Detailed step-by-step solutions for each question in Quiz Set 2.
Question: Which of these represents "the sum of \(y\) and seven"?
Solution: "The sum of \(y\) and seven" means we add 7 to \(y\). Therefore, the expression is \(y + 7\).
Correct Answer: (c) \( y + 7 \)
Question: Solve for \( m \): \( m - 4 = 5 \)
Solution: To solve for \(m\), add 4 to both sides of the equation: \( m - 4 + 4 = 5 + 4 \), which simplifies to \( m = 9 \).
Correct Answer: (c) \( 9 \)
Question: Which inequality is the same as "\(x\) is no more than three"?
Solution: "No more than three" means \(x\) can be three or less than three. This is represented by the inequality \( x \leq 3 \).
Correct Answer: (c) \( x \leq 3 \)
Question: Evaluate: \( 6 + (-9) \)
Solution: Adding a negative number is the same as subtraction. \( 6 + (-9) = 6 - 9 = -3 \).
Correct Answer: (b) \( -3 \)
Question: Simplify: \( y^6 \div y^2 \)
Solution: Using the exponent rule \( \frac{a^m}{a^n} = a^{m-n} \), we subtract the exponents: \( y^{6-2} = y^4 \).
Correct Answer: (b) \( y^4 \)
Question: Solve for \( a \): \( 2a + 5 = 11 \)
Solution: First, subtract 5 from both sides: \( 2a + 5 - 5 = 11 - 5 \), which gives \( 2a = 6 \). Then, divide both sides by 2: \( \frac{2a}{2} = \frac{6}{2} \), resulting in \( a = 3 \).
Correct Answer: (b) \( a = 3 \)
Question: Which number line shows \( x \geq -1 \)?
Solution: The inequality \( x \geq -1 \) means \(x\) can be -1 or any number greater than -1. On a number line, this is represented by a closed circle at -1 (to include -1) and shading to the right (for all numbers greater than -1).
Correct Answer: (b) Number line with closed circle at -1, shading to the right
Question: Evaluate: \( -\frac{2}{5} - \frac{1}{5} \)
Solution: Since the denominators are the same, we can directly subtract the numerators: \( -\frac{2}{5} - \frac{1}{5} = \frac{-2 - 1}{5} = \frac{-3}{5} = -\frac{3}{5} \).
Correct Answer: (a) \( -\frac{3}{5} \)
Question: Simplify: \( (3x^2)^3 \)
Solution: Using the power of a product rule \( (ab)^n = a^n b^n \) and the power of a power rule \( (a^m)^n = a^{m \cdot n} \), we get \( (3x^2)^3 = 3^3 \cdot (x^2)^3 = 27 \cdot x^{2 \times 3} = 27x^6 \).
Correct Answer: (d) \( 27x^6 \)
Question: Does the set of points \( \{(1, 2), (2, 4), (3, 6), (4, 8)\} \) represent a function? Why?
Solution: For a set of points to represent a function, each \(x\) value must correspond to exactly one \(y\) value. In this set, each \(x\) value (1, 2, 3, 4) is paired with only one \(y\) value (2, 4, 6, 8), and no \(x\) value is repeated with a different \(y\) value.
Correct Answer: (c) Yes, because each \(x\) value has a unique \(y\) value
Question: Solve for \( y \): \( -2y + 6 = 3y - 4 \)
Solution: Add \(2y\) to both sides: \( 6 = 3y + 2y - 4 \Rightarrow 6 = 5y - 4 \). Add 4 to both sides: \( 6 + 4 = 5y \Rightarrow 10 = 5y \). Divide by 5: \( \frac{10}{5} = y \Rightarrow y = 2 \).
Correct Answer: (b) \( y = 2 \)
Question: Solve the inequality: \( -3x \leq 12 \)
Solution: Divide both sides by -3. Remember to reverse the inequality sign when dividing by a negative number: \( \frac{-3x}{-3} \geq \frac{12}{-3} \), which simplifies to \( x \geq -4 \).
Correct Answer: (b) \( x \geq -4 \)
Question: Evaluate: \( \frac{4 - (-2)}{-3 \times 2} \)
Solution: First, simplify the numerator: \( 4 - (-2) = 4 + 2 = 6 \). Then, simplify the denominator: \( -3 \times 2 = -6 \). Finally, divide: \( \frac{6}{-6} = -1 \).
Correct Answer: (b) \( -1 \)
Question: Simplify: \( \frac{b^2 \cdot b^{-5}}{b^{-2}} \)
Solution: First, simplify the numerator: \( b^2 \cdot b^{-5} = b^{2 + (-5)} = b^{-3} \). Then, divide by \( b^{-2} \): \( \frac{b^{-3}}{b^{-2}} = b^{-3 - (-2)} = b^{-3 + 2} = b^{-1} \), which is also \( \frac{1}{b} \).
Correct Answer: (b) \( b^{-1} \) or \( \frac{1}{b} \)
Question: Given \( f(x) = x^2 + 2x - 3 \), find \( f(-3) \)
Solution: Substitute \( x = -3 \) into the function: \( f(-3) = (-3)^2 + 2(-3) - 3 = 9 + (-6) - 3 = 9 - 6 - 3 = 0 \).
Correct Answer: (c) \( 0 \)