. In which interval does f(x) = -2x2 - 3x - 4 have a maximum value?

    A) (0 , 1)
    B) (-3 , -2)
    C) (-0.5 , 0)
    D) (-2 , -1)
    E) (-1 , 0)

    Solution

    f is a quadratic function with leading coefficient -2 (negative) and therefore has a maximum at the vertex with coordinates (h , k)

    h = - b / 2a = - (-3) / - 4 = - 3 / 4 = - 0.75

    which is in the interval (-1 , 0), answer E.

  16. Simplify x3/4 x1/3 x-2/3.



    Solution

    Apply product rule of exponential expressions: x m x n = x m + n.

    x3/4 x1/3 x-2/3 = x 3/4 + 1/3 - 2/3

    = x 9/12 + 4/12 - 8/12 = x 5/12

  17. Find x if (2/7)2x = (7/2)3x + 5.

    Solution

    Use the exponential rule: (a / b) n = (b / a) -n to rewrite the given equation as follows

    (2/7)2x = (2/7)- (3x + 5)

    The right and left sides are exponential expressions with the same base, hence

    2x = - (3x + 5)

    Solve for x

    x = - 1

  18. 2x2 + 3x - 5 is the product of (2x + 5) and another factor. What is the other factor?

    Solution

    Factor 2x2 + 3x - 5 taking into account that one factor is 2x + 5. Hence

    2x2 + 3x - 5 = (2x + 5)(x - 1)

    Hence, the other factor is x - 1.

  19. If the operator ** is defined by x**y = 2xy + x + y. What is 2**3?

    Solution

    Evaluate x**y for x = 2 and y = 3

    2**3 = 2(2)(3) + (2) + (3) = 17

  20. If i = sqrt(-1), then i 2 + i 3 + i 4 + i 5 =

    Solution

    Simplify each term in the above expression.

    i2 = -1

    i3 = i2 i = - i

    i4 = i3 i = (- i)(i) = 1

    i5 = i4 i = i

    We now find the sum

    i 2 + i 3 + i 4 + i 5 = -1 + (-i) + 1 + i = 0

  21. For all x, (2x - 4)2 =

    Solution

    Use identity (formula) or expand as follows.

    (2x - 4)2 = (2x - 4)(2x - 4)

    = 4x2 - 8x - 8x + 16 = 4x2 - 16 x + 16

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