Test: Algebra - GMAT MCQ

I. ac=b²
This is not necessarily true. If A & C are non zero integers and B=0 then this is not true.
II. b=0
If a=b=c=1 then, the above is false.
III. ac=1
Similarly from statement 1 a & c could be any infinite value. 
Therefore, all the above statements are not necessarily true.
Correct answer is A. None

Case 1: x + 3 ≥ 0 (when x + 3 is non-negative)
In this case, the equation becomes:
x² + 6x + 9 = x + 3

Rearranging the terms, we have:
x² + 5x + 6 = 0

Factoring the quadratic equation, we get:
(x + 3)(x + 2) = 0

This gives us two possible solutions:
x + 3 = 0 → x = -3
x + 2 = 0 → x = -2

Case 2: x + 3 < 0 (when x + 3 is negative)
In this case, the absolute value changes the sign of x + 3, so the equation becomes:
x² + 6x + 9 = -(x + 3)

Simplifying this equation, we have:
x² + 6x + 9 = -x - 3

Rearranging the terms, we get:
x² + 7x + 12 = 0

Factoring the quadratic equation, we have:
(x + 3)(x + 4) = 0

This gives us two additional solutions:
x + 3 = 0 → x = -3
x + 4 = 0 → x = -4

Now, let's find the sum of all the unique solutions:

Sum = -3 + (-2) + (-4) = -9

Among the given answer choices, the sum of all unique solutions is -9 (B).

Therefore, the answer is B.

To find the possible values of k in the quadratic equation x² − 63x + k = 0 with prime roots, we need to determine the conditions for the roots to be prime numbers.

In a quadratic equation ax² + bx + c = 0, the sum of the roots is given by −b/a and the product of the roots is given by c/a.

In this case, the sum of the roots is −(−63)/1 = 63. For the roots to be prime, both roots must be prime numbers that add up to 63.

Let's check the prime numbers less than or equal to 63:

2 + 61 = 63 (Prime + Prime = 63)
3 + 60 = 63 (Prime + Composite ≠ 63)
5 + 58 = 63 (Prime + Composite ≠ 63)
7 + 56 = 63 (Prime + Composite ≠ 63)
11 + 52 = 63 (Prime + Composite ≠ 63)
...
53 + 10 = 63 (Prime + Composite ≠ 63)
59 + 4 = 63 (Prime + Composite ≠ 63)

From the calculations, we can see that there is only one pair of prime numbers (2 and 61) that adds up to 63.
Therefore, there is only one possible value of k, which is the product of the roots: k = 2 × 61 = 122.
The answer is B: 1.

To find the number of pairs (m, n) of integers that satisfy the equation m + n = mn, we can rearrange the equation as follows:

mn - m - n = 0
mn - m - n + 1 = 1
(m - 1)(n - 1) = 1

For the product of two integers to be 1, either both integers must be 1 or both must be -1. Therefore, we have two cases to consider:

Case 1: (m - 1) = 1 and (n - 1) = 1
This gives us m = 2 and n = 2.

Case 2: (m - 1) = -1 and (n - 1) = -1
This gives us m = 0 and n = 0.

So, there are two pairs (2, 2) and (0, 0) that satisfy the equation m + n = mn.

The answer is B: 2.

We are given that the shoe sizes range from 8 to 17, with each unit increase in size corresponding to a 1/4-inch increase in length.

Let's calculate the difference in length between the largest and smallest sizes:

Length difference = (17 - 8) * (1/4)
Length difference = 9/4 inches

We are also given that the largest size is 20% longer than the smallest size. Let's calculate this increase:

20% of the length of the smallest size = (20/100) * (9/4)
Increase = 9/20 inches

To find the length of the shoe in size 15, we need to add the increase to the length of the smallest size:

Length of size 15 = Length of smallest size + Increase
Length of size 15 = (8 * 1/4) + (9/20)
Length of size 15 = 2 + 9/20
Length of size 15 = (40/20 + 9/20)
Length of size 15 = 49/20

To convert this fraction into a decimal, we divide the numerator by the denominator:

Length of size 15 = 2.45 inches

Among the given answer choices, the closest value to 2.45 inches is 13 inches (E).

Therefore, the answer is E.

Given that the quadratic equation x² + bx + c = 0 has roots 4a and 3a, we can use the sum and product of roots formulas to relate these values to the coefficients:

Sum of roots = -b/a = 4a + 3a = 7a
Product of roots = c/a = (4a)(3a) = 12a²

We know that the sum of roots is equal to -b/a, so we have:
-b/a = 7a

From this equation, we can deduce that b = -7a².

Now, let's find the value of b² + c:
b² + c = (-7a²)² + c
b² + c = 49a⁴ + c

Since we are given that a is an integer, let's substitute some values for a and evaluate the expression 49a⁴ + c:

For a = 1:
49(1)⁴ + c = 49 + c

For a = 2:
49(2)⁴ + c = 784 + c

For a = 3:
49(3)⁴ + c = 6561 + c

From the options, only option C (549) can be expressed as 49a⁴ + c, where a is an integer.

Therefore, the answer is C.

Step 1: Find the midpoint of AB The midpoint of AB can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Midpoint = ((2 + 0)/2, (2 + (-2))/2)
Midpoint = (1, 0)

Step 2: Determine the slope of AB The slope of AB can be calculated using the formula:
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (-2 - 2)/(0 - 2)
Slope (m) = (-4)/(-2)
Slope (m) = 2

Step 3: Find the negative reciprocal of the slope from step 2
The negative reciprocal of 2 is -1/2.

Step 4: Use the midpoint and the negative reciprocal slope to form the equation in slope-intercept form
Using the midpoint (1, 0) and the negative reciprocal slope of -1/2, the equation of the perpendicular bisector is:
y = (-1/2)x + b

To find the value of b, substitute the coordinates of the midpoint (1, 0):
0 = (-1/2)(1) + b
0 = -1/2 + b
b = 1/2

Therefore, the equation of the perpendicular bisector is:
y = -1/2x + 1/2

The correct answer is D.

Let the cost = 100x
i.e. Profit = 5% of 100x = 5x

A's profit = (2/3)*5x
B's profit = (1/2)*(1/3)*5x = (1/6)*5x
C's profit = (1/2)*(1/3)*5x = (1/6)*5x

New Profit = 7% of 100x = 7x
i..e. Now A's profit = (2/3)*7x
Change in A's profit = (2/3)*7x - (2/3)*5x = (4/3)x
Given, (4/3)*x = Rs.400
i.e. x = Rs.300
B's profit = (1/6)*7x = 7*300/6 = $350

x and z are positive constants

xy² = zy⁴
y²(x − zy²) = 0
Case 1: y = 0Case 2: x−zy² = 0
x = zy²
y² = x/z

y can either take a positive or negative value.
Hence, three values of y are possible.

When a quadratic equation is tangent to the x-axis, it has exactly one real root (or zero). In such cases, the discriminant (b² - 4ac) of the quadratic equation is equal to zero.

Let's equate the discriminant to zero:

b² - 4ac = 0

Substituting the values from the given equation f(x) = 3x² - tx + 5:

(-t)² - 4(3)(5) = 0

t² - 60 = 0

t² = 60

t = ±√60

Since we are looking for a positive value of t, t = √60 = √(4*15) = 2√15.

Therefore, the correct answer is A: 2√15.