Question 1 :
If ⁽²ˣ ⁻ ³⁾⁄₂ = k - 1 and k = 5, what is the value of 2x?
(A) 4
(B) 5.5
(C) 8
(D) 11
Answer :
⁽²ˣ ⁻ ³⁾⁄₂ = k - 1
Substitute k = 5.
⁽²ˣ ⁻ ³⁾⁄₂ = 5 - 1
⁽²ˣ ⁻ ³⁾⁄₂ = 4
Multiply both sides by 2.
2x - 3 = 8
Add 3 to both sides.
2x = 11
Therefore, the correct answer choice is (D).
Question 2 :
(5 + 3i) - (8 - 2i) = a + bi
In the equation above, a and b are real numbers. If i = √-1, what is the value of b?
(A) -1
(B) 1
(C) -5
(D) 5
Answer :
(5 + 3i) - (8 - 2i) = a + bi
5 + 3i - 8 + 2i = a + bi
5 + 3i - 8 + 2i = a + bi
-3 + 5i = a + bi
Comparing the coefficients of like terms,
b = 5
Therefore, the correct answer choice is (D).
Question 3 :
If Claire paid k dollars for a computer that was only 20 dollars more than half the original price, what was the original price, in dollar?
(A) k + 20
(B) k - 40
(C) 2k - 20
(D) 2k - 40
Answer :
Let x be the original price of the computer.
(½)x + 20 = k
Subtract 20 from both sides.
(½)x = k - 20
Multiply both sides by 2.
x = 2(k - 20)
x = 2k - 40
Therefore, the correct answer choice is (D).
Question 4 :
Jenny is on the school swim team and has swim practice m hours in the morning and p hours in the evening each day. The schedule is the same each day. If she swims k hours for five days, which of the following expression for m?
(A) ⁽ᵏ ⁻ ᵖ⁾⁄₅
(B) ⁽ᵏ ⁻ ⁵ᵖ⁾⁄₅
(C) k - 5p
(D) 5(k - p)
Answer :
5(m + p) = k
5m + 5p = k
Subtract 5p from both sides.
5m = k - 5p
Divide both sides by 5.
m = ⁽ᵏ ⁻ ⁵ᵖ⁾⁄₅
Therefore, the correct answer choice is (B).
Question 5 :
A certain business is marketing its product and has determined that, when it raised the selling price of its product, its sales went down. The number of units sold, P, is modeled by the equation P = 1200 - 20s, where s is the selling price, in dollars. Based on this model, what is the decrease in selling price from 700 units sold to 900 units?
(A) 5
(B) 10
(C) 15
(D) 20
Answer :
P = 1200 - 20s ----(1)
Substitute P = 700 into (1) and solve for s.
700 = 1200 - 20s
20s = 500
s = 25
Substitute P = 900 into (1) and solve for s.
900 = 1200 - 20s
20s = 300
s = 15
Decrease in selling price :
= 25 - 15
= 10
Therefore, the correct answer choice is (B).
Question 6 :
(x^{2} + y^{2})^{2} - (x^{2} - y^{2})^{2}
Which of the following is equivalent to the expression above?
(A) x^{4} - y^{4}
(B) 2(x^{2} + y^{2})
(C) 2x^{2}y^{2}
(D) 4x^{2}y^{2}
Answer :
Consider the following algebraic identity.
a^{2} - b^{2 }= (a + b)(a - b)
If we consider (x^{2} + y^{2}) for a and (x^{2} - y^{2}) for b, we have
= (x^{2} + y^{2})^{2} - (x^{2} - y^{2})^{2}
= [(x^{2} + y^{2}) + (x^{2} - y^{2})][(x^{2} + y^{2}) - (x^{2} - y^{2})]
= [x^{2} + y^{2} + x^{2} - y^{2}][x^{2} + y^{2} - x^{2} + y^{2}]
= (2x^{2})(2y^{2})
= 4x^{2}y^{2}
Therefore, the correct answer choice is (D).
Question 7 :
Kimberly earns k dollars per week.At this rate how maany weeks will it take her to earn p dollars?
(A) ᵖ⁄k
(B) ᵏ⁄p
(C) kp
(D) ¹⁰ᵖ⁄k
Answer :
k dollars ----> 1 week
1 dollar ----> ¹⁄k weeks
p(1 dollar) ----> p(¹⁄k weeks)
p dollars ----> ᵖ⁄k weeks
Therefore, the correct answer choice is (A).
Question 8 :
If ²ᵃ⁄b = 5, what is the value of ⁵^{b}⁄ₐ?
(A) 2
(B) 4
(C) 10
(D) 12.5
Answer :
²ᵃ⁄b = 5
2(ᵃ⁄b) = 5
Divide both sides by 2.
ᵃ⁄b = ⁵⁄₂
Take reciprocal on both sides.
^{b}⁄ₐ = ⅖
Multiply both sides by 5.
5(^{b}⁄ₐ) = 5(⅖)
⁵^{b}⁄ₐ = 2
Therefore, the correct answer choice is (A).
Question 9 :
2x + by = 10
ax + 4y = 15
In the system of equations above, a and b are constants and a = 2b. If the system has no solution, which of the following could be a possible value of a?
(A) -2
(B) ½
(C) 4
(D) 8
Answer :
Write the given system of equations in slope-intercept form.
2x + by = 10 by = -2x + 10 y = (-²⁄b)x + ¹⁰⁄b |
ax + 4y = 15 4y = -ax + 15 y = (-ᵃ⁄₄)x + ¹⁵⁄₂ |
Since the system has no solution, the lines must be parallel.
If the lines are parallel, then the slopes must be equal.
-²⁄b = -ᵃ⁄₄
-8 = -ab
ab = 8
Since a = 2b, we have b = .
a(ᵃ⁄₂) = 8
a^{2} = 16
Taking square root on both sides,
a = ±4
a = 4 or -4
Therefore, the correct answer choice is (C).
Question 10 :
f(x) = ax^{2} - 15
In the function f(x) above, a is a constant and f(3) = 10. Which of the following is equal to the value of f(5)?
(A) f(0)
(B) f(3)
(C) f(-3)
(D) f(-5)
Answer :
f(3) = 10
a(3^{2}) - 15 = 10
9a - 15 = 10
9a = 25
a = ²⁵⁄₉
f(x) = (²⁵⁄₉)x^{2} - 15
f(5) = (²⁵⁄₉)(5^{2}) - 15
f(5) = (²⁵⁄₉)(25) - 15
If x = -5,
f(-5) = (²⁵⁄₉)(-5)^{2} - 15
f(-5) = (²⁵⁄₉)(25) - 15
f(5) = f(-5)
Therefore, the correct answer choice is (D).
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