Question 1 :
Solve for x, y and z in the equation shown below.
(x - 2)^{2} + (y - 3)^{2 }+ (z + 4)^{2 }= 0
Answer :
In the equation (x - 2)^{2} + (y - 3)^{2 }+ (z + 4)^{2 }= 0, on the left side, we have the addition of squares of three binomials.
The square of each binomial can never be a negative value.
The only option to get zero for the addition of the squares of three binomials is that the value of square of each binomial must be zero.
(x - 2)^{2} = 0
x - 2 = 0
x = 2
(y - 3)^{2} = 0
x - 3 = 0
y = 3
(z + 4)^{2} = 0
z + 4 = 0
z = -4
Question 2 :
The equation shown above is true for all values of x ≠ 2/3, where k is a constant. What is the value of k?
Answer :
Multiply each side by (3x - 2).
kx^{2} + 14x - 20 = (3x - 2)(5x + 8) - 4
kx^{2} + 14x - 20 = 15x^{2} + 24x - 10x - 4
kx^{2} + 14x - 20 = 15x^{2} + 14x - 4
On the both sides of the above equation, we have quadratic expressions.
If two expressions are equal, then the coefficients of like terms are also equal.
Equating the coefficients of x^{2},
k = 15
Question 3 :
If |1 - x| > 4 and x is positive, what is one possible value of x?
Answer :
|1 - x| > 4
1 - x > 4 or 1 - x < -4
1 - x > 4 (1 - x) - 1 > 4 - 1 1 - x - 1 > 4 - 1 -x > 3 x < -3 |
1 - x < -4 (1 - x) - 1 < -4 - 1 1 - x - 1 < -5 -x < -5 x > 5 |
Since x is positive, one possible value of x satisfying x > 5 is 6. That is
x = 6
Questions 4-5 refer to the following information.
An ambulance is moving at a velocity of v_{a}, in meters per second, towards an observer standing still on a sidewalk. Because of this movement, the actual frequency of the sound waves emitted by the ambulance's siren f_{s}, in hertz, is perceived by the observer to be a different frequency f_{obs}. The siren's sound waves travel at a velocity v_{w}, in meters per second. The formula above show the relationship between these variables.
Question 4 :
Which of the following expresses the velocity of the ambulance in terms of the other variables?
Answer :
Let
a = v_{a}
b = v_{w}
c = f_{s}
d = f_{obs}
Then, the given formula becomes as shown below.
We have to find the velocity of the ambulance (v_{a}).
So, we have to solve for 'a'.
In the formula above, multiply both sides by (b - a).
(b - a)d = bc
bd - ad = bc
Subtract bd from both sides.
-ad = bc - bd
Multiply both sides by -1.
ad = bd - bc
Divide both sides by d.
Replace a, b and c by the original variables.
The correct answer choice is (D).
Question 5 :
If the velocity of the siren's sound waves is 340 meters per second, the velocity of the ambulance is 22 meters per second, and the observer perceives the frequency of the siren's sound waves to be 500 hertz, which of the following is closest to the actual frequency of the siren's sound waves?
A) 468
B) 496
C) 507
D) 535
Answer :
Substitute v_{w} = 340, v_{a} = 22 and f_{obs} = 500.
The correct answer choice is (A).
Question 6 :
x^{2} + kx + 9 = (x + a)^{2}
In the equation above, k and a are positive constants. If the equation is true for all values of x, what is the value of k?
A) 0
B) 3
C) 6
D) 9
Answer :
x^{2} + kx + 9 = (x + a)^{2}
Expand the right side.
x^{2} + kx + 9 = (x + a)(x + a)
x^{2} + kx + 9 = x^{2} + ax + ax + a^{2}
x^{2} + kx + 9 = x^{2} + 2ax + a^{2}
Equate the coefficients of x and constant terms.
k = 2a |
a^{2} = 9 √a^{2} = √9 a = ±3 |
Since 'a' is a positive constant, a = 3.
Substitute a = 3 in k = 2a.
k = 2(3)
k = 6
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