IIT M QUALIFIER AN EXAM QDQ2 27 Oct

Mathematics for Data Science

Question 1

Question: Consider the polynomials $ p(x) = x^5 - 3x^4 + 100x - 1 $ and $ q(x) = x^5 + x + 5 $. Which of the following statements are correct?

• $ p(x) + q(x) \to \infty $ as $ x \to -\infty $
• $ p(x) - q(x) \to -\infty $ as $ x \to \infty $
• $ -5p(x) \to -\infty $ as $ x \to -\infty $
• $ \frac{1}{2}q(x) \to \infty $ as $ x \to -\infty $

Step-by-Step Solution:

1. Leading Terms Analysis:
   • Both $ p(x) $ and $ q(x) $ have the leading term $ x^5 $.
   • For large $ |x| $, the highest degree term dominates.
2. Option 1: $ p(x) + q(x) = 2x^5 - 3x^4 + 100x - 1 + x + 5 $
   • For $ x \to -\infty $, $ 2x^5 \to -\infty $.
   • So, $ p(x) + q(x) \to -\infty $, not $ +\infty $.
   • Incorrect.
3. Option 2: $ p(x) - q(x) = (x^5 - x^5) - 3x^4 + 100x - 1 - x - 5 = -3x^4 + 99x - 6 $
   • For large $ x \to \infty $, $ -3x^4 $ dominates and $ \to -\infty $.
   • So, $ p(x) - q(x) \to -\infty $ as $ x \to \infty $.
   • Correct.
4. Option 3: $ -5p(x) = -5x^5 + 15x^4 - 500x + 5 $
   • For $ x \to -\infty $, $ -5x^5 \to +\infty $.
   • So, $ -5p(x) \to +\infty $, not $ -\infty $.
   • Incorrect.
5. Option 4: $ \frac{1}{2}q(x) = \frac{1}{2}x^5 + \frac{1}{2}x + \frac{5}{2} $
   • For $ x \to -\infty $, $ \frac{1}{2}x^5 \to -\infty $.
   • So, $ \frac{1}{2}q(x) \to -\infty $, not $ +\infty $.
   • Incorrect.

Question 2

Question: Given $ p(x) = -0.3x^3(x^2-1)(x-2)^2(x-3) $, which figure represents the polynomial $ p(x) $ among Figures 1 to 4?

Step-by-Step Solution:

1. Roots Analysis:
   • $ x^2 - 1 = 0 \implies x = 1, -1 $
   • $ x-2 = 0 \implies x = 2 $ (multiplicity 2)
   • $ x-3 = 0 \implies x = 3 $
   • $ x^3 $ gives a root at $ x=0 $ (multiplicity 3)
2. Behavior at Roots:
   • At $ x = 2 $: Even multiplicity (touches x-axis, does not cross)
   • At others: Odd multiplicity (crosses x-axis)
3. End Behavior:
   • Leading term: $ -0.3x^7 $ (degree 7, negative coefficient)
   • As $ x \to -\infty $, $ p(x) \to +\infty $
   • As $ x \to +\infty $, $ p(x) \to -\infty $
4. Figure Matching:
   • Figure 4 matches the described root behavior and end behavior.

Question 3

Question: Let $ A $ be the set of odd positive integers $ \leq 30 $, $ B $ the set of positive integers $ \leq 20 $ divisible by 6. Relations on $ A $ to $ B $:

• $ R_1 = { (a, b) | a \in A, b \in B, a is a factor of b } $
• $ R_2 = { (a, b) | a \in A, b \in B, (a+b) \mod 15 = 0 } $

Which statements are correct?

• (6, 14) is an element in $ R_2 $
• $ R_2 $ is not symmetric
• $ R_1 $ is transitive
• $ R_2 $ is reflexive

Step-by-Step Solution:

1. (6, 14) in $ R_2 $?
   • $ 6 \notin A $ (A is odd numbers only), so not in $ R_2 $.
2. Is $ R_2 $ symmetric?
   • Symmetric if $ (a, b) \in R_2 \implies (b, a) \in R_2 $.
   • $ (a+b) \mod 15 = 0 \implies (b+a) \mod 15 = 0 $, but $ a \in A, b \in B $, so not symmetric as domain/codomain differ.
   • Correct: Not symmetric.
3. Is $ R_1 $ transitive?
   • If $ a $ divides $ b $ and $ b $ divides $ c $, then $ a $ divides $ c $.
   • Correct: Transitive.
4. Is $ R_2 $ reflexive?
   • Reflexive if $ (a, a) \in R_2 $ for all $ a \in A \cap B $.
   • But $ a + a = 2a $, check if $ 2a \mod 15 = 0 $ for all $ a $. Not true for all.
   • Not reflexive.

Question 4

Question: Which of the following options is/are true?

• If $ T = {a, b, c, d} $, then cardinality of $ T \times T $ is 16.
• The minimum value of the quadratic $ f(x) = 3x^2 - 18x + 7 $ is -20.
• For a quadratic equation $ ax^2 + bx + c = 0 $, if $ b^2 - 4ac > 0 $ and a perfect square, then there exists a rational root.
• A line with an undefined slope is parallel to the X-axis.

Step-by-Step Solution:

1. Cardinality of $ T \times T $:
   • $ |T| = 4 $, so $ |T \times T| = 4 \times 4 = 16 $.
   • True.
2. Minimum of $ f(x) $:
   • Vertex at $ x = -\frac{b}{2a} = \frac{18}{6} = 3 $
   • $ f(3) = 3(9) - 18(3) + 7 = 27 - 54 + 7 = -20 $.
   • True.
3. Quadratic roots:
   • If discriminant is a positive perfect square, roots are rational.
   • True.
4. Line with undefined slope:
   • Undefined slope means vertical line, which is parallel to Y-axis, not X-axis.
   • False.

Question 5

Question: In a college of 500 students:

• 285 took Mathematics
• 195 took Statistics
• 115 took English
• 70 took Mathematics and Statistics
• 45 took Mathematics and English
• 50 took Statistics and English
• 10 took all three courses

How many students took none of these three subjects?

Step-by-Step Solution:

Let $ n(M) = 285 $, $ n(S) = 195 $, $ n(E) = 115 $ $ n(M \cap S) = 70 $, $ n(M \cap E) = 45 $, $ n(S \cap E) = 50 $, $ n(M \cap S \cap E) = 10 $

Using Inclusion-Exclusion Principle:

Number who took none:

Question 6

Question: Given the table:

The best-fit line is $ y = 4x + 2 $. What is the value of SSE (sum of squared errors) with respect to this line?

Step-by-Step Solution:

1. Calculate predicted $ y $ for each $ x $:
   • For $ x=20 $: $ y_{pred} = 4 \times 20 + 2 = 82 $
   • For $ x=15 $: $ y_{pred} = 4 \times 15 + 2 = 62 $
   • For $ x=16 $: $ y_{pred} = 4 \times 16 + 2 = 66 $
   • For $ x=25 $: $ y_{pred} = 4 \times 25 + 2 = 102 $
   • For $ x=14 $: $ y_{pred} = 4 \times 14 + 2 = 58 $
2. Compute squared errors:
   • $ (80-82)^2 = 4 $
   • $ (60-62)^2 = 4 $
   • $ (62-66)^2 = 16 $
   • $ (100-102)^2 = 4 $
   • $ (55-58)^2 = 9 $
3. Sum of squared errors (SSE):
   • $ 4 + 4 + 16 + 4 + 9 = 37 $

English

15. Instructional MCQ Q: THIS IS QUESTION PAPER FOR THE SUBJECT “QUALIFIER LEVEL : ENGLISH I (COMPUTER BASED EXAM)” A: No solution required (instructional).

Comprehension: What a Comb Can Tell Us about the History of the Written Word

16. Where was the ancient comb discovered? A: Lachish[^1].
17. Edward Maher is a _____ by profession A: Zooarchaeologist[^1].
18. Microscopic examination of the comb revealed _____ A: Fossilized remains of lice[^1].
19. The ancient comb is made of _____ A: Ivory[^1].
20. The comb confirms the presence of elephants in Canaan in 1700 BC. State true or false. A: FALSE (Ivory was imported)[^1].
21. The inscription on the comb was: A: May this tusk root out the lice of the hair and the beard[^1].
22. Daniel Vainstub’s occupation is ______. A: To study words that are written or cut into hard materials (Epigraphist)[^1].
23. What is the relevance of the sentence on the comb? A: It is the oldest-known complete sentence written in a Canaanite script[^1].
24. Which of the following is a feature of the Canaanite writing system? A: All of these (User friendly, Emerged between 1900 and 1400 B.C., Basis for Western alphabets)[^1].
25. In this sentence, ‘faintly’ is an example of a/an? A: Adverb[^1].
26. Which word has the same underlined sound as in “Shape”? A: Charade[^1].
27. Choose the odd one out: pet, bet, threat. A: Threat[^1].
28. The vowel sound between b and n in the word bun is the same as: A: That which occurs between r and m in the word crumb[^1].
29. Choose the word with a “ng” sound: A: Both Song and Hang[^1].
30. Which among the following words has the same sound as that which is underlined in “pain”? A: Weight[^1].
31. Identify the sequence of consonant and vowel sounds in the word verdict: A: CVCCVC[^1].
32. Identify the part of speech of the underlined word. John lives in a red house on the hill. A: Class noun[^1].
33. Which one of the following words is a noun? A: Book[^1].
34. Identify the part of speech of the underlined word. They killed him. A: Pronoun[^1].
35. He was advised ________ the seminar without preparation. A: Not to attend[^1].
36. Identify the part of speech of the underlined word: She kicked the ball powerfully. A: Verb[^1].
37. These speakers were the _______ he had ever heard. A: Loudest[^1].
38. Which of the following is NOT an adjective? A: Foot[^1].
39. I found my wallet, it was _____ the chair. A: Under[^1].
40. Identify part of speech of the underlined word: …sat beside his best friend. A: Preposition[^1].
41. _____ crow entered my kitchen through the window. A: A[^1].
42. In this sentence, the phrase ‘a nine days wonder’ means ‘something that arouses great interest, but only for a short period.’ A: True[^1].
43. To discover exact details about something. (phrasal verb) A: Pin down[^1].
44. After several attempts, she finally _______ how to solve the complex puzzle. A: figured out[^1].
45. You _____ obtain a refund from the office. (formal permission) A: May[^1].

Vocabulary

46. Acquire A: Attain[^1].
47. Banter A: Repartee[^1].
48. Acquiesce A: Submit[^1].
49. Bargain A: Haggle[^1].
50. Atrocity A: Brutality[^1].

Telephonic conversation

51. HR: Yes. (i)__________ A: May I know who I am speaking to?[^1].
52. Ramya: (ii)__________ A: Sure, I will send my resume[^1].
53. Ramya: (iii)__________ A: Can you say that again, please?[^1].
54. Ramya: (iv)__________ A: Sure, I will send my resume by Monday[^1].
55. Ramya: (v)__________ A: Thanks for your help[^1].

Computational Thinking

56. Instructional MCQ A: No solution required (instructional).
57. Instructional MCQ A: No solution required (instructional).
58. Let X and Y be two rows in the “Scores” table. We call X and Y partially matching if student X and Y are either from the same city or have the same total marks but not both. Let partialMatch(X, Y) be a procedure to find whether X and Y are matching. Choose the correct implementation of the procedure partialMatch. A: Not enough information to provide a solution.
59. What will Q represent at the end of execution? A: Number of bills from “Big Bazaar” or “SV Stores” with total bill amount greater than the average total bill amount.[^1].
60. What will Count represent at the end of execution? A: Number of sentences with at least two longest words[^1].
61. What will count represent at the end of the execution? A: Number of male students whose Physics marks are greater than Mathematics marks[^1].
62. What will count represent at the end of the execution of pseudocode? A: Number of students who scored greater than 75 marks in exactly two subjects.[^1].
63. What will Count represent at the end of execution? A: Number of sentences with equal number of vowels and consonants[^1].
64. At the end of the execution, A captures the number of non-fictional books which were published between 2005 and 2010 (including 2005 and 2010) and are written in English. Choose the correct code for procedure doSomething to complete the pseudocode. A: Not enough information to provide a solution.
65. Which of the following pseudocode(s) is/are executed on the ‘Words’ dataset, such that by the end of execution, A stores the maximum letter count of a word that is a noun? A: Not enough information to provide a solution.
66. Identify all rows with such mistakes. A: Not enough information to provide a solution.
67. Identify all such mistakes (if any). A: Not enough information to provide a solution.
68. At the end of the execution, A captures the highest Chemistry marks scored by a female student from Vellore. Choose the correct code fragment(s) to complete the pseudocode. A: Not enough information to provide a solution.
69. Short Answer Q: The following pseudocode is executed using a dataset similar to the “Words” dataset, based on the following paragraph. … What would be the value of count at the end of the execution of the above pseudocode? A: 4[^1].

Statistics for Data Science

70. Instructional MCQ A: No solution required (instructional).
71. Which of the following statements is/are true? A:

• A sample is the subset of a population.
• The correlation coefficient measures the strength of the linear association between two numerical variables.[^1]

72. If a categorical variable is measured on an ordinal scale, which of the following statistical measures is appropriate? A: Median, Mode[^1].
73. Given bar chart represent the T-Shirt sizes worn by the members of a sports club. Which of the following option(s) is(are) the best way to represent the data? A: Not enough information to provide a solution.
74. Based on the data collected from an organisation, an analyst made a statement that the average salary of an employee is 50,000 rupees in different organizations in the city. The given statement of analyst is based on which kind of statistical analysis? A: Descriptive statistics[^1].
75. What is the sample standard deviation of salary (in thousand rupees)? (Enter the answer correct to 2 decimal accuracy) A: 12.22 to 12.28[^1].
76. What is the mode of the dataset? A: Strawberry[^1].
77. What percentage of the total votes is represented by Butterscotch and Strawberry ice creams combined? A: 31.81%[^1].
78. What is the median of the data set represented by the stem-and-leaf plot? A: 28[^1].
79. Calculate the range of the data set. A: 34[^1].
80. What is the value of y (frequency of Thai cuisine)? A: 15[^1].
81. What is the value of x (relative frequency of Mexican cuisine)? (write correct upto 2 decimal places) A: 0.15 to 0.17[^1].
82. Find the sample covariance between X and Y for the dataset given in Table 2. A: 48[^1].
83. Find the sample correlation coefficient(r) between X and Y for the dataset given in Table 2. (Write correct upto 3 digits after the decimal) A: 0.921 to 0.925[^1].
84. In an exam, students’ scores have an interquartile range(IQR) of 20. The teacher decides to first add 5 marks to each student’s score and then multiply each adjusted score by 2. What will be the interquartile range now? A: 40 (IQR is multiplied by 2; adding a constant does not affect IQR)[^1].

If you need step-by-step explanations for any specific question, please specify the question number.