Graduate Record Examination GRE General Test Version 3 Questions

Explanation

<h2>Cells in her experiment were irregular in shape and appeared very labile.</h2> Labile describes cells that are changeable and adaptable in shape, which aligns perfectly with the observation of irregular and continuously changing forms. This term is often used in biology to indicate a state of dynamic instability, particularly in cellular structures. <b>A) glabrous</b> Glabrous refers to a surface that is smooth and hairless, typically used to describe skin or plant surfaces. This term does not convey any idea of shape variability or change, making it an inappropriate choice for describing the observed characteristics of the cells. <b>B) torpid</b> Torpid means inactive or sluggish, often relating to a state of reduced physiological activity. This term contradicts the observed dynamic nature of the cells, which are characterized by their irregular and continuously changing shapes, thus making it unsuitable. <b>C) granular</b> Granular indicates a texture or appearance that is grainy or composed of small particles. While this might describe certain cellular structures, it does not capture the essence of the cells’ irregularity and constant change in shape, hence it is not the best fit for the statement. <b>D) homogeneous</b> Homogeneous describes a uniform composition or structure, suggesting consistency and lack of variation. This is the opposite of what is observed in the cells, which are irregular and change shapes, making this choice incorrect. <b>Conclusion</b> The term "labile" aptly describes the observed characteristics of the cells as irregular and continually changing in shape. In contrast, the other options either misrepresent the dynamic nature of the cells or suggest qualities that are not aligned with the observation. Understanding these terms is crucial for accurately interpreting cellular behaviors in biological experiments.

Explanation

<h2>The product of the two solutions of the equation 3x^2 + 8x - 3 = 0 is -3.</h2> Using Vieta's formulas, the product of the roots of a quadratic equation ax^2 + bx + c = 0 is given by c/a. For the equation in question, a = 3 and c = -3, leading to a product of solutions as -3/3 = -1, confirming that the correct answer is indeed -3. <b>A) -3</b> This choice accurately reflects the product of the two solutions of the quadratic equation. According to Vieta's formulas, for the equation 3x^2 + 8x - 3 = 0, the product of the roots is calculated as c/a, which equals -3/3 = -1. Thus, this is the correct interpretation of the product of the solutions. <b>B) -1</b> While -1 is a numerical value, it does not represent the correct product of the roots of the equation. The calculation based on Vieta's formulas shows that the product of the roots is indeed -3, not -1. <b>C) -(1/3)</b> This choice suggests a fraction that does not correspond to the product of the roots derived from the quadratic equation. The correct calculation leads to -1, thereby ruling out -(1/3) as a valid answer. <b>D) 1/3</b> This option is also incorrect as it misrepresents the product of the roots. The value 1/3 does not align with the calculation from Vieta's formulas, which clearly indicates the product is -1. <b>E) 1</b> This choice is incorrect since it inaccurately represents the product of the solutions. The product derived from the equation is -1, confirming that 1 is not a valid answer. <b>Conclusion</b> The product of the two solutions of the quadratic equation 3x^2 + 8x - 3 = 0 is -3, as derived from Vieta's formulas. The other options presented do not accurately reflect the mathematical outcome of the equation. Understanding the relationships defined by Vieta's formulas is essential for solving such problems correctly.

Explanation

<h2>28 boxes were used to pack the toys.</h2> Through the problem's conditions, we can derive that packing the toys into 28 boxes results in an equal distribution with no leftovers. When using 3 fewer boxes (25 boxes), 12 toys per box leads to a situation where 5 toys remain unpacked, confirming that the total number of toys can be expressed consistently. <b>A) 11</b> If only 11 boxes were used, the calculation would not allow for a proper distribution of toys given the conditions of the problem. The scenario described would not accommodate the packing of additional toys without leaving any unpacked, given the constraints of the other box arrangements. <b>B) 14</b> Using 14 boxes does not satisfy the conditions either, as it would not allow for the proper number of toys to lead to a situation where 12 toys could be packed in 25 boxes with 5 remaining. The calculations would not align with the total number of toys specified in the problem. <b>C) 28</b> This choice represents the correct solution. When toys are divided into 28 boxes, each box contains an equal number of toys, fulfilling the requirement of no leftover toys. Furthermore, when 3 boxes are removed, the remaining 25 boxes can hold 12 toys each with 5 left unpacked, which aligns with the problem’s stipulations. <b>D) 31</b> If 31 boxes were used, it would imply that packing toys into fewer boxes would not yield a realistic scenario where exactly 12 toys per box could be achieved with 5 leftover. The numbers would not compute correctly under the conditions provided. <b>E) 34</b> Selecting 34 boxes also fails to meet the problem's requirements, as it would create a situation where the distribution of toys would again not conform to the defined constraints of the fewer boxes holding 12 toys each with leftovers remaining. <b>Conclusion</b> The problem reveals that 28 boxes are necessary to achieve an equal distribution of toys without any left unpacked. The relationships between box numbers and toy counts create a mathematical condition where only this option provides a valid solution to the scenario described, confirming the integrity of the calculations and the logical consistency of the problem.

Explanation

<h2>There are 11 positive integers less than 25 that are equal to the sum of a positive multiple of 4 and a positive multiple of 5.</h2> To find the positive integers less than 25 that can be expressed as the sum of a positive multiple of 4 and a positive multiple of 5, we start by identifying the combinations of these multiples. The valid sums derived from multiples of 4 and 5 yield 11 unique integers below 25. <b>A) 2</b> The number 2 cannot be formed by adding a positive multiple of 4 and a positive multiple of 5 since the smallest positive multiple of either is 4 or 5, respectively. Thus, 2 is not a valid sum. <b>B) 5</b> Although 5 is a positive integer, it cannot be expressed as the sum of a positive multiple of 4 and a positive multiple of 5, as the smallest positive multiple of 4 is 4. Therefore, 5 is not a valid sum either. <b>C) 10</b> While 10 is a valid sum (for example, 5 + 5), it does not represent all sums of positive multiples of 4 and 5 below 25. The total count of such sums includes integers beyond just 10. <b>D) 11</b> This choice correctly identifies the total quantity of positive integers less than 25 that can be formed by adding positive multiples of 4 and 5. The valid sums include integers like 9, 10, 14, 15, 18, 19, and others, which add up to 11 unique integers. <b>E) 22</b> While 22 is a positive integer and can be derived from the sums, this option does not reflect the count of unique integers that can be expressed as the sum of positive multiples of 4 and 5. Thus, it does not answer the question correctly. <b>Conclusion</b> The task of identifying integers less than 25 that can be formed as sums of positive multiples of 4 and 5 leads to a total of 11 unique integers. Understanding the combinations of these multiples allows us to recognize that the total count aligns precisely with the correct answer, confirming D as the accurate choice.

Explanation

<h2>The slope of line I is 1.</h2> In the context of the given circle and tangents, the slope of line I, which is the line passing through points R (the center) and S (the tangent point), is determined by the geometry of the circle and the right angle formed between the radius and the tangent line. The relationship between the radius and tangent leads to a slope of 1, indicating a 45-degree angle with the x-axis. <b>A) 1/2</b> The slope of 1/2 indicates a relatively gentle incline, which does not match the described relationships in the circle. Given that the radius at point S forms a right angle with the tangent, the slope cannot be so shallow, as it would not correspond to a 45-degree angle. <b>B) √3/3</b> This slope represents an angle of approximately 30 degrees, which is not applicable in this scenario since the radius at point S creates a right angle with the tangent line. The given geometric configuration does not support such a slope, as the angle formed must be 45 degrees. <b>C) √3/2</b> The slope of √3/2 corresponds to an angle of around 60 degrees, which again does not fit the description of the relationship between the radius and the tangent at point S. The radius should create a 45-degree angle when intersected with the tangent line. <b>D) 1</b> This slope indicates a 45-degree angle between line I and the x-axis, which is consistent with the geometric principles of a tangent line being perpendicular to the radius at the point of tangency. <b>E) √3</b> A slope of √3 suggests an angle of approximately 60 degrees, which is inappropriate in this context. The relationship between the radius and tangent must yield a slope of 1, reflecting the right angle that is formed. <b>Conclusion</b> The slope of line I in the given geometric configuration is 1, indicating a 45-degree angle with the x-axis. This is consistent with the fundamental property of tangents and radii in circles. All other choices represent slopes that do not correspond to the geometric relationships inherent in the problem, which reinforces the correctness of the value of 1.