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Solutions with detailed explanations to compass math test practice questions in sample 3.
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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
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(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
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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
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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
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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
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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
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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
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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
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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
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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 |
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2x 2 - 6x + 3 2x 2 + 4 |
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- 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)
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If i is the imaginary unit such that sqrt(-1) = i, then (7i)2 = ?
Solution
(7i)2 = 72(i)2
= 49*(-1) = -49
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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)
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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
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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
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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.
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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
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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
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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.
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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
- 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
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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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