Question

Which word or phrase, if inserted into the blank space of the passage, best defines the relationship of the last sentence in the passage to the one preceding it?

• A. For example
• B. Despite this
• C. However
• D. In other words
• A. It is predictable and uncommon.
• B. It is supercilious and abnormal.
• C. It is unpredictable and erratic.
• D. It is extraordinary and erratic.
• A. Understanding Mozarts and Einsteins
• B. Predicting the Life of a Genius
• C. The Uncanny Patterns in the Lives of Geniuses
• D. Pattern and Disorder in the Lives of Geniuses
• A. \(220 \)
• B. \(210 \)
• C. \(200 \)
• D. \(105 \)

Answer 1

The Correct Option is A

Step 1: Understand the connection between the sentences.
The passage talks about predictable patterns among geniuses. The next sentence provides a specific example from Findley’s study to illustrate that point. Step 2: Choose the correct linking phrase.
"For example" correctly introduces an illustration of the general idea just discussed. Thus, it best defines the relationship between the two sentences. Therefore, the answer is (A) For example.

Answer 2

Step 1: Analyze the passage.
The passage mentions that people see genius as a "good abnormality" and as "completely unpredictable." Psychologists also regarded the quirks of genius as "too erratic." Step 2: Match with the options.
The words "unpredictable" and "erratic" match perfectly with the general public’s perception described in the passage. Thus, the correct answer is (C) It is unpredictable and erratic.

Answer 3

Step 1: Understand the main idea of the passage.
The passage talks about how genius, though seen as unpredictable, actually shows certain recurring patterns, as per Anna Findley’s study. Step 2: Match with the given options.
Option (C) clearly captures the essence — "uncanny patterns" reflects both the strange and the recurring nature described in the passage. Thus, the best title is (C) The Uncanny Patterns in the Lives of Geniuses.

Answer 4

Step 1: Using the given condition \( a_1 + a_3 = 10 \). The general terms of an arithmetic progression are: \[ a_1, \, a_1 + d, \, a_1 + 2d, \, a_1 + 3d, \, a_1 + 4d, \, a_1 + 5d. \] From \( a_1 + a_3 = 10 \): \[ a_1 + (a_1 + 2d) = 10 \quad \Rightarrow \quad 2a_1 + 2d = 10 \quad \Rightarrow \quad a_1 + d = 5. \quad \cdots (1) \] This means the second term \( a_2 = 5 \). Step 2: Using the mean information. The mean is \( \mu = \frac{19}{2} \). The sum of the 6 terms is \( S_6 = 6 \times \mu = 6 \times \frac{19}{2} = 57 \). Also, \( S_6 = \frac{6}{2} (2a_1 + (6-1)d) = 3(2a_1 + 5d) \). So, \( 3(2a_1 + 5d) = 57 \), which simplifies to \( 2a_1 + 5d = 19 \). \( \cdots (2) \) Step 3: Solving for \( a_1 \) and \( d \). We have the system: \begin{align} a_1 + d &= 5 \quad &(1)
2a_1 + 5d &= 19 \quad &(2) \end{align} From (1), \( a_1 = 5 - d \). Substituting into (2): \[ 2(5 - d) + 5d = 19 \implies 10 - 2d + 5d = 19 \implies 10 + 3d = 19 \implies 3d = 9 \implies d = 3. \] Then \( a_1 = 5 - d = 5 - 3 = 2 \). The terms are \( 2, 5, 8, 11, 14, 17 \). Step 4: Calculating the variance \( \sigma^2 \). The variance \( \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} a_i^2 - \mu^2 \). \( \sum a_i^2 = 2^2 + 5^2 + 8^2 + 11^2 + 14^2 + 17^2 = 4 + 25 + 64 + 121 + 196 + 289 = 699 \). \[ \sigma^2 = \frac{699}{6} - \left(\frac{19}{2}\right)^2 = \frac{233}{2} - \frac{361}{4} = \frac{466}{4} - \frac{361}{4} = \frac{105}{4}. \] Alternatively, using the formula \( \sigma^2 = d^2 \frac{n^2-1}{12} \): \[ \sigma^2 = 3^2 \frac{6^2-1}{12} = 9 \frac{35}{12} = \frac{3 \times 35}{4} = \frac{105}{4}. \] Step 5: Calculating \( 8\sigma^2 \). \[ 8\sigma^2 = 8 \times \frac{105}{4} = 2 \times 105 = 210. \]