Unit 1: Number System and Average - Subjective Questions

PEA515 — Analytical Skills-I • Practice Questions with Detailed Answers

20 questions

Define natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, prime numbers, and composite numbers. Explain the relationship among these classifications with suitable examples.

State and explain the divisibility rules for $2$ , $3$ , $4$ , $5$ , $6$ , $8$ , $9$ , $10$ , and $11$ , giving one example for each rule.

Explain the meaning of factorial notation. Derive the relationship between $n!$ and $(n-1)!$ , and evaluate $\frac{8!}{6!}+\frac{7!}{5!}$ .

Determine the unit digit of $7^{2025}+3^{2026}+9^{2027}$ using the cyclicity method.

Describe the general method for finding the number of factors of a positive integer. Apply it to find the number of factors of $7,560$ .

Find the number of odd factors and even factors of $3,600$ .

State the remainder theorem for division by a linear expression and use it to find the remainder when $f(x)=2x^4-3x^3+5x^2-7x+9$ is divided by $x-2$ .

Explain how the remainder theorem can be used to test whether a polynomial is exactly divisible by $x-a$ . Illustrate your answer using $f(x)=x^3-4x^2+x+6$ and the divisor $x-3$ .

Define HCF and LCM. Find the HCF and LCM of $84$ , $126$ , and $210$ using prime factorization, and verify the relationship between HCF and LCM for two numbers.

Two bells ring at intervals of $18$ minutes and $24$ minutes. If they ring together at 9:00 a.m., at what time will they ring together again? Explain the method used.

Derive the formula for the average of $n$ observations and use it to find the missing number if the average of $12$ , $18$ , $25$ , $x$ , and $35$ is $22$ .

The average age of $8$ students is $16$ years. When a teacher joins them, the average becomes $20$ years. Find the teacher's age.

The average marks of $20$ students in a test is $64$ . Later, it is found that one mark was recorded as $46$ instead of $64$ . Find the correct average.

Explain the principle of inclusion and exclusion for two finite sets. In a class of $60$ students, $35$ study Mathematics, $28$ study Science, and $15$ study both. Find how many study at least one of the two subjects and how many study neither.

State the inclusion-exclusion formula for three sets and solve the following: In a survey of $100$ people, $60$ like tea, $ fifty$ like coffee, and $40$ like juice. Also, $25$ like tea and coffee, $20$ like coffee and juice, $15$ like tea and juice, and $10$ like all three. How many like at least one beverage?

Define weighted average and distinguish it from the simple average. Find the weighted average of scores $70$ , $80$ , and $90$ having weights $2$ , $3$ , and $5$ , respectively.

A student scores $72$ marks in a test carrying $100$ marks, $80$ marks in a test carrying $120$ marks, and $90$ marks in a test carrying $150$ marks. Find the overall percentage using a weighted approach.

Find the least number that must be added to $1,234$ so that the result is divisible by $7$ , $9$ , and $11$ . Explain the role of the LCM in your solution.

Find the greatest number that divides $245$ , $325$ , and $445$ , leaving the same remainder in each case.

Derive a method for finding the unit digit of a product involving large powers. Use it to find the unit digit of $13^{101}\times 27^{52}\times 46^{37}$ .

Question 1

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Answer 1

Classification of numbers:

Natural numbers: Positive counting numbers such as $1, 2, 3, \ldots$ .
Whole numbers: Natural numbers including zero, such as $0, 1, 2, 3, \ldots$ .
Integers: Positive and negative whole numbers, including zero, such as $\ldots, -2, -1, 0, 1, 2, \ldots$ .
Rational numbers: Numbers that can be written as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \ne 0$ . Examples are $\frac{3}{4}$ and $-5$ .
Irrational numbers: Numbers that cannot be expressed as the ratio of two integers. Examples are $\sqrt{2}$ and $\pi$ .
Real numbers: The set containing both rational and irrational numbers.
Prime numbers: Natural numbers greater than $1$ having exactly two factors: $1$ and the number itself. Examples are $2, 3,$ and $5$ .
Composite numbers: Natural numbers greater than $1 having more than two factors. Examples are $4, 6,$ and $12$.

The relationship is represented by $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ , while irrational numbers are also part of $\mathbb{R}$ but not of $\mathbb{Q}$ .

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Question 2

State and explain the divisibility rules for $2$ , $3$ , $4$ , $5$ , $6$ , $8$ , $9$ , $10$ , and $11$ , giving one example for each rule.

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Answer 2

Divisibility rules:

Divisibility by $2$ : The last digit must be even. Example: $348$ is divisible by $2$ .
Divisibility by $3$ : The sum of the digits must be divisible by $3$ . Example: $123$ has digit sum $6$ , so it is divisible by $3$ .
Divisibility by $4$ : The number formed by the last two digits must be divisible by $4$ . Example: $516$ is divisible by $4$ because $16$ is divisible by $4$ .
Divisibility by $5$ : The last digit must be $0$ or $5$ . Example: $245$ is divisible by $5$ .
Divisibility by $6$ : The number must be divisible by both $2$ and $3$ . Example: $126$ is divisible by $6$ .
Divisibility by $8$ : The number formed by the last three digits must be divisible by $8$ . Example: $1,024$ is divisible by $8$ because $024$ is divisible by $8$ .
Divisibility by $9$ : The sum of the digits must be divisible by $9$ . Example: $729$ has digit sum $18$ , so it is divisible by $9$ .
Divisibility by $10$ : The last digit must be $0$ . Example: $450$ is divisible by $10$ .
Divisibility by $11$ : The difference between the sums of alternate digits must be $0$ or a multiple of $11$ . For $1,452$ , $(1+5)-(4+2)=0$ , so it is divisible by $11$ .

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Question 14

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Answer 14

For two sets $A$ and $B$ , the principle of inclusion and exclusion states

$n(A \cup B)=n(A)+n(B)-n(A \cap B).$

The intersection is subtracted because students in both sets are counted twice when $n(A)$ and $n(B)$ are added.

Let $A$ represent students studying Mathematics and $B$ represent students studying Science. Then

$n(A \cup B)=35+28-15=48.$

Therefore, $48$ students study at least one subject.

The number studying neither subject is

$60-48=12.$

Hence:

Students studying at least one subject: $48$
Students studying neither subject: $12$

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Question 17

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Answer 17

The overall percentage must be calculated from total marks obtained divided by total maximum marks. This is a weighted calculation because the tests have different maximum marks.

Total marks obtained:

$72+80+90=242.$

Total maximum marks:

$100+120+150=370.$

Therefore, the overall percentage is

$\frac{242}{370}\times 100=65.4054\ldots\%.$

Hence, the student's overall percentage is approximately $65.41\%$ .

A simple average of the three percentages would not be exact because the tests do not carry equal marks.

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Unit2

Unit 1: Number System and Average - Subjective Questions

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