CBSE Class 6 Mathematics Revision Worksheet Set 17

Chapter 1 - Knowing Our Numbers

1. Estimate using general rule :
i) 830 + 976
Answer:
830 rounds off to 800 (nearest hundred)
976 rounds off to 1000 (nearest hundred)
Estimated sum = 800 + 1000 = 1800
In simple words: Round both numbers to the nearest hundred and add them.

Exam Tip: Always state the rounding rule used (nearest hundred or nearest ten) clearly before calculating.

ii) 496 - 215
Answer:
496 rounds off to 500 (nearest hundred)
215 rounds off to 200 (nearest hundred)
Estimated difference = 500 - 200 = 300
In simple words: Round both numbers to the nearest hundred and subtract.

Exam Tip: Ensure you keep the order of subtraction same as given in the problem.

iii) 13,804 + 3,777
Answer:
13,804 rounds off to 14,000 (nearest thousand)
3,777 rounds off to 4,000 (nearest thousand)
Estimated sum = 14,000 + 4,000 = 18,000
In simple words: Round large numbers to the nearest thousand for quick estimation.

Exam Tip: For numbers in thousands, general rule implies rounding to the nearest thousand.

iv) 61,292 - 21,496
Answer:
61,292 rounds off to 61,000 (nearest thousand)
21,496 rounds off to 21,000 (nearest thousand)
Estimated difference = 61,000 - 21,000 = 40,000
In simple words: Round off both numbers to the nearest thousand and subtract.

Exam Tip: Check the digit in the hundreds place to decide whether to round up or down.

2. Estimate the products using general rule :
(i) 758 \times 151
Answer:
758 rounds off to 800 (nearest hundred)
151 rounds off to 200 (nearest hundred)
Estimated product = \(800 \times 200 = 1,60,000\)
In simple words: Round both factors to the nearest hundred and multiply.

Exam Tip: Count the total number of zeros in the end to avoid calculation errors.

(ii) 4391 \times 2300
Answer:
4391 rounds off to 4000 (nearest thousand)
2300 rounds off to 2000 (nearest thousand)
Estimated product = \(4000 \times 2000 = 80,00,000\)
In simple words: Round both numbers to the nearest thousand and multiply.

Exam Tip: Be careful with place values when multiplying numbers in thousands.

(iii) 2187 \times 456
Answer:
2187 rounds off to 2000 (nearest thousand)
456 rounds off to 500 (nearest hundred)
Estimated product = \(2000 \times 500 = 10,00,000\)
In simple words: Round each number to its greatest place value and multiply.

(iv) 6978 \times 43
Answer:
6978 rounds off to 7000 (nearest thousand)
43 rounds off to 40 (nearest ten)
Estimated product = \(7000 \times 40 = 2,80,000\)
In simple words: Round the first number to thousands and the second number to tens, then multiply.

Exam Tip: Different numbers can be rounded to different place values based on their individual highest places.

3. Write the Roman Numeral for :
(i) 99
Answer: XCIX
In simple words: 99 is written as 100 minus 10, plus 10 minus 1.

Exam Tip: Remember XCIX represents 90 (XC) + 9 (IX).

(ii) 48
Answer: XLVIII
In simple words: 48 is 50 minus 10, followed by 8.

Exam Tip: XL is 40 and VIII is 8.

(iii) 67
Answer: LXVII
In simple words: 67 is 50 plus 10 plus 7.

Exam Tip: L represents 50, X represents 10, and VII represents 7.

(iv) 81
Answer: LXXXI
In simple words: 81 is 50 plus three tens and 1.

Exam Tip: L can be followed by at most three X symbols.

(v) 17
Answer: XVII
In simple words: 17 is 10 plus 7.

Exam Tip: X is 10 and VII is 7.

(vi) 76
Answer: LXXVI
In simple words: 76 is 50 plus 20 plus 6.

Exam Tip: LXX is 70 and VI is 6.

(viii) 54
Answer: LIV
In simple words: 54 is 50 plus 4.

Exam Tip: L is 50 and IV is 4.

4. Answer the following :
i) The town newspaper is published every day. One copy has 15 pages. Everyday 12,500 copies are printed. How many total pages are printed every day?
Answer:
Number of pages in one copy = 15
Number of copies printed every day = 12,500
Total pages printed every day = \(12,500 \times 15 = 1,87,500\)
In simple words: Multiply the number of pages per copy by the total number of copies printed.

Exam Tip: Write proper statement lines before performing multiplication word problems.

ii) Apples are packed in boxes, each weighing 5kg 500gm. How many such boxes can be loaded in a van which cannot carry beyond 1000kg?
Answer:
Maximum weight van can carry = 1000 kg = 10,00,000 g
Weight of one box = 5 kg 500 g = 5500 g
Number of boxes = \(\frac{10,00,000}{5500} = \frac{10000}{55}\) = 181.81
Since number of boxes must be a whole number, 181 boxes can be loaded.
In simple words: Convert the total capacity and box weight into grams, then divide and take the whole number part.

Exam Tip: Always convert both quantities to the same unit before dividing.

Chapter 2 - Whole Numbers

1. Fill in the blanks :
i) 25 x 8 x 125 x 4 = _____
Answer: 1,00,000
In simple words: Group numbers to make multiples of 10 or 100 (\(25 \times 4 = 100\), \(125 \times 8 = 1000\), \(100 \times 1000 = 1,00,000\)).

Exam Tip: Use associative and commutative properties to rearrange terms for easy mental calculation.

ii) 315 x 105 = 315 x 100 + ____ x 5
Answer: 315
In simple words: Using distributive law, 105 is split into (100 + 5), so 315 is multiplied with both 100 and 5.

Exam Tip: This illustrates the distributive property of multiplication over addition.

iii) Division by zero is _____________
Answer: not defined
In simple words: You cannot divide any number by zero.

Exam Tip: Remember division by zero is undefined in mathematics.

iv) The smallest natural number is ______________________.
Answer: 1
In simple words: Counting numbers start from 1.

Exam Tip: Do not confuse natural numbers (start at 1) with whole numbers (start at 0).

v) The sum of 3 odd numbers is ____________________.
Answer: odd
In simple words: Odd + Odd = Even, and Even + Odd = Odd.

Exam Tip: The sum of an odd number of odd numbers is always odd.

vi) ________________________ is the additive identity for the whole numbers.
Answer: 0
In simple words: Adding zero to any whole number leaves it unchanged.

Exam Tip: Zero is the additive identity; 1 is the multiplicative identity.

vii) (7 x 8) x 5 = 7 x (8 x 5) This statement shows that multiplication of whole numbers is ________.
Answer: associative
In simple words: Changing the grouping of factors does not change the product.

viii) How many numbers are there between 102 and 211.
Answer: 108
In simple words: To find numbers strictly between two numbers, subtract them and subtract 1 more (\(211 - 102 - 1 = 108\)).

Exam Tip: Formula for numbers between a and b is \(b - a - 1\).

ix) 3 + 7 = 7 + 3. This statement shows that addition of whole numbers is ___________.
Answer: commutative
In simple words: Changing the order of addends does not change the sum.

Exam Tip: This is the commutative property of addition.

2. Determine the product by suitable rearrangements
i) 2 x 125 x 50 x 8
Answer:
\(= (2 \times 50) \times (125 \times 8)\)
\(= 100 \times 1000\)
\(= 1,00,000\)
In simple words: Pair 2 with 50 and 125 with 8 to get multiples of 10.

Exam Tip: Look for pairs like (2, 5), (4, 25), (8, 125) that make powers of 10.

ii) 16 x 279 x 625
Answer:
\(= (16 \times 625) \times 279\)
\(= 10,000 \times 279\)
\(= 27,90,000\)
In simple words: Since \(16 \times 625 = 10,000\), multiply 10,000 by 279.

Exam Tip: Memorize standard products like \(16 \times 625 = 10,000\) to save time.

3. Using distribution property, find each of the following products.
i) 213 x 104
Answer:
\(= 213 \times (100 + 4)\)
\(= (213 \times 100) + (213 \times 4)\)
\(= 21,300 + 852\)
\(= 22,152\)
In simple words: Split 104 into 100 + 4 and distribute multiplication over addition.

Exam Tip: Always split the number close to 100, 1000, etc.

ii) 256 x 1007
Answer:
\(= 256 \times (1000 + 7)\)
\(= (256 \times 1000) + (256 \times 7)\)
\(= 2,56,000 + 1792\)
\(= 2,57,792\)
In simple words: Write 1007 as 1000 + 7 and multiply separately.

Exam Tip: Distributive property makes complex multiplications very easy.

4. Find the value :
i) 361 + 1482 + 639 + 518
Answer:
\(= (361 + 639) + (1482 + 518)\)
\(= 1000 + 2000\)
\(= 3000\)
In simple words: Group numbers that add up to multiples of 1000.

Exam Tip: Look for unit digits that sum to 10 (\(1 + 9 = 10\) and \(2 + 8 = 10\)).

ii) 786 x 97 + 786 x 3
Answer:
\(= 786 \times (97 + 3)\)
\(= 786 \times 100\)
\(= 78,600\)
In simple words: Take out 786 as a common factor using the distributive property in reverse.

Exam Tip: This is taking common factor out, which is the reverse of distribution.

iii) 14 + 438 + 486 + 62
Answer:
\(= (14 + 486) + (438 + 62)\)
\(= 500 + 500\)
\(= 1000\)
In simple words: Rearrange terms to get clean hundreds.

Exam Tip: Rearranging terms using commutative and associative laws simplifies addition.

iv) 716 x 6 + 716 x 4
Answer:
\(= 716 \times (6 + 4)\)
\(= 716 \times 10\)
\(= 7160\)
In simple words: Factor out 716 and add 6 and 4 to make 10.

v) 8062 x 169 - 8062 x 69
Answer:
\(= 8062 \times (169 - 69)\)
\(= 8062 \times 100\)
\(= 8,06,200\)
In simple words: Factor out 8062 and subtract 69 from 169.

Exam Tip: Spotting common terms saves extensive calculation time.

5. A teacher purchases 42 Mathematics books and 42 English books for his class. If the cost of a Mathematics book is Rs 52 and the cost of an English book is Rs 48. Find the total amount paid by the teacher to the shopkeeper.
Answer:
Number of Math books = 42, Cost of 1 Math book = Rs 52
Number of English books = 42, Cost of 1 English book = Rs 48
Total amount = \((42 \times 52) + (42 \times 48)\)
\(= 42 \times (52 + 48)\)
\(= 42 \times 100\)
\(= \text{Rs } 4200\)
In simple words: Use distributive property to multiply 42 by the sum of both book prices (52 + 48).

Exam Tip: Factoring out common quantities like 42 simplifies word problems greatly.

6. If the cost of a pack of mango drink is Rs 14. Then how many packs of the drink can be purchased for Rs 76 and what is the balance ?
Answer:
Total money available = Rs 76
Cost of 1 pack = Rs 14
Number of packs purchased = \(76 \div 14\) = 5 packs (with a remainder)
Cost of 5 packs = \(5 \times 14 = \text{Rs } 70\)
Balance amount = \(76 - 70 = \text{Rs } 6\)
In simple words: Divide total money by the cost per pack to get the quantity, and find the remainder for the balance.

Exam Tip: Division gives the quotient as quantity and the remainder as balance money.

Chapter 3 - Playing With Numbers

1. Fill in the blanks:
a. _______________________ is a factor of every number.
Answer: 1
In simple words: 1 divides every number completely.

Exam Tip: 1 is a universal factor.

b. The factor of a prime number is _____________________ and ______________.
Answer: 1, itself
In simple words: Prime numbers have exactly two distinct factors: 1 and the number itself.

c. A number which has more than two factor is called _____________________.
Answer: a composite number
In simple words: Numbers with 3 or more factors are composite.

Exam Tip: Composite numbers have factors other than 1 and themselves.

d. The smallest perfect number is _____________________.
Answer: 6
In simple words: Factors of 6 are 1, 2, 3 and their sum (\(1 + 2 + 3 = 6\)).

Exam Tip: Next perfect numbers are 28 and 496.

e. If a number ends with 0, it is divisible by _______________________.
Answer: 2, 5, and 10
In simple words: Numbers ending in 0 are multiples of 10, and hence divisible by 2, 5, and 10.

Exam Tip: Divisibility rules for 10 apply whenever the units digit is 0.

f. The sum of all the factors of a perfect number is equal to _______________ the number.
Answer: twice
In simple words: For 6, factors are 1, 2, 3, 6 and their sum is 12, which is twice of 6.

Exam Tip: Alternative definition of a perfect number.

g. ______________________________ is neither prime nor composite.
Answer: 1
In simple words: 1 has only one factor (itself), so it doesn't fit prime or composite definitions.

Exam Tip: 1 is a unique number in number theory.

h. A number is divisible by 6, if it is divisible by both ________ and _______________.
Answer: 2, 3
In simple words: 6 is the product of co-prime numbers 2 and 3.

Exam Tip: Test divisibility by 6 by checking both evenness and sum of digits divisible by 3.

i. The smallest even numbers is _______ and the smallest odd numbers is a _________.
Answer: 2, 1
In simple words: 2 is the first even whole/natural number and 1 is the first odd number.

Exam Tip: 0 is even, but for natural numbers 2 is the smallest even number.

j. Sum of any two even numbers is _______________________.
Answer: even
In simple words: Even + Even = Even.

Exam Tip: Basic property of even numbers.

k. Sum of two odd numbers is ______________________.
Answer: even
In simple words: Odd + Odd = Even.

Exam Tip: Sum of two odd numbers always yields an even number.

l. The only one even prime is __________________________.
Answer: 2
In simple words: All other even numbers are divisible by 2, making them composite.

Exam Tip: 2 is the only even prime number.

m. The greatest two digit prime number is __________________________.
Answer: 97
In simple words: Checking downwards from 99, 98 is even, but 97 has no factors except 1 and 97.

Exam Tip: Memorize prime numbers up to 100.

n. The smallest two digit prime numbers is _____________________________.
Answer: 11
In simple words: 10 is composite, 11 is prime.

Exam Tip: 11 is the first two-digit prime.

o. The difference between two twin prime is ________________________.
Answer: 2
In simple words: Twin primes are prime numbers that differ by 2.

Exam Tip: Example: (3, 5), (11, 13).

p. A prime number has only __________________ factors.
Answer: two
In simple words: Exactly 1 and the number itself.

q. ____________________ is the unique number.
Answer: 1
In simple words: 1 is neither prime nor composite.

Exam Tip: Frequently asked conceptual question.

r. The smallest digit in the blank space of _____9853. So that the number so formed is divisible by 3.
Answer: 2
In simple words: Sum of digits of 9853 is \(9 + 8 + 5 + 3 = 25\). Adding 2 gives 27, which is divisible by 3.

Exam Tip: Sum of digits must be a multiple of 3 for divisibility by 3.

s. The L.C.M of two numbers in which one is a factor of the other is _______________opharmaceuticals.
Answer: the larger number
In simple words: If a is a factor of b, their LCM is b.

Exam Tip: Example: LCM of 4 and 12 is 12.

t. The L.C.M of two co-prime numbers ________________________.
Answer: is their product
In simple words: Co-prime numbers have no common factor other than 1, so their LCM is their multiplication.

Exam Tip: LCM of (3, 5) is \(3 \times 5 = 15\).

u. The smallest factor of 856 is __________________________.
Answer: 1
In simple words: 1 is a factor of every number.

Exam Tip: 1 is always the smallest factor.

v. The smallest multiple of 856 is _____________________.
Answer: 856
In simple words: The smallest multiple of any number is the number itself.

Exam Tip: Number multiplied by 1 gives itself.

w. The greatest factor of 856 is _____________________.
Answer: 856
In simple words: The greatest factor of a number is the number itself.

Exam Tip: Factors are finite and bounded by the number.

x. The perfect numbers below 100 are ________________ and _________________.
Answer: 6, 28
In simple words: The only perfect numbers under 100 are 6 and 28.

Exam Tip: Memorize 6, 28, and 496 as the first three perfect numbers.

y. The smallest prime number is ______________________________.
Answer: 2
In simple words: 2 is the only even prime and the smallest prime.

Exam Tip: Never say 1 is prime.

z. The smallest composite number is ____________________________.
Answer: 4
In simple words: 1 is neither, 2 is prime, 3 is prime, 4 has factors 1, 2, 4.

Exam Tip: 4 is the first composite number.

2. Do the following :
a. Express the smallest 5-digit number in the form of prime factor.
Answer:
Smallest 5-digit number = 10,000
Prime factorization of 10,000:
\(10,000 = 10^4 = (2 \times 5)^4 = 2^4 \times 5^4 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5\)
In simple words: Break 10,000 down into prime factors 2 and 5.

Exam Tip: Use factor tree or repeated division by prime numbers.

b. Determine if 9130 is divisible by 110.
Answer:
\(110 = 11 \times 10\)
1. Since 9130 ends in 0, it is divisible by 10.
2. Check divisibility of 9130 by 11: Sum of digits at odd places from right = \(0 + 1 = 1\). Sum of digits at even places = \(3 + 9 = 12\). Difference = \(12 - 1 = 11\), which is divisible by 11. Thus 9130 is divisible by 11.
Since 9130 is divisible by both 10 and 11, it is divisible by 110.
In simple words: Check if the number is divisible by both 10 and 11.

Exam Tip: For composite divisors, check divisibility by co-prime factors.

c. Using divisibility test check whether the following are divisible by 2, 3, 4, 5, 6, 8, 9, 10 and 11
(a) 91800 (b) 31956 (c) 81615 (d) 61042 (e) 48400 (f) 99909
Answer:
- (a) 91800:
Divisible by 2 (ends in 0)
Divisible by 3 (sum of digits = 18, div by 3)
Divisible by 4 (last two digits 00)
Divisible by 5 (ends in 0)
Divisible by 6 (divisible by both 2 and 3)
Divisible by 8 (last 3 digits 800 is div by 8)
Divisible by 9 (sum of digits 18 is div by 9)
Divisible by 10 (ends in 0)
Divisible by 11 (Alt sum difference = 0)
- (b) 31956: Divisible by 2, 3, 4, 6, 9 (Not by 5, 8, 10, 11)
- (c) 81615: Divisible by 3, 5, 9 (Not by 2, 4, 6, 8, 10, 11)
- (d) 61042: Divisible by 2 (Not by 3, 4, 5, 6, 8, 9, 10, 11)
- (e) 48400: Divisible by 2, 4, 5, 8, 10, 11 (Not by 3, 6, 9)
- (f) 99909: Divisible by 3, 9, 11 (Not by 2, 4, 5, 6, 8, 10)
In simple words: Apply standard divisibility rules for each number step-by-step.

Exam Tip: Master rules for 2, 3, 5, 9, and 11 as they cover most test cases.

d. Write all the twin primes below 100.
Answer:
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)
In simple words: Pairs of prime numbers with a difference of 2.

Exam Tip: Remember there are 8 pairs of twin primes below 100.

e. Write all the prime numbers below 70.
Answer:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67
In simple words: Numbers less than 70 having only two factors (1 and themselves).

Exam Tip: Practice writing primes using Eratosthenes sieve method.

f. Find the smallest number when divided by 28, 40 and 44 leave a remainder 8 in each case.
Answer:
First, find LCM of 28, 40, and 44.
\(28 = 2^2 \times 7\)
\(40 = 2^3 \times 5\)
\(44 = 2^2 \times 11\)
LCM = \(2^3 \times 5 \times 7 \times 11 = 8 \times 5 \times 7 \times 11 = 3080\)
Required number = LCM + remainder = \(3080 + 8 = 3088\)
In simple words: Find the LCM of the divisors and add the common remainder.

Exam Tip: Whenever a common remainder is required, add it to the LCM.

g. Write two prime numbers whose sum is 100.
Answer:
3 and 97 (since \(3 + 97 = 100\), other pairs like 11 and 89, 17 and 83, 29 and 71, 41 and 59 also work).
In simple words: Find any two prime numbers that add up to 100.

Exam Tip: Goldbach's conjecture states every even number greater than 2 is the sum of two primes.

h. Write three pairs of prime numbers whose sum is an odd number.
Answer:
Since the sum of two primes is odd, one prime must be even (which is 2) and the other must be odd.
Pairs: (2, 3), (2, 5), (2, 11)
In simple words: Pair the even prime 2 with any odd prime.

Exam Tip: Sum of two odd numbers is even; hence an odd sum requires one even prime (2).

i. Find the smallest four digit number which is exactly divisible by 12, 16, 24 and 36.
Answer:
Find LCM of 12, 16, 24, and 36.
\(12 = 2^2 \times 3\)
\(16 = 2^4\)
\(24 = 2^3 \times 3\)
\(36 = 2^2 \times 3^2\)
LCM = \(2^4 \times 3^2 = 16 \times 9 = 144\)
Smallest 4-digit number = 1000
Divide 1000 by 144: \(1000 \div 144 = 6\) with remainder 136.
Next multiple = \(144 \times 7 = 1008\)
In simple words: Find the LCM, then find the smallest multiple of this LCM that has 4 digits.

Exam Tip: Always check multiples of the LCM to find the required digit-length number.

j. Write all the composite numbers between 30 and 50.
Answer:
32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49
In simple words: Numbers between 30 and 50 that have more than two factors.

Exam Tip: Exclude primes like 31, 37, 41, 43, 47 from the range.

k. The length, breadth and height of a room are 8m25cm, 6m75cm and 4m50cm respectively.
l. Determine the longest tape which can measure the three dimension of the room exactly.
Answer:
Length = 825 cm
Breadth = 675 cm
Height = 450 cm
The length of the longest tape is the HCF of 825, 675, and 450.
Prime factorization:
\(825 = 3 \times 5^2 \times 11\)
\(675 = 3^3 \times 5^2\)
\(450 = 2 \times 3^2 \times 5^2\)
HCF = \(3^1 \times 5^2 = 3 \times 25 = 75 \text{ cm}\) (or 0.75 m)
In simple words: Find the HCF of the three dimensions converted into centimeters.

Exam Tip: 'Longest tape' or 'greatest capacity' problems always require finding the HCF.

Chapter 4 - Basic Geometrical Ideas

1. (a) Name all the different angles shown in the figures :
(b) Count the number of angles.
(c) List the acute angles
(d) List the obtuse angles
(e) Identify the right angles and straight angles.
Answer:
(a) Angles formed by intersecting rays and lines in the given diagrams.
(b) Total count depends on the sub-parts of the figure (typically 4 to 6 main angles per figure).
(c) Acute angles: Angles measuring less than 90 degrees.
(d) Obtuse angles: Angles measuring greater than 90 and less than 180 degrees.
(e) Right angles = 90 degrees, Straight angles = 180 degrees.
In simple words: Measure or identify angles based on their opening size compared to a right angle.

Exam Tip: Use a protractor or visual geometry rules to classify angles correctly.

2. List all the points which are in the exterior and interior of the given angle
Answer:
Interior points: Points lying inside the arms of the angle (e.g., r, m depending on the diagram).
Exterior points: Points lying outside the region enclosed by the arms (e.g., q, k, p, x).
In simple words: Interior points are inside the V-shape, exterior points are outside.

Exam Tip: Points lying directly on the ray arms are considered on the boundary, neither strictly interior nor exterior.

3. Draw a rough sketch of a quadrilateral PQRS state
a) two pairs of opposite sides.
b) two pairs of opposite angles.
c) four pairs of adjacent angles
d) four pairs of adjacent sides.
e) Draw the diagonals and name them.
Answer:
a) Opposite sides: PQ and SR, PS and QR.
b) Opposite angles: Angle P and Angle R, Angle Q and Angle S.
c) Adjacent angles: (Angle P, Angle Q), (Angle Q, Angle R), (Angle R, Angle S), (Angle S, Angle P).
d) Adjacent sides: (PQ, QR), (QR, RS), (RS, SP), (SP, PQ).
e) Diagonals: PR and QS.
In simple words: Opposite sides/angles face each other; adjacent sides/angles share a common vertex or side.

Exam Tip: Label vertices in cyclic order (P-Q-R-S) to easily identify sides and diagonals.

4. a) Identify the triangles in the figure :
b) Write the names of angles
c) Write the names of line segments.
Answer:
a) Triangles formed by vertices in the polygon/figure (e.g., Triangle ABC, Triangle ABD, etc.).
b) Angles: Angle CAB, Angle ABC, Angle BCA, etc.
c) Line segments: AB, BC, CD, DA, AC, etc.
In simple words: List every 3-sided closed figure as a triangle and identify their bounding segments and corner angles.

Exam Tip: Use 3 letters to name an angle with the vertex letter always in the middle.

5. Draw any circle and mark
(a) its centre
(b) three radii
(c) a diameter
(d) shade a minor sector
(e) colour a minor arc
(f) a chord
(g) two points in its interior
(h) two points in its exterior
(i) three points on the circle
Answer:
(a) Centre: Point O.
(b) Radii: OA, OB, OC.
(c) Diameter: AB (passing through centre O).
(d) Minor sector: Region bounded by two radii and an arc (shaded).
(e) Minor arc: Smaller curved boundary part between two points on the circle.
(f) Chord: Line segment joining any two points on the circle (e.g., DE).
(g) Interior points: P, Q.
(h) Exterior points: X, Y.
(i) Points on circle: A, B, C.
In simple words: Draw a circle using a compass and label each requested part clearly.

Exam Tip: Ensure the diameter passes strictly through the center.

Chapter 5 - Understanding Elementary Shapes

1. Given below are the lengths of the sides of triangles. Classify them as equilateral, isosceles or scalene
a) 6cm, 2.4cm, 6cm b) 7cm, 9cm, 5.5cm c) 5.4cm, 7cm, 6.1cm
d) 7.2cm, 7.2cm, 7.2cm e) 10.1cm, 8.6cm, 8.6cm f) 3.5cm, 4.5cm, 5.1cm
g) 4.8cm, 4.8cm, 4.8cm h) 6.8cm, 6.8cm, 8.6cm i) 3cm, 4cm, 5cm
Answer:
a) Isosceles (two equal sides)
b) Scalene (all unequal sides)
c) Scalene (all unequal sides)
d) Equilateral (all equal sides)
e) Isosceles (two equal sides)
f) Scalene (all unequal sides)
g) Equilateral (all equal sides)
h) Isosceles (two equal sides)
i) Scalene (all unequal sides)
In simple words: Equilateral = 3 equal sides, Isosceles = 2 equal sides, Scalene = 0 equal sides.

Exam Tip: Check the number of matching side lengths to classify.

2. Given below are the measures of the angles of some triangles. Classify them as acute-angled, obtuse-angled or right-angled.
i) \(60^\circ, 90^\circ, 30^\circ\) ii) \(40^\circ, 100^\circ, 40^\circ\) iii) \(60^\circ, 60^\circ, 60^\circ\)
iv) \(20^\circ, 40^\circ, 120^\circ\) v) \(50^\circ, 60^\circ, 70^\circ\) vi) \(45^\circ, 45^\circ, 90^\circ\)
Answer:
i) Right-angled (contains a \(90^\circ\) angle)
ii) Obtuse-angled (contains an angle greater than \(90^\circ\))
iii) Acute-angled (all angles less than \(90^\circ\))
iv) Obtuse-angled (contains \(120^\circ\))
v) Acute-angled (all angles less than \(90^\circ\))
vi) Right-angled (contains \(90^\circ\))
In simple words: Classification depends on the largest angle in the triangle.

3. Write the number of sides of the following Polygons: Triangles, Pentagon, Quadrilateral, Heptagon, Hexagon, Nonagon, Octogon, Decagon.
Answer:
- Triangle: 3 sides
- Pentagon: 5 sides
- Quadrilateral: 4 sides
- Heptagon: 7 sides
- Hexagon: 6 sides
- Nonagon: 9 sides
- Octagon: 8 sides
- Decagon: 10 sides
In simple words: Match polygon names with their corresponding number of sides.

Exam Tip: Learn Greek numerical prefixes (penta=5, hexa=6, hepta=7, octa=8, nona=9, deca=10).

4. Fill in:
[Table: Name of Solid | Faces | Edges | Vertices]
Answer:
- Cube: Faces = 6, Edges = 12, Vertices = 8
- Cuboid: Faces = 6, Edges = 12, Vertices = 8
- Square Pyramid: Faces = 5, Edges = 8, Vertices = 5
- Triangular Pyramid: Faces = 4, Edges = 6, Vertices = 4
- Triangular Prism: Faces = 5, Edges = 9, Vertices = 6
In simple words: Count flat faces, straight edges, and corner vertices for each 3D shape.

Exam Tip: Verify using Euler's formula: Faces + Vertices - Edges = 2.

5. Write yes or No
[Table: Quadrilateral properties for Trapezium, Parallelogram, Rhombus, Rectangle, Square]
Answer:
- Trapezium: Opposite Sides Parallel = Yes (one pair), Equal = No, All Sides Equal = No, Opposite Angles Equal = No, All Angles Equal = No, Diagonals Equal = No, Perpendicular = No, Bisect = No.
- Parallelogram: Opposite Sides Parallel = Yes, Equal = Yes, All Sides Equal = No, Opposite Angles Equal = Yes, All Angles Equal = No, Diagonals Equal = No, Perpendicular = No, Bisect = Yes.
- Rhombus: Opposite Sides Parallel = Yes, Equal = Yes, All Sides Equal = Yes, Opposite Angles Equal = Yes, All Angles Equal = No, Diagonals Equal = No, Perpendicular = Yes, Bisect = Yes.
- Rectangle: Opposite Sides Parallel = Yes, Equal = Yes, All Sides Equal = No, Opposite Angles Equal = Yes, All Angles Equal = Yes, Diagonals Equal = Yes, Perpendicular = No, Bisect = Yes.
- Square: Opposite Sides Parallel = Yes, Equal = Yes, All Sides Equal = Yes, Opposite Angles Equal = Yes, All Angles Equal = Yes, Diagonals Equal = Yes, Perpendicular = Yes, Bisect = Yes.
In simple words: Check geometric properties for each special quadrilateral.

Exam Tip: A square is a special type of rectangle, rhombus, and parallelogram.

6. Name each of the following 3-D shapes. Write number of Curved Surfaces and no of flat faces.
Answer:
i) Cylinder: 1 curved surface, 2 flat faces
ii) Cone: 1 curved surface, 1 flat face
iii) Sphere / Hemisphere: Hemisphere has 1 curved surface, 1 flat face
In simple words: Identify the 3D solid and count curved vs flat surfaces.

Exam Tip: A cylinder has two circular flat faces and one curved rectangular wrap.

7. Fill in the blanks :
a. An angle whose measure is greater than that of a right angle is ____________
Answer: obtuse angle (or reflex angle)
In simple words: Angles bigger than 90 degrees are obtuse or reflex.

Exam Tip: Between 90 and 180 is obtuse; greater than 180 is reflex.

b. Three edges meet at a point called a ________________________.
Answer: vertex
In simple words: Corners where edges meet are vertices.

Exam Tip: In 3D shapes, vertices are corner points.

c. A ___________________________ is larger than a straight angle.
Answer: reflex angle
In simple words: Reflex angles measure between 180 and 360 degrees.

d. A Polygon with 5 sides is called a ___________________________.
Answer: pentagon
In simple words: 5-sided polygon is a pentagon.

Exam Tip: 'Penta' means five.

e. A triangle having all three unequal sides is called a ____________________.
Answer: scalene triangle
In simple words: Triangles with different side lengths are scalene.

Exam Tip: Scalene triangles have no equal sides and no equal angles.

8. Name the types of following triangles
a) \(\triangle LMN\) with \(m\angle L = 80^\circ, m\angle M = 70^\circ, m\angle N = 30^\circ\)
b) \(\triangle ABC\) with \(m\angle A = 90^\circ\)
c) \(\triangle PQR\) such that \(PQ = QR = PR = 8\text{ cm}\)
d) \(\triangle XYZ\) with \(AB = 8\text{ cm}, BC = 5\text{ cm}, CA = 5\text{ cm}\)
e) Triangle with lengths of sides \(7\text{ cm}, 8\text{ cm}\) and \(9\text{ cm}\)
f) \(\triangle PQR\) with \(m\angle Q = 90^\circ\) and \(PQ = QR\)
Answer:
a) Acute-angled triangle (all angles less than \(90^\circ\))
b) Right-angled triangle (contains a \(90^\circ\) angle)
c) Equilateral triangle (all sides are equal)
d) Isosceles triangle (two sides are equal)
e) Scalene triangle (all sides are unequal)
f) Right-angled isosceles triangle (contains \(90^\circ\) and two equal sides)
In simple words: Classify based on given side lengths or angle measures.

Exam Tip: Check both angle measures and side lengths when identifying triangle types.

9. Classify each one of the following angles as right, straight, acute, obtuse or reflex.
Answer:
- Acute angle: Between \(0^\circ\) and \(90^\circ\)
- Right angle: Exactly \(90^\circ\)
- Obtuse angle: Between \(90^\circ\) and \(180^\circ\)
- Straight angle: Exactly \(180^\circ\)
- Reflex angle: Between \(180^\circ\) and \(360^\circ\)
In simple words: Identify the angle type based on its visual opening.

Exam Tip: Use a protractor for precise measurement if needed.

10. Find the angle measure between the hands of the clock in each figure
Answer:
- Clock a) 3 o'clock: \(90^\circ\) (right angle)
- Clock b) 5 o'clock: Each hour space is \(30^\circ\), so \(5 \times 30^\circ = 150^\circ\) (obtuse angle)
- Clock c) 6 o'clock: \(180^\circ\) (straight angle)
In simple words: Each hour interval on a clock face represents \(30^\circ\) (\(360^\circ \div 12\)).

Exam Tip: Multiply the number of hour spaces between hands by \(30^\circ\).

11. Name each polygon
Answer:
a) Pentagon (5 sides)
b) Triangle (3 sides)
c) Quadrilateral (4 sides)
d) Hexagon (6 sides)
e) Quadrilateral / Trapezium (4 sides)
In simple words: Count the number of straight sides to name the polygon.

Exam Tip: Polygons are closed figures made entirely of line segments.

12.
a) A cuboid has ______________________ faces.
b) Each face has ______________________ edges.
c) Each face has _________________ vertices.
Answer:
a) 6 faces
b) 4 edges
c) 4 vertices
In simple words: A cuboid has 6 rectangular faces, and each rectangular face has 4 sides (edges) and 4 corners (vertices).

Exam Tip: Pay attention to whether the question asks about the whole solid or each individual face.

Chapter 6 - Integers

1. Fill in the blanks:
a. -5 + (-11) = ______________
Answer: -16
In simple words: Add two negative numbers and keep the negative sign (\(-(5 + 11)\)).

Exam Tip: Same signs add up and retain the common sign.

b. 8 + (-6) = ____________
Answer: 2
In simple words: Subtract 6 from 8 since they have opposite signs.

Exam Tip: Take the sign of the number with the larger absolute value.

c. (-26) + (-37) = ___________
Answer: -63
In simple words: Add 26 and 37 and make the result negative.

Exam Tip: Sum of two negative integers is always negative.

d. Write the greatest negative integer _____________
Answer: -1
In simple words: -1 is closest to zero on the number line, making it the largest negative number.

Exam Tip: As negative numbers increase in magnitude, their value decreases.

e. Write all integers between -30 and -20 ___________
Answer: -29, -28, -27, -26, -25, -24, -23, -22, -21
In simple words: List integers strictly between -30 and -20.

Exam Tip: Do not include -30 and -20 in a 'between' range.

f. Find the sum of -45 and 30
Answer:
\(-45 + 30 = -(45 - 30) = -15\)
In simple words: Subtract 30 from 45 and give it a negative sign because 45 is larger.

Exam Tip: Difference of two integers with opposite signs takes the sign of the larger absolute value.

g. Which is greater : -65 or -56 ?
Answer: -56
In simple words: -56 is to the right of -65 on the number line, so it is greater.

Exam Tip: For negative numbers, the one with the smaller digits is greater.

h. Which integer is neither positive nor negative ?
Answer: 0
In simple words: Zero is neutral; it has no sign.

Exam Tip: 0 is an integer, but neither positive nor negative.

2. Draw a number line and answer the following :
a. Which number will we reach if we move 4 numbers to the right of -2 ?
Answer: 2
In simple words: Start at -2 and move 4 units right: \(-2 + 4 = 2\).

Exam Tip: Moving right on the number line means adding.

b. If we are at -6 on the number line, in which direction should we move to reach -1 ?
Answer: Right (by 5 units)
In simple words: Since -1 is greater than -6, move to the right.

Exam Tip: Moving towards larger numbers means moving right.

c. Using the number line write the integer
i) 4 less than -1
ii) 6 more than -6
Answer:
i) \(-1 - 4 = -5\)
ii) \(-6 + 6 = 0\)
In simple words: 'Less than' means move left, 'more than' means move right.

Exam Tip: Translate words like 'less than' to minus and 'more than' to plus.

d. Use number line and add the following integers :
i) (-1) + (-8)
ii) (-1) + (-2) + (-4)
iii) -8 - (-10)
Answer:
i) \(-1 - 8 = -9\)
ii) \(-1 - 2 - 4 = -7\)
iii) \(-8 + 10 = 2\)
In simple words: Combine negatives by adding magnitudes; subtracting a negative is the same as adding a positive.

Exam Tip: Two minus signs side by side become a plus (\(-(-10) = +10\)).

e. Fill in the blanks with >, < or =
i) 54 - (-11) _________ 57 + (-4)
ii) (-35) - (-52) _________ (-52) - (-35)
Answer:
i) \(54 - (-11) = 54 + 11 = 65\); \(57 + (-4) = 53\). So, \(65 > 53\) (>)
ii) \((-35) - (-52) = -35 + 52 = 17\); \((-52) - (-35) = -52 + 35 = -17\). So, \(17 > -17\) (>)
In simple words: Simplify both sides first, then compare the resulting integers.

Exam Tip: Always simplify double negatives to a single positive sign before comparing.

6. Find:
i) 60 - (-20) - (+10)
ii) (-15) + 12 - 9 + 1
Answer:
i) \(60 - (-20) - (+10) = 60 + 20 - 10 = 80 - 10 = 70\)
ii) \((-15) + 12 - 9 + 1 = -3 - 9 + 1 = -12 + 1 = -11\)
In simple words: Open brackets carefully and perform addition and subtraction from left to right.

Exam Tip: Pay close attention to signs when opening brackets.