This page provides clear, step-by-step solutions to the GMAT-style questions presented in Sample Test 1. Each explanation emphasizes logical reasoning and efficient problem-solving techniques.
For any triangle, the length of the third side must be:
\[ |a - b| < c < a + b \]Here, the given sides are \(3\) and \(5\).
\[ \text{Sum } = 3 + 5 = 8, \quad \text{Difference } = 5 - 3 = 2 \]Thus, the third side must satisfy:
\[ 2 < AC < 8 \]None of the proposed values satisfy this condition.
Correct answer: E
Given:
\[ \sqrt{x} = 3 \]Square both sides:
\[ x = 9 \]Raise both sides to the fourth power:
\[ x^4 = 9^4 = 6561 \]Correct answer: A
Since \(n\) is an odd integer, it can be written as:
\[ n = 2k + 1 \quad (k \in \mathbb{Z}) \]Then:
\[ n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 \]So \(n^2\) is odd.
Now:
\[ n^2 + 1 = 2(2k^2 + 2k + 1) \]which is even.
Finally:
\[ 3n^2 - 1 = 3[2(2k^2 + 2k) + 1] - 1 = 6(2k^2 + 2k) + 2 \]which is also even.
Correct answer: E
Rewrite constants as powers of 2:
\[ 8 = 2^3, \quad 4 = 2^2 \]Then:
\[ 8 \cdot 2^{100} + 4 \cdot 2^{101} = 2^{103} + 2^{103} = 2 \cdot 2^{103} = 2^{104} \]Correct answer: D
Add the two equations:
\[ (3x + 5y) + (x + 3y) = 5 + 20 \] \[ 4x + 8y = 25 \]Divide both sides by 2:
\[ 2x + 4y = \frac{25}{2} \]Correct answer: B
The condition \(0 < n < 1\) implies:
\[ 0 < n^2 < n \]Thus:
\[ n^2 - n < 0 \]Statement (I) is true.
Multiplying again by \(n\):
\[ 0 < n^3 < n^2 < n \]Statement (II) is also true.
Statement (III) is false since \(n + 1 > 1\).
Correct answer: D
Convert all expressions to decimals:
The greatest value is \(10\).
Correct answer: D
Factor numerator and denominator:
\[ 4x^2 - 4 = 4(x - 1)(x + 1) \] \[ -3x + 3 = -3(x - 1) \]Simplify:
\[ \frac{4(x - 1)(x + 1)}{-3(x - 1)} = -\frac{4}{3}(x + 1) \]Correct answer: A
Substitute the solutions into \(x^2 + bx = c\):
\[ 2^2 + 2b = c \Rightarrow 4 + 2b = c \] \[ (-3)^2 - 3b = c \Rightarrow 9 - 3b = c \]Equate and solve:
\[ 4 + 2b = 9 - 3b \Rightarrow 5b = 5 \Rightarrow b = 1 \]Then:
\[ c = 4 + 2(1) = 6 \]Correct answer: D
Correct answer: B