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The World of Algorithms – Class 9 Mathematics notes, important questions and MCQs

NCERT Class 9 Mathematics, chapter 11. Notes, key terms, important questions with answers and MCQs.

Chapter overview

An algorithm is a finite list of clear steps that solves a problem. Euclid's division algorithm for HCF is a classic example; flowcharts help to show the steps.

What makes an algorithm

Clear input and output.
Each step is precise and doable.
It must stop after a finite number of steps.

Examples

Euclid's algorithm: HCF(a, b) = HCF(b, r), where a = bq + r; repeat until r = 0.
Finding the largest of three numbers by comparing two at a time.
Binary search: halve the search range each time in a sorted list.
Flowchart symbols: oval (start/stop), parallelogram (input/output), rectangle (process), diamond (decision).

Key terms

Algorithm: step-by-step method that ends with the answer
Flowchart: a diagram showing the steps of an algorithm
Division lemma: a = bq + r with 0 ≤ r < b

Formulas

a = bq + r, 0 ≤ r < b
HCF(a, b) = HCF(b, r)

Important questions with answers

What are two properties every algorithm must have?2 marks

It must end after a finite number of steps, and each step must be clear and unambiguous (it should also produce the correct output for valid input).

Use Euclid's division algorithm to find the HCF of 135 and 225.3 marks

225 = 135 × 1 + 90; 135 = 90 × 1 + 45; 90 = 45 × 2 + 0. The last non-zero remainder is 45, so HCF = 45.

Write an algorithm (in steps) to find the largest of three numbers, and describe how a flowchart would show it.5 marks

Step 1: Read A, B, C. Step 2: Set Max = A. Step 3: If B > Max, set Max = B. Step 4: If C > Max, set Max = C. Step 5: Print Max and stop. In a flowchart, an oval shows Start/Stop, a parallelogram shows input/output (Read A, B, C; Print Max), rectangles show processes (Max = A), and diamonds show decisions (Is B > Max?) with Yes/No arrows.

Quick revision

Algorithms must be finite and precise.
Euclid's algorithm stops when the remainder is 0; the last divisor is the HCF.
Binary search needs sorted data.

Exam tips

Show each division step neatly in Euclid's algorithm.
Use a diamond only where a yes/no decision is made.

Practice MCQs

An algorithm must
1. never stop
2. stop after finitely many steps
3. use a computer
4. have no input
Show answer
Answer: (b) stop after finitely many steps. Finiteness.
In a flowchart, a decision is shown by a
1. rectangle
2. oval
3. diamond
4. circle
Show answer
Answer: (c) diamond. Diamond.
In a = bq + r, r is the
1. quotient
2. divisor
3. dividend
4. remainder
Show answer
Answer: (d) remainder. Remainder.
Using Euclid's algorithm, HCF(36, 48) is
1. 6
2. 12
3. 18
4. 24
Show answer
Answer: (b) 12. 48 = 36 × 1 + 12; 36 = 12 × 3 + 0.
In Euclid's algorithm, r must satisfy
1. r > b
2. r < 0
3. r = b
4. 0 ≤ r < b
Show answer
Answer: (d) 0 ≤ r < b. Division lemma condition.
Binary search in a sorted list of 64 items needs at most about
1. 64 checks
2. 7 checks
3. 32 checks
4. 2 checks
Show answer
Answer: (b) 7 checks. Halving: 64 → 32 → … → 1 takes about 6–7 checks.
HCF(455, 42) by Euclid's algorithm is
1. 14
2. 7
3. 21
4. 13
Show answer
Answer: (b) 7. 455 = 42 × 10 + 35; 42 = 35 × 1 + 7; 35 = 7 × 5.
Which is NOT a property of a good algorithm?
1. ambiguity
2. finiteness
3. definiteness
4. effectiveness
Show answer
Answer: (a) ambiguity. Steps must not be ambiguous.

All chapters of Class 9 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.