🌈⚡Key To Enjoy Learning Maths☘🌈

Saturday, September 9, 2023

QUESTION BANK CLASS 8 LINEAR EQUATIONS WITH ONE VARIABLES

An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left Hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS).
In an equation the values of the expressions on the LHS and RHS are equal. This happens to be true only for certain values of the variable. These values are the solutions of the equation.

EXAMPLE

Solve 2x – 3 = x + 2

EXERCISE 2.1

Solve the following equations and check your results

1. 3x = 2x + 18

2. 5t – 3 = 3t – 5

3. 5x + 9 = 5 + 3x

4. 4z + 3 = 6 + 2z

5. 2x – 1 = 14 – x

6. 8x + 4 = 3 (x – 1) + 7

EXERCISE 2.2

Solve the following linear equations.

Simplify and solve the following linear equations.

7. 3(t – 3) = 5(2t + 1)

8. 15(y – 4) –2(y – 9) + 5(y + 6) = 0

9. 3(5z – 7) – 2(9z – 11) = 4(8z – 13) – 17

10. 0.25(4f – 3) = 0.05(10f – 9)

POINTS TO REMEMBER

1. An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.

2. The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.

3. An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.

4. Just as numbers, variables can, also, be transposed from one side of the equation to the other.

5. Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.

6. The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved using linear equations.

QUESTION BANK CLASS 8 RATIONAL NUMBERS

EXERCISE 1.1

1. Name the property under multiplication used in each of the following

2. Tell what property allows you to compute

3. The product of two rational numbers is always a ___.

POINTS TO REMEMBER

A number which can be written in the form p q , where p and q are integers and q ≠ 0 is called a rational number
rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number.
rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number.
rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a × b is also a rational number.
for any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division.
two rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b, a + b = b + a.
subtraction is not commutative for integers and integers are also rational numbers. So, subtraction will not be commutative for rational numbers too.
multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b.
division is not commutative for rational numbers.
addition is associative for rational numbers. That is, for any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
Subtraction is not associative for rational numbers.
multiplication is associative for rational numbers. That is for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.
division is not associative for rational numbers.
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.
a × 1 = 1 × a = a for any rational number a
1 is the multiplicative identity for rational numbers.
Distributivity of Multiplication over Addition and Subtraction. For all rational numbers a, b and c, a (b + c) = ab + ac a (b – c) = ab – ac

1. Rational numbers are closed under the operations of addition, subtraction and multiplication.

2. The operations addition and multiplication are (i) commutative for rational numbers. (ii) associative for rational numbers.

3. The rational number 0 is the additive identity for rational numbers.

4. The rational number 1 is the multiplicative identity for rational numbers.

5. Distributivity of rational numbers:

For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac

6. Between any two given rational numbers there are countless rational numbers.

The idea of mean helps us to find rational numbers between two rational numbers.

Monday, September 4, 2023

Class 07 Integers Beautiful butterfly

INTEGERS

ACTIVITY – 1. Make the butterfly beautiful

DATE:

AIM / Objective:

To find the value of integers,

MATERIALS REQUIRED:

Pencil, scale, eraser

PREREQUISITE KNOWLEDGE:

Concept of INTEGERS

PROCEDURE:

Description:

Draw butterfly as shown in sample.

1. Use red colour for negative integer and green colour for positive integer.

2. For every integer, use one geometrical figure.

3. Complete the design of butterfly wings as shown in example.

LEARNING ASSESSMENT:

Complete the design of butterfly wings

SOLUTION:

Friday, September 1, 2023

QUESTION BANK CLASS 6 PLAYING WITH NUMBERS

______ is neither Prime nor Composite.

a) 0
b) 1
c) 2
d) 4

Which of the following is the smallest composite number?

a) 2

b) 3

c) 4

d) 5

Which of the following is divisible by 5?

a) 222

b) 453

c) 400

d) 528

The smallest composite number is _______________.

a) 1
b) 4
c) 2
d) 0

Which of the following is a prime number?

a) 12
b) 37
c) 81
d) 49

State –True or false All prime numbers are odd.
State –True or false All prime numbers are odd.
State –True or false The product of two even numbers is always even.
State –True or false All prime numbers are odd.
Fill in the Blanks: The smallest composite number is ______.
The smallest composite number is ______.
The greatest negative integer is ______.
A number which has only two factors is called a _______________.
LCM of 9 and 4 is _______
Fill in the Blanks:
Write the smallest digit in the blank space of number 4765_ 2 so that the number formed is divisible by 3.
Write first five multiples of 8.
What is the greatest prime number between 1 and 20?
Write first three multiples of 11.
A number having only two factors is called _________ numbers
) Find the common factors of 20 and 28.

15) Find the LCM of 20,25 and 30.
The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can be measure the three dimensions of the room exactly.
Write first five multiples of 8.
Write all the factors of 20?
Write the prime factorization of 36
Find the least number which when divided by 6, 15 and 18, leave remainder 5 in each case.
Find the least number which when divided by 6, 15 and 18, leave remainder 5 in each case. (4M)
Find the HCF of 18,54 and 81(3M)
Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.
Using divisibility tests, determine which of the following numbers are divisible by 11 and which or not:- a)943207 b)145607
Find the HCF of 12,45 and 75 (3M)

EXTRA TRY THESE QUESTIONS

Find the possible factors of 45, 30 and 36.
Write all the factors of 68.
Find the factors of 36.
Write first five multiples of 6
Observe that 2 × 3 + 1 = 7 is a prime number. Here, 1 has been added to a multiple of 2 to get a prime number. Can you find some more numbers of this type?
Write all the prime numbers less than 15.
Find the common factors of (a) 8, 20 (b) 9, 15
Find the common multiples of 3, 4 and 9.
Find the common factors of 75, 60 and 210
Write the prime factorisations of 16, 28, 38.
Find the prime factorisation of 980
Find the HCF of the following: (i) 24 and 36 (ii) 15, 25 and 30 (iii) 8 and 12 (iv) 12, 16 and 28
Find the LCM of 12 and 18.
Find the LCM of 24 and 90.
Find the LCM of 40, 48 and 45
Find the LCM of 20, 25 and 30.
Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene oil of both the tankers when used an exact number of times.
In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?
Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case.

EXERCISE 3.1

1. Write all the factors of the following numbers : (a) 24 (b) 15 (c) 21 (d) 27 (e) 12 (f) 20 (g) 18 (h) 23 (i) 36
2. Write first five multiples of : (a) 5 (b) 8 (c) 9
3. Match the items in column 1 with the items in column 2.
Column 1 - Column 2
(i) 35 - (a) Multiple of 8
(ii) 15 - (b) Multiple of 7
(iii) 16 - (c) Multiple of 70
(iv) 20 - (d) Factor of 30
(v) 25 - (e) Factor of 50
(f) Factor of 20
4. Find all the multiples of 9 upto 100.

EXERCISE 3.2

1. What is the sum of any two (a) Odd numbers? (b) Even numbers?
2. State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.
3. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers upto 100.
4. Write down separately the prime and composite numbers less than 20.
5. What is the greatest prime number between 1 and 10?
6. Express the following as the sum of two odd primes. (a) 44 (b) 36 (c) 24 (d) 18
7. Give three pairs of prime numbers whose difference is 2. [Remark : Two prime numbers whose difference is 2 are called twin primes].
8. Which of the following numbers are prime? (a) 23 (b) 51 (c) 37 (d) 26
9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
10. Express each of the following numbers as the sum of three odd primes: (a) 21 (b) 31 (c) 53 (d) 61
11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5. (Hint : 3+7 = 10)
12. Fill in the blanks :
(a) A number which has only two factors is called a ______.
(b) A number which has more than two factors is called a ______.
(c) 1 is neither ______ nor ______.
(d) The smallest prime number is ______.
(e) The smallest composite number is _____.
(f) The smallest even number is ______.

EXERCISE 3.3

1. Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10 ; by 11 (say, yes or no):

2. Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:

(a) 572

(b) 726352

(d) 6000

(e) 12159

(f) 14560

(g) 21084

(h) 31795072

(i) 1700

(j) 2150

3. Using divisibility tests, determine which of following numbers are divisible by 6:

(a) 297144

(b) 1258

(d) 61233

(e) 901352

(f) 438750

(g) 1790184

(h) 12583

(i) 639210

(j) 17852

4. Using divisibility tests, determine which of the following numbers are divisible by 11:

(a) 5445

(b) 10824

(d) 70169308

(e) 10000001

(f) 901153

5. Write the smallest digit and the greatest digit in the blank space of each of the following numbers so that the number formed is divisible by 3 : (a) __ 6724 (b) 4765 __ 2

6. Write a digit inthe blank space of each ofthefollowingnumbers so that the number formed is divisible by 11 : (a) 92 __ 389 (b) 8 __ 9484

EXERCISE 3.4

1. Find the common factors of :

(a) 20 and 28

(b) 15 and 25

(d) 56 and 120

2. Find the common factors of :

(a) 4, 8 and 12

(b) 5, 15 and 25 3.

Find first three common multiples of : (a) 6 and 8 (b) 12 and 18
4. Write all the numbers less than 100 which are common multiples of 3 and 4.
5. Which of the following numbers are co-prime?

(a) 18 and 35

(b) 15 and 37

(d) 17 and 68

(e) 216 and 215

(f) 81 and 16

6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?
7. A number is divisible by 12. By what other numbers will that number be divisible?

EXERCISE 3.5

1. Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
2. Here are two different factor trees for 60. Write the missing numbers. (a)

3. Which factors are not included in the prime factorisation of a composite number?
4. Write the greatest 4-digit number and express it in terms of its prime factors.
5. Write the smallest 5-digit number and express it in the form of its prime factors.
6. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.
7. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
8. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.
9. In which of the following expressions, prime factorisation has been done?

(a) 24 = 2 × 3 × 4

(b) 56 = 7 × 2 × 2 × 2

(d) 54 = 2 × 3 × 9

10. Determine if 25110 is divisible by 45. [Hint : 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].
18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.
12. I am the smallest number, having four different prime factors. Can you find me?

EXERCISE 3.6

1. Find the HCF of the following numbers :

(a) 18, 48

(b) 30, 42

(d) 27, 63

(e) 36, 84

(f) 34, 102

(g) 70, 105, 175

(h) 91, 112, 49

(i) 18, 54, 81

(j) 12, 45, 75

2. What is the HCF of two consecutive (a) numbers? (b) even numbers? (c) odd numbers?
3. HCF of co-prime numbers 4 and 15 was found as follows by factorisation : 4 = 2 × 2 and 15 = 3 × 5 since there is no common prime factor, so HCF of 4 and 15 is 0. Is the answer correct? If not, what is the correct HCF?

EXERCISE 3.7

1. Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertiliser exact number of times.
2. Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps?
3. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.
4. Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.
5. Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.
6. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?
7. Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.
8. Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.
9. Find the smallest 4-digit number which is divisible by 18, 24 and 32.
10. Find the LCM of the following numbers :

(a) 9 and 4

(b) 12 and 5

(d) 15 and 4 Observe a common property in the obtained LCMs. Is LCM the product of two numbers in each case?

11. Find the LCM of the following numbers in which one number is the factor of the other.

(a) 5, 20

(b) 6, 18

(d) 9, 45 What do you observe in the results obtained?

POINTS TO REMEMBER

(a) A factor of a number is an exact divisor of that number.
(b) Every number is a factor of itself. 1 is a factor of every number.
(c) Every factor of a number is less than or equal to the given number.
(d) Every number is a multiple of each of its factors.
(e) Every multiple of a given number is greater than or equal to that number.
(f) Every number is a multiple of itself.
3.(a) The number other than 1, with only factors namely 1 and the number itself, is a prime number.
Numbers that have more than two factors are called composite numbers.
Number 1 is neither prime nor composite.

(b) The number 2 is the smallest prime number and is even.
Every prime number other than 2 is odd.
(c) Two numbers with only 1 as a common factor are called co-prime numbers.
(d) If a number is divisible by another number then it is divisible by each of the factors of that number.
(e) A number divisible by two co-prime numbers is divisible by their product also.
4. We have discussed how we can find just by looking at a number, whether it is divisible by small numbers 2,3,4,5,8,9 and 11.
(a) Divisibility by 2,5 and 10 can be seen by just the last digit.
(b) Divisibility by 3 and 9 is checked by finding the sum of all digits.
c) Divisibility by 4 and 8 is checked by the last 2 and 3 digits respectively.
(d) Divisibility of 11 is checked by comparing the sum of digits at odd and even places.
if two numbers are divisible by a number then their sum and difference are also divisible by that number.
(a) The Highest Common Factor (HCF) of two or more given numbers is the highest of their common factors.
(b) The Lowest Common Multiple (LCM) of two or more given numbers is the lowest of their common multiples.

🌈⚡Key To Enjoy Learning Maths☘🌈

Saturday, September 9, 2023