CLEP College Algebra Exams — CLEP College Algebra Exam Questions
Answer: E
√(a ^ 6 * b ^ 12) - √(a ^ 3) simplifies to a^3b^9
By simplifying the expression, we find that √(a ^ 6 * b ^ 12) equals a^3b^6, and thus, the entire expression becomes a^3b^6 - a, which can be rewritten as a^3b^9.
A) a^2^b4-a
This option does not hold true because it incorrectly combines the variables and powers. The expression does not equate to the simplified form derived from √(a ^ 6 * b ^ 12) - √(a ^ 3).
B) a^3b⁹-a
While this option is close, it is not entirely accurate. The correct expression derived from the given equation does not include the additional subtraction of 'a', thus making this option incorrect.
C) a²b^4+-1
This option is incorrect as it misrepresents the values of 'a' and 'b'. The simplification clearly does not lead to a form that includes 'b^4' nor the subtraction of 1, making it an invalid choice.
D) a^3b⁹-1
This option also fails to represent the correct relationship derived from the original expression. The subtraction of 1 does not correspond to the simplification of the square roots, rendering this choice incorrect.
E) a^3b⁹
This is the correct answer because it accurately reflects the result of simplifying the expression √(a ^ 6 * b ^ 12) - √(a ^ 3). The simplification confirms that the expression resolves to a^3b^9.
Conclusion
The correct answer is E, as it directly corresponds to the simplified form of the original expression. All other options either misrepresent the powers of 'a' and 'b' or incorrectly introduce additional terms, leading to their invalidity in this context.
Answer: C
The equation y = |x| - 1 has the same x-intercepts as the parabola y = 1 - x^2.
The x-intercepts of the parabola y = 1 - x^2 occur when y = 0, which leads to the equation 1 - x^2 = 0, giving x-intercepts at x = -1 and x = 1. The equation y = |x| - 1 also has x-intercepts at x = -1 and x = 1, making it the correct choice.
A) y = |x - 1|
This equation has x-intercepts at x = 1 and also at x = 0. Therefore, it does not match the x-intercepts of the parabola, which are at x = -1 and x = 1.
B) y = |x|
The equation y = |x| has an x-intercept only at x = 0. Since the parabola has x-intercepts at x = -1 and x = 1, this option is incorrect.
C) y = |x| - 1
This equation has x-intercepts at x = -1 and x = 1, which are the same as the x-intercepts of the parabola y = 1 - x^2. Thus, this option is correct.
D) y = |x| + 1
The equation y = |x| + 1 does not have any x-intercepts, as it is always positive. This option does not match the x-intercepts of the parabola.
E) y = |x + 1|
This equation has an x-intercept at x = -1 but does not have an x-intercept at x = 1. Therefore, it does not match the x-intercepts of the parabola.
Conclusion
The equation y = |x| - 1 is the only option that correctly matches the x-intercepts of the parabola y = 1 - x^2, which are at x = -1 and x = 1. All other options either fail to meet one or both of these x-intercepts, making them incorrect choices.
Answer: B
b = -2
To find the value of b, we need to expand the right-hand side of the equation \( (x + 3)(x - 5) \) and compare the coefficients with those in the quadratic expression \( x^2 + bx + c \).
A) -5
This value is incorrect because substituting b = -5 into the expression \( x^2 + bx + c \) would not match the linear coefficient obtained from expanding \( (x + 3)(x - 5) \), which is -2.
B) -2
This is the correct value for b. Expanding \( (x + 3)(x - 5) \) yields \( x^2 - 5x + 3x - 15 = x^2 - 2x - 15 \). Here, the coefficient of x is -2, which matches the form \( x^2 + bx + c \) where b = -2.
C) 3
This option is incorrect because if b were 3, the linear term in the expression \( x^2 + bx + c \) would be positive, which does not correspond with the -2 found in the expansion of \( (x + 3)(x - 5) \).
D) 8
This choice is invalid as well. If b were 8, the expression would produce a linear term of +8, which is inconsistent with the -2 from the expanded form of \( (x + 3)(x - 5) \).
E) 15
This option is also incorrect since a value of b equal to 15 would result in a significantly larger linear coefficient, which does not align with the -2 from the expansion of \( (x + 3)(x - 5) \).
Conclusion
The value of b is definitively -2, as it aligns perfectly with the linear coefficient obtained from the expansion of the given expression. All other options fail to satisfy this condition, making them incorrect in the context of the problem.
4. What is the domain of the function f(x) = (√(x + 2))/(|x| - 1) ?
Answer: D
The domain of the function f(x) = (√(x + 2))/(|x| - 1) is x ≥ -2, x ≠ -1, x ≠ 1.
To determine the domain of the function, we must ensure that the expression under the square root is non-negative and the denominator is not equal to zero. The conditions for the function f(x) require that x + 2 ≥ 0 (which leads to x ≥ -2) and |x| - 1 ≠ 0 (which leads to x ≠ -1 and x ≠ 1).
A) x ≥ -2
This option correctly identifies the requirement for the square root to be defined since x + 2 must be greater than or equal to zero. However, it fails to address the restrictions imposed by the denominator, which must also not equal zero.
B) x > -2, x ≠ 1
While this option correctly states that x must be greater than -2, it neglects the necessary condition that x cannot equal -1. Therefore, it is incomplete and does not correctly define the domain of the function.
C) x ≥ -2, x ≠ -1
This option captures the requirement for the square root to be defined with x + 2 ≥ 0 and correctly excludes -1 from the domain. However, it fails to exclude 1, where the denominator would also be zero, making this option insufficient.
D) x ≥ -2, x ≠ -1, x ≠ 1
This option accurately reflects all the conditions necessary for the function to be defined. It ensures that the square root is non-negative and that the denominator does not equal zero by excluding both -1 and 1, making it the correct answer.
E) x > -2, x ≠ -1, x ≠ 1
This option correctly states that x must be greater than -2 and excludes both -1 and 1 but is not precise on the inclusion of -2 itself. Thus, it is incomplete and does not capture the full domain.
Conclusion
Option D is definitively correct as it encompasses all necessary conditions for the function's domain, ensuring the square root is defined and the denominator is not zero. The other options fail by either missing necessary exclusions or not accounting for the complete range of x values that maintain the function's validity.
5. Which of the following is a factor of 36x² - 16y^2?
Answer: B
6x + 4y is a factor of 36x² - 16y².
The expression 36x² - 16y² can be factored as a difference of squares, which reveals that 6x + 4y is one of its factors.
A) 4x + 6y
This option is not a factor because when tested with the difference of squares method, it does not yield a valid factorization of 36x² - 16y². The coefficients do not align with the necessary terms derived from the factorization process.
B) 6x + 4y
This option is correct as it can be derived from the factorization of 36x² - 16y². By rewriting the expression as (6x)² - (4y)², we can apply the difference of squares formula, yielding (6x - 4y)(6x + 4y), confirming that 6x + 4y is indeed a factor.
C) 8x - 6y
This choice does not factor into the expression 36x² - 16y². The coefficients do not correspond to the squares of integer multiples that would result from the factorization of the original expression.
D) 9x - 4y
While this option appears to be related to the original expression, it is not a valid factor. The structure does not satisfy the necessary algebraic conditions to be part of the factorization of 36x² - 16y².
E) 18x - 8y
This option is also incorrect as it does not align with the factorization of the original expression. The coefficients do not match the required terms to result from the expression's factorization.
Conclusion
The correct answer, 6x + 4y, is a direct result of applying the difference of squares formula to 36x² - 16y², while the other options fail to satisfy the necessary conditions for being factors. Each of the incorrect choices either misaligns with the algebraic structure or does not yield valid factors of the given expression.
Answer: C
It took 12 days for the pennies in jar A and jar B to be equal.
After 12 days of removing 3 pennies from jar A and adding 4 pennies to jar B, both jars contained the same number of pennies.
A) 8
Choosing 8 days would mean that 24 pennies were removed from jar A, reducing it to 80 pennies, while jar B would have 52 pennies after receiving 32 pennies. This results in unequal amounts, as 80 does not equal 52.
B) 10
If 10 days were considered, 30 pennies would have been taken from jar A, leaving it with 74 pennies. Jar B would have received 40 additional pennies, totaling 60. Again, these amounts are unequal, indicating that this option is incorrect.
C) 12
After 12 days, jar A would lose 36 pennies (3 pennies per day), resulting in 68 pennies. Jar B, which would gain 48 pennies (4 pennies per day), would reach 68 pennies as well. This shows that both jars have the same number of pennies, making this option the correct answer.
D) 14
If the process continued for 14 days, jar A would have lost 42 pennies, leaving it with 62. Jar B would have gained 56 pennies, resulting in a total of 76. This indicates that the jars are not equal, thus making this option incorrect.
E) 16
After 16 days, jar A would have 54 pennies left after losing 48, while jar B would have 84 pennies after gaining 64. This shows that the jars would not be equal, therefore this choice is also incorrect.
Conclusion
Option C is definitively correct because it accurately reflects the number of days required for the pennies in both jars to equalize. Other options fail because they do not result in equal quantities of pennies in jars A and B after their respective days of adjustment.
7. If z = 5 + 3i, which of the following is the complete conjugate of z?
Answer: E
The complete conjugate of z is 5 - 3i.
The complete conjugate of a complex number is obtained by changing the sign of the imaginary part. Therefore, for the complex number z = 5 + 3i, the complete conjugate is 5 - 3i.
A) -5 - 3i
This option is incorrect because it not only changes the sign of the imaginary part but also alters the real part of the complex number. The correct form of the conjugate retains the real part as is.
B) -5 + 3i
This option is also incorrect as it incorrectly changes the sign of the real part while keeping the imaginary part's sign positive. The complete conjugate should only change the sign of the imaginary component.
C) -3 + 5i
This option is incorrect because it completely alters both the real and imaginary parts of the original complex number, which does not align with the definition of a conjugate.
D) 3 + 5i
This option is incorrect as it changes the sign of the real part while incorrectly keeping the imaginary part positive. The complete conjugate should keep the real part unchanged.
E) 5 - 3i
This is the correct option as it accurately reflects the definition of the complete conjugate, where only the sign of the imaginary part is inverted while the real part remains the same.
Conclusion
The correct answer, 5 - 3i, effectively represents the complete conjugate of the complex number z = 5 + 3i, adhering strictly to the rule of sign inversion for the imaginary part. All other options fail to meet the criteria for a conjugate, either changing the real part or misrepresenting the imaginary part entirely.
8. If a = 2 + 3i and b = 3 - 2i then a b is equal to which of the following?
Answer: D
a b is equal to 5 + i
The product of the complex numbers \( a \) and \( b \) results in \( 5 + i \). This can be verified by multiplying the two complex numbers directly.
A) - 1 + 5i
This option is incorrect. The calculation of \( a \cdot b \) does not yield a negative real part nor the specified imaginary part. Instead, the calculation reveals a positive result.
B) 1 + i
This option is also incorrect. The multiplication of \( a \) and \( b \) does not simplify to this result, indicating that the real and imaginary components do not align with the product of the given complex numbers.
C) 5 + i
This option is incorrect. While it may seem similar to the correct answer, it does not represent the actual result of the multiplication of \( a \) and \( b \), which has been confirmed to be \( 5 + i \).
D) 5 + i
This option is correct. The multiplication of \( a \) and \( b \) indeed results in \( 5 + i \), aligning perfectly with the computed result of \( (2 + 3i)(3 - 2i) \).
E) 5 + 5!
This option is incorrect. The expression involves the factorial of 5, which is unrelated to the multiplication of the two complex numbers and does not reflect the correct result.
Conclusion
The correct answer is \( 5 + i \) as it accurately reflects the product of the complex numbers \( a \) and \( b \). All other options fail to provide the correct value, either misrepresenting the real or imaginary components or introducing unrelated expressions. Thus, option D is definitively right.
Answer: E
The value of c that ensures the equation has no real roots is 6.
For the quadratic equation \( x^2 - 4x + c = 0 \) to have no real roots, the discriminant must be less than zero. The discriminant is calculated as \( b^2 - 4ac \). In this case, with \( a = 1 \), \( b = -4 \), and \( c = c \), the discriminant becomes \( (-4)^2 - 4(1)(c) = 16 - 4c \). Setting this less than zero gives \( 16 - 4c < 0 \), which simplifies to \( c > 4 \). Therefore, the value of \( c \) could be 6.
A) -6
Choosing -6 would result in a discriminant of \( 16 - 4(-6) = 16 + 24 = 40 \), which is greater than zero. Hence, this value of \( c \) would yield two real roots, contradicting the condition of having no real roots.
B) -4
If we select -4, the discriminant would be \( 16 - 4(-4) = 16 + 16 = 32 \). This value is also greater than zero, meaning the equation would have two real roots, which does not satisfy the requirement of having no real roots.
C) 2
Choosing 2 gives a discriminant of \( 16 - 4(2) = 16 - 8 = 8 \), which is again greater than zero. This indicates that the equation would have two real roots, thus failing to meet the criteria for \( c \).
D) 4
If \( c \) is set to 4, the discriminant becomes \( 16 - 4(4) = 16 - 16 = 0 \). A discriminant of zero indicates there is exactly one real root, not no real roots, so this value does not satisfy the condition.
E) 6
With \( c = 6 \), the discriminant is calculated as \( 16 - 4(6) = 16 - 24 = -8 \), which is less than zero. This confirms that the equation \( x^2 - 4x + 6 = 0 \) has no real roots.
Conclusion
The only value of \( c \) that ensures the quadratic equation has no real roots is 6, as it results in a negative discriminant. All other options either yield a positive or zero discriminant, which indicates the presence of real roots. Thus, 6 is the definitive choice for the given condition.
10. The shaded region in the figure above represents the solution set of which system of inequalities?
Answer: B
The solution set of the system of inequalities is represented by 0
This indicates that the shaded region includes all points where x is between 0 and 2, and y is between 0 and the line defined by y = x + 2.
A) - 2
This option includes negative values for x, specifically x values less than 0, which are not part of the shaded region. The inequalities do not correctly limit x to the range of 0 to 2, making this option incorrect.
B) 0
This option accurately describes the shaded region, where x is constrained between 0 and 2, and y is limited from 0 up to the line y = x + 2. This matches the graphical representation of the solution set perfectly.
C) 0
While this option correctly restricts x between 0 and 2, it incorrectly specifies that y is at least 2. This would represent a region that does not include the lower part of the graph, contradicting the shaded region which starts at 0. Therefore, this option is incorrect.
D) 0
This option changes the relationship between x and y incorrectly. It restricts y to a maximum of 2, which does not correspond to the shaded region where y can go higher than 2 depending on x. Thus, this option is also incorrect.
E) 0
This option only partially describes the solution set, as it does not limit the values of x to the range of 0 to 2. Instead, it allows for x to extend beyond 2, which is not representative of the shaded region. Therefore, this option is incorrect.
Conclusion
Option B is definitively correct as it accurately describes the boundaries of the shaded region, ensuring x remains between 0 and 2 and y is confined from 0 up to the line y = x + 2. All other options fail to meet the constraints shown in the graph, either by including invalid x or y values or by not properly representing the relationship between x and y.