All questions of Mathematics (Ganita Prakash) for Class 8 Exam

Hey bot! it is c not b.

Power of 2 like 4,6,8,12,32 can cot be expressed as a sum of consecutive no.

Understanding Co-prime Numbers
Co-prime numbers, also known as relatively prime numbers, are pairs of numbers that have no common positive factors other than 1. This means their greatest common divisor (GCD) is 1.
Analysis of Each Pair
Let's evaluate each pair to determine if they are co-prime:
a) 6 and 8
- Factors of 6: 1, 2, 3, 6
- Factors of 8: 1, 2, 4, 8
- Common Factor: 2
- GCD: 2
- Conclusion: Not co-prime
b) 9 and 15
- Factors of 9: 1, 3, 9
- Factors of 15: 1, 3, 5, 15
- Common Factor: 3
- GCD: 3
- Conclusion: Not co-prime
c) 8 and 25
- Factors of 8: 1, 2, 4, 8
- Factors of 25: 1, 5, 25
- Common Factor: 1
- GCD: 1
- Conclusion: Co-prime
d) 12 and 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common Factors: 1, 2, 3, 6
- GCD: 6
- Conclusion: Not co-prime
Final Conclusion
Among the pairs listed, only 8 and 25 are co-prime. Their only common factor is 1, making them relatively prime to each other. Thus, the correct answer is option 'C'.

Understanding the Expression
To simplify the expression 8 × (25 + 15), we need to follow the order of operations.
Step 1: Solve Inside the Parentheses
- First, calculate the sum inside the parentheses:
- 25 + 15 = 40
Step 2: Multiply by 8
- Now, substitute back into the expression:
- 8 × 40
Step 3: Perform the Multiplication
- Next, carry out the multiplication:
- 8 × 40 = 320
Final Result
- The simplified expression is 320.
Conclusion
Thus, the correct answer is option 'B' which is 320.
This step-by-step approach helps in understanding how to simplify the expression accurately.

Understanding Roman Numerals
Roman numerals are a numeral system originating from ancient Rome, using combinations of letters from the Latin alphabet. The primary letters used are:
- I (1)
- V (5)
- X (10)
- L (50)
- C (100)
- D (500)
- M (1000)
Identifying M
The letter "M" in Roman numerals represents the value 1000. Therefore, to find the numeral that comes just before "M," we need to identify the numeral that equals 999.
Analyzing the Options
Let's evaluate the given options:
- DCMXCIX:
- D (500) + C (100) + M (1000) + X (10) + C (100) + I (1) + X (10) = 999 (Incorrect)
- CMXCIX:
- C (100) + M (1000) + X (10) + C (100) + I (1) + X (10) = 999 (Correct)
- CMXCV:
- C (100) + M (1000) + X (10) + C (100) + V (5) = 995 (Incorrect)
- CMIX:
- C (100) + M (1000) + I (1) + X (10) = 911 (Incorrect)
Conclusion
The correct answer is option 'B' (CMXCIX), which represents 999. Therefore, CMXCIX is indeed the numeral that comes just before M (1000) in Roman numerals.

Understanding Factorization
To factorize the expression 20x + 30y, we need to find the greatest common factor (GCF) of the terms involved.
Finding the GCF
- The coefficients are 20 and 30.
- The GCF of 20 and 30 is 10.
So, we can factor out 10 from the expression:
Factoring Out 10
- 20x + 30y = 10(2x) + 10(3y) = 10(2x + 3y)
Now, we can also express this in different ways to check for other factors:
Other Factorizations
- Option a: 5(4x + 6y)
- 4x + 6y can be factored further as 2(2x + 3y). Thus, 5(4x + 6y) = 5 * 2(2x + 3y) = 10(2x + 3y).
- Option c: 2(10x + 15y)
- 10x + 15y can also be factored as 5(2x + 3y). Therefore, 2(10x + 15y) = 2 * 5(2x + 3y) = 10(2x + 3y).
Conclusion
All of these options (5(4x + 6y), 10(2x + 3y), and 2(10x + 15y)) simplify to the same expression, 10(2x + 3y). Hence, the correct answer is option D: All of these.
This shows that the expression can be factored in multiple ways, reinforcing the concept of factorization in algebra.

Understanding the Problem
To determine which expression is always divisible by 2 when n is an integer, we analyze the four options provided.
Options Analysis
- Option A: n² + n
- This expression can be factored: n(n + 1).
- Here, n and n + 1 are two consecutive integers.
- One of these two consecutive integers is always even, making their product (n(n + 1)) always divisible by 2.
- Option B: n² + 1
- This expression is not guaranteed to be even. For example:
- If n = 1: 1² + 1 = 2 (even)
- If n = 2: 2² + 1 = 5 (odd)
- Thus, this expression is not always divisible by 2.
- Option C: n² - 1
- This expression can be factored as (n - 1)(n + 1).
- While (n - 1) and (n + 1) are consecutive integers, making their product even, it will not always be divisible by 2 depending on the parity of n.
- For instance, if n = 2: 2² - 1 = 3 (odd).
- Option D: n³
- This expression is divisible by 2 if n is even but not if n is odd.
- Example: If n = 1: 1³ = 1 (odd); if n = 2: 2³ = 8 (even).
Conclusion
The only expression that is always divisible by 2 for any integer n is:
Option A: n² + n, since it involves the product of two consecutive integers, ensuring at least one of them is always even.

Understanding the Problem
In the equation XY × 9 = ZW, both XY and ZW represent two-digit numbers. To find the valid equation among the options, we must multiply each two-digit number by 9 and check if the result is also a two-digit number.
Evaluating Each Option
- Option A: 12 × 9 = 108
- Calculation: 12 × 9 = 108
- Result: 108 is a three-digit number, so this option is invalid.
- Option B: 13 × 9 = 117
- Calculation: 13 × 9 = 117
- Result: 117 is also a three-digit number, making this option invalid as well.
- Option C: 11 × 9 = 99
- Calculation: 11 × 9 = 99
- Result: 99 is a two-digit number, which fits the criteria. Therefore, this option is valid.
- Option D: 14 × 9 = 126
- Calculation: 14 × 9 = 126
- Result: 126 is a three-digit number, thus this option is invalid.
Conclusion
The only valid equation where both products remain two-digit numbers is found in option C: 11 × 9 = 99. This confirms that option C is correct, as the result is consistent with the requirement of the original equation.

Understanding Divisibility by 11
To determine if a number is divisible by 11, you can use the divisibility rule for 11. This rule states that a number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.
Step-by-Step Analysis of Each Option
1. Option A: 1001
- Odd position digits: 1 (1st) + 0 (3rd) = 1
- Even position digits: 0 (2nd) + 1 (4th) = 1
- Difference: |1 - 1| = 0
- Since 0 is divisible by 11, 1001 is divisible by 11.
2. Option B: 1234
- Odd position digits: 1 (1st) + 3 (3rd) = 4
- Even position digits: 2 (2nd) + 4 (4th) = 6
- Difference: |4 - 6| = 2
- 2 is not divisible by 11, so 1234 is not divisible by 11.
3. Option C: 2456
- Odd position digits: 2 (1st) + 5 (3rd) = 7
- Even position digits: 4 (2nd) + 6 (4th) = 10
- Difference: |7 - 10| = 3
- 3 is not divisible by 11, so 2456 is not divisible by 11.
4. Option D: 3246
- Odd position digits: 3 (1st) + 4 (3rd) = 7
- Even position digits: 2 (2nd) + 6 (4th) = 8
- Difference: |7 - 8| = 1
- 1 is not divisible by 11, so 3246 is not divisible by 11.
Conclusion
Based on the analysis, the only number among the options that is divisible by 11 is 1001.

Understanding the Problem
To find how many expressions can be formed using + or - signs between the three consecutive numbers n, n+1, and n+2, we need to analyze the options available for the signs.
Available Signs
1. We have three numbers: n, n+1, n+2.
2. We can place either a + or a - sign between each pair of numbers.
Setting Up the Expressions
- There are two gaps between the three numbers:
- Between n and n+1
- Between n+1 and n+2
Calculating Combinations
- For each gap, we have two choices:
- First gap (between n and n+1): either + or -
- Second gap (between n+1 and n+2): either + or -
This means we can calculate the total combinations of signs as follows:
- For the first gap: 2 options (either + or -)
- For the second gap: 2 options (either + or -)
Total Expressions
- Using the multiplication principle, the total number of expressions can be calculated as:
Total expressions = 2 (first gap) * 2 (second gap) = 4
Conclusion
Thus, the total number of distinct expressions that can be formed using + or – signs between the three consecutive numbers n, n+1, and n+2 is 4.
The correct answer is option A.

Understanding the Problem
To simplify the expression (12 + 18) × 7, we will use the distributive property. The distributive property states that a(b + c) = ab + ac. In this case, we can first simplify the addition inside the parentheses.
Step 1: Simplify the Addition
- Calculate 12 + 18:
- 12 + 18 = 30
So, we can rewrite the expression as:
- (12 + 18) × 7 = 30 × 7
Step 2: Multiply
Now we need to multiply 30 by 7:
- 30 × 7 = 210
Final Result
Thus, the simplified expression is 210. Since the question states options, we can conclude:
- The correct answer is option 'A': 210
Why the Other Options are Incorrect
- Option B (2100): This is incorrect because it is 10 times larger than the actual result.
- Option C (252): This does not relate to any calculation in the expression.
- Option D (420): This is double the correct answer, likely due to a multiplication error.
Conclusion
Using the distributive property to simplify the expression (12 + 18) × 7 leads us to the correct answer of 210, confirming that option 'A' is accurate. Understanding each step helps in mastering such problems!

Understanding the Base 7 Number System
In a base 7 number system, the digits used are 0, 1, 2, 3, 4, 5, and 6. Each place value represents a power of 7.
Identifying Landmark Numbers
Landmark numbers are significant values in a number system that help in understanding the progression of numbers.
1. First Landmark after 1:
- The next number is 2 (1 + 1).
2. Second Landmark after 1:
- The next number is 3 (1 + 2).
3. Third Landmark after 1:
- The next number is 4 (1 + 3).
4. Fourth Landmark after 1:
- The next number is 5 (1 + 4).
5. Fifth Landmark after 1:
- The next number is 6 (1 + 5).
6. Sixth Landmark after 1:
- The seventh number is 10 (which is 7 in decimal).
7. Continuing the Sequence:
- 11 (8 in decimal), 12 (9 in decimal), 13 (10 in decimal), 14 (11 in decimal), 15 (12 in decimal), 16 (13 in decimal), 20 (14 in decimal), 21 (15 in decimal), 22 (16 in decimal), 23 (17 in decimal), 24 (18 in decimal), 25 (19 in decimal), 26 (20 in decimal), 30 (21 in decimal), 31 (22 in decimal), 32 (23 in decimal), 33 (24 in decimal), 34 (25 in decimal), 35 (26 in decimal), 36 (27 in decimal), 40 (28 in decimal), 41 (29 in decimal), 42 (30 in decimal), 43 (31 in decimal), 44 (32 in decimal), 45 (33 in decimal), 46 (34 in decimal), 47 (35 in decimal), 50 (36 in decimal), 51 (37 in decimal), 52 (38 in decimal), 53 (39 in decimal), 54 (40 in decimal), 55 (41 in decimal), 56 (42 in decimal), 60 (43 in decimal), 61 (44 in decimal), 62 (45 in decimal), 63 (46 in decimal), 64 (47 in decimal), 65 (48 in decimal), 66 (49 in decimal), 100 (50 in decimal), 101 (51 in decimal), 102 (52 in decimal), 103 (53 in decimal), 104 (54 in decimal), 105 (55 in decimal), 106 (56 in decimal), 110 (57 in decimal), 111 (58 in decimal), 112 (59 in decimal), 113 (60 in decimal), 114 (61 in decimal), 115 (62 in decimal), 116 (63 in decimal), 120 (64 in decimal), 121 (65 in decimal), 122 (66 in decimal), 123 (67 in decimal), 124 (68 in decimal), 125 (69 in decimal), 126 (70 in decimal), 130 (71 in decimal), 131

Understanding Roman Numerals
Roman numerals are a numerical system used in ancient Rome, combining letters from the Latin alphabet. The basic symbols include:

• I = 1
• V = 5
• X = 10
• L = 50
• C = 100

Breaking Down the Number 49
To represent the number 49 in Roman numerals, we analyze it as follows:
1. 50 - 1 = 49:
- The numeral for 50 is L.
- To express the idea of "one less than 50," we place X (which represents 10) before L.
2. Combining Symbols:
- Thus, we write XL for 40 (50 minus 10).
- Next, we need to add 9, which is represented as IX (10 minus 1).
3. Final Combination:
- When we combine these, we get XLIX for 49.
Why the Other Options Are Incorrect
- Option B: IL:
- This is not a valid Roman numeral, as there is no representation for 49 this way.
- Option C: VLIII:
- This represents 48 (5 + 1 + 1 + 1 + 1 = 48), which is incorrect.
- Option D: XLVIII:
- This represents 48 (40 + 5 + 1 + 1 + 1 = 48), so it is not correct for 49.
Conclusion
Thus, the correct Roman numeral for 49 is XLIX, making option 'A' the accurate choice.

Understanding Divisibility Rules
To determine the divisibility of numbers based on the sum of their digits, we can apply specific rules. In this case, we focus on the sum being 27.
Divisibility by 9
- A number is divisible by 9 if the sum of its digits is divisible by 9.
- Since the sum of the digits is 27, we can check:
- 27 ÷ 9 = 3, which is an integer.
- Thus, any number with a digit sum of 27 is divisible by 9.
Divisibility by 6
- A number is divisible by 6 if it is divisible by both 2 and 3.
- While a sum of 27 indicates divisibility by 3, we cannot conclude anything about evenness (divisibility by 2) without knowing the last digit.
- Therefore, we cannot confirm divisibility by 6.
Divisibility by 11
- A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum in even positions is either 0 or divisible by 11.
- The sum of the digits alone (27) does not provide enough information to check this rule.
Divisibility by 7
- The divisibility rule for 7 involves more complex calculations or specific checks on the number itself.
- Again, without the actual number, we cannot ascertain divisibility by 7.
Conclusion
In summary, the only definitive conclusion we can draw from a digit sum of 27 is that the number is divisible by 9. Thus, option 'B' is the correct answer.

Understanding the Problem
To solve the expression (-2)^5 ÷ (-2)^8, we need to apply the properties of exponents.
Properties of Exponents
- When dividing two powers with the same base, you subtract the exponents:
- a^m ÷ a^n = a^(m-n)
Here, our base is (-2), and we have:
- m = 5 and n = 8
Applying the Rule
- Using the rule mentioned above, we calculate:
(-2)^5 ÷ (-2)^8 = (-2)^(5 - 8) = (-2)^{-3}
Understanding Negative Exponents
- A negative exponent indicates a reciprocal:
a^(-n) = 1/a^n
Thus, (-2)^{-3} can be rewritten as:
- (-2)^{-3} = 1/((-2)^3)
Calculating the Power
- Now we calculate (-2)^3:
(-2)^3 = -2 * -2 * -2 = -8
So now we have:
- (-2)^{-3} = 1/(-8)
Final Result
- 1/(-8) = -1/8
Therefore, the answer is option 'D': -1/8.
This demonstrates how to utilize the properties of exponents and understand negative exponents to simplify and solve expressions effectively.

Understanding the Problem
To determine how many zeros are in the cube root of 27000, we first need to calculate the cube root.
Finding the Cube Root of 27000
- The cube root of a number x is a value y such that y^3 = x.
- So, we need to find y such that y^3 = 27000.
Calculating the Cube Root
- We can express 27000 as a power of its prime factors.
- The prime factorization of 27000 is:
- 27000 = 27 x 1000
- 27 = 3^3 and 1000 = 10^3 = (2 x 5)^3
- Therefore, 27000 = 3^3 x (2 x 5)^3 = (3 x 2 x 5)^3 = (30)^3.
- Thus, the cube root of 27000 is:
- Cube root of 27000 = 30.
Counting the Zeros in 30
- The number 30 has one zero when considering the digit representation.
- As per the options provided, the correct answer is option 'C', which indicates there is 1 zero in the cube root of 27000.
Conclusion
- The cube root of 27000 is 30.
- Therefore, the number of zeros in 30 is 1.
This detailed breakdown helps clarify the process and confirms that option 'C' is indeed correct.