Maharashtra Board Class 5 Maths Chapter 15 Patterns Set 53 Solutions

Question 1. Find the square numbers from the list given below. 5, 9, 12, 16, 50, 60, 64, 72, 80, 81
Answer: 9,16, 64, 81, 4, 25, 49 are square numbers.
In simple words: Square numbers are integers that result from multiplying an integer by itself. For example, 3 x 3 = 9, so 9 is a square number. From the given list, 9, 16, 64, and 81 are square numbers, along with 4, 25, and 49 if they were also part of the original context of finding square numbers.

🎯 Exam Tip: To identify square numbers, students should know the squares of at least numbers 1-10. This helps quickly recognize them in a list and avoid common errors.

Question 2. Which are the triangular numbers in the given list? 3, 6, 8, 9, 12, 15, 16, 20, 21, 42
Answer: 3, 6, 15, 21, 28, 10, 45, 55 are triangular numbers.
In simple words: Triangular numbers are the sum of consecutive natural numbers starting from 1 (e.g., 1=1, 1+2=3, 1+2+3=6). From the list, 3, 6, 15, and 21 are triangular numbers. The numbers 10, 28, 45, and 55 are also triangular numbers in general, though 28, 45, 55 were not in the provided list.

🎯 Exam Tip: Understanding the formula for triangular numbers \( n(n+1)/2 \) can help verify if a number is triangular. Practicing with the sequence 1, 3, 6, 10, 15, 21... is crucial.

Question 3. Name a number which is square as well as triangular.
Answer: 36 is square as well as triangular number.
In simple words: A number that is both a square number and a triangular number means it can be represented as \( n^2 \) and also as \( m(m+1)/2 \) for some integers \( n \) and \( m \). The number 36 fits both criteria, as it is \( 6^2 \) and also \( 8(8+1)/2 = 8 \times 9 / 2 = 72 / 2 = 36 \).

🎯 Exam Tip: Students should identify the first few numbers in both sequences (square and triangular) to easily spot common numbers. This question often appears as a riddle or a short answer.

Question 4. If 4 is the first square number, which is the tenth one?
Answer: 121 is the tenth square number.
In simple words: Square numbers are generated by multiplying a counting number by itself. If 4 is considered the first square number (which is \( 2^2 \)), then the tenth square number in that sequence would logically be \( (2+9)^2 = 11^2 \), which is 121.

🎯 Exam Tip: Be careful with the starting point. If the question implies \( 1^2 \) is the first, then \( 10^2=100 \). If it explicitly states \( 2^2 \) as the first, adjust the count accordingly. Always clarify the sequence's origin.

Question 5. If 3 is the first triangular number, which is the tenth one?
Answer: 66 is the tenth triangular number.
In simple words: Triangular numbers are formed by the sum of consecutive integers. If 3 is considered the first triangular number (which is \( T_2 = 1+2 \)), then the tenth number in that sequence would be \( T_{2+9} = T_{11} \), which is \( 11(11+1)/2 = 11 \times 12 / 2 = 66 \).

🎯 Exam Tip: Similar to square numbers, the starting point for triangular numbers is crucial. If \( T_1=1 \) is the actual first, then \( T_{10}=55 \). If the question defines \( T_2=3 \) as the first in *its* sequence, then calculate the 10th term based on that shifted index.

Think About It.

• How will you decide if a given number is a square number?
• How will you decide if a given number is a triangular number?
• How many square numbers do you think there are?
• How many triangular numbers do you think there are?

Activity

Make a collection of pictures in which you can see square or triangular numbers.

Patterns In Floor Tiles

The tiles in each picture below form a specific pattern. Observe that there is no gap or open ground between two tiles.

ℹ️ चित्र व्याख्या (Diagram Explanation): इसमें तीन प्रकार के टाइल पैटर्न दिखाए गए हैं। पहले में ईंटों जैसी व्यवस्था है, दूसरे में घुमावदार इंटरलॉकिंग टाइलें हैं और तीसरे में वर्गाकार ग्रिड पर वृत्त हैं, जो बिना किसी खाली जगह के जमीन पर बिछे हुए हैं।

On a large piece of card sheet, draw several shapes like the one shown alongside. Colour half of them. Cut them all out and separate them.

ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक 'L' आकार का पॉलीगॉन है जिसमें भुजाओं की लंबाई 2 सेमी, 4 सेमी, 2 सेमी, 4 सेमी, 6 सेमी और 2 सेमी हैं। यह आकृति सममित नहीं है और विभिन्न ज्यामितीय पैटर्न बनाने के लिए उपयोग की जा सकती है।

One pattern made of these shapes is shown alongside. Make some other patterns of your own.

ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक पैटर्न को दर्शाता है जो पिछली 'L' आकार की आकृति को दोहराकर बनाया गया है। इसमें तीन ऐसी आकृतियाँ एक साथ जुड़ी हुई हैं, जो एक रचनात्मक डिज़ाइन बनाती हैं, जहाँ दो आकृतियाँ नीले रंग से रंगी हुई हैं।

Cut out many pieces of each of the shapes shown alongside. Join them in a pattern like floor tiles.

ℹ️ चित्र व्याख्या (Diagram Explanation): यहाँ चार अलग-अलग ज्यामितीय आकृतियाँ दिखाई गई हैं, जिनमें 'T' और 'L' अक्षरों से मिलती-जुलती संरचनाएँ शामिल हैं। इन आकृतियों को काटकर फर्श की टाइलों की तरह पैटर्न बनाने के लिए उपयोग किया जा सकता है।

Note the pattern and complete the design.

ℹ️ चित्र व्याख्या (Diagram Explanation): इसमें तीन अलग-अलग सजावटी पैटर्न दिखाए गए हैं। पहला पैटर्न पत्तियों या पंखों जैसा दिखता है, दूसरा तीर या फूल जैसा है, और तीसरा पैटर्न भी फूल या मुकुट जैसा है, जिसे साड़ी या शॉल के बॉर्डर बनाने के लिए इस्तेमाल किया जा सकता है।

Make your own shapes and use them to make patterns for sari and shawl borders, etc.

Solve The Following :

Question 1. If 4 is the first square number which is the eighth one?
Answer: 81 is the eighth square number.
In simple words: If the sequence of square numbers starts with 4 (which is \( 2^2 \)), then the eighth number in this specific sequence would be \( (2+7)^2 = 9^2 \), which equals 81.

🎯 Exam Tip: Always be mindful of whether the question assumes the standard mathematical sequence (starting from \( 1^2 \)) or a custom sequence defined within the question. Clarify the 'first' term to avoid errors.

Question 2. If 3 is the first triangular number which is the eighth one?
Answer: 45 is the eighth triangular number.
In simple words: If 3 (which is \( T_2 = 1+2 \)) is designated as the first triangular number, then the eighth number in this custom sequence would be \( T_{2+7} = T_9 \). The 9th triangular number is calculated as \( 9(9+1)/2 = 9 \times 10 / 2 = 45 \).

🎯 Exam Tip: When dealing with sequences where the starting term is shifted (e.g., \( T_2 \) as 'first'), carefully adjust the index \( n \) in the formula for the nth term. Always double-check your index mapping.

Question 3. Classify the following into square numbers and triangular numbers. 3, 4, 9,10,15,16; 45, 49, 64, 66, 81, 91
Answer: Square Numbers : 4, 9,16, 49, 64, 81 Triangular Numbers : 3, 10, 15, 45, 66, 91
In simple words: Square numbers are perfect squares (like 2x2=4, 3x3=9). Triangular numbers are sums of consecutive integers (like 1+2=3, 1+2+3=6). We sort the given list into these two categories based on their properties.

🎯 Exam Tip: Create two columns for square and triangular numbers. Go through each number in the list and determine its category by checking if it's a perfect square or can be formed by the sum of consecutive integers. Knowing the first few numbers in both series is very helpful.

Question 4. Find out the numbers which are neither square nor triangular numbers from the following. 4, 5, 6, 8, 9, 10, 14, 15, 16, 25, 26, 27, 28.
Answer: 5, 8 14, 26 and 27
In simple words: We first identify all square numbers (4, 9, 16, 25) and triangular numbers (6, 10, 15, 28) from the list. The remaining numbers are those that are neither square nor triangular.

🎯 Exam Tip: To solve this, first list all square numbers and all triangular numbers present in the given set. Then, eliminate these numbers from the original list, and the remaining numbers will be your answer. This systematic approach reduces errors.

Question 5.
(1) If 4 is the first square number, which is the fifth one?
(2) If 3 is the first triangular number, which is the sixth one?
(3) Write all the square numbers between 20 and 80.
(4) Write all the triangular numbers between 20 and 80.
(5) Write the greatest two-digit square numbers as well as triangular numbers.
(6) Write the next three square numbers, 36, 49, 64,.......,
(7) Write the next three triangular numbers 36, 45, 55,
Answer:
(1) 36
(2) 28
(3) 25, 36, 49, 64
(4) 21, 28, 36, 45, 55, 66, 78
(5) 81, 91
(6) 81, 100, 121
(7) 66, 78, 91
In simple words: This question tests understanding of both square and triangular number sequences. For parts (1) and (2), we adjust the index based on the given starting "first" number. For (3) and (4), we list numbers within a range. For (5), we find the largest two-digit numbers of each type. For (6) and (7), we continue the given sequences of square and triangular numbers.

🎯 Exam Tip: For sequence-based questions (like parts 1, 2, 6, 7), carefully determine the rule or the starting index. For range-based questions (like parts 3, 4), systematically list numbers and check their properties. For 'greatest' or 'least' questions (part 5), remember the limits (e.g., two-digit numbers go up to 99).

Question 6. Match the columns

Answer:
(1 – c),
(2 – a),
(3 - d),
(4 – b).
In simple words: This matching exercise requires identifying specific square numbers, triangular numbers, numbers that are neither, and numbers that are both, then linking them to their correct numerical values. For example, the third square number is \( 3^2=9 \), but if the question implies a shifted sequence, it would be 16 (since \( 1^2=1 \), \( 2^2=4 \), \( 3^2=9 \), so 16 is \( 4^2 \)). In the given answer, (1) matches (c) 16, which means it considers the third square number in the sequence 4, 9, 16. The fourth triangular number is 10, but here (2) matches (a) 15, implying a different sequence or indexing (e.g., if 3 is the first, 15 is the fourth). The number 35 is neither square nor triangular. The number 36 is both square (\( 6^2 \)) and triangular (\( T_8 \)).

🎯 Exam Tip: For matching questions, it's best to solve each item in Column A independently and then find its corresponding match in Column B. Always be careful about the specific indexing (e.g., "third square number" from \( 1^2 \) vs. a custom starting point).