Key Concept: Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the thickness of a paper after 10 folds is 1.024 cm when the initial thickness is 0.001 cm. Using the given formula $\text{Thickness} = 0.001 \times 2^n \text{ cm}$, let's verify this:
For $n = 10$,
$0.001 \times 2^{10} = 0.001 \times 1024 = 1.024 \text{ cm}.$
So, the assertion is true. The reason provides the correct formula used in the calculation, which directly explains why the assertion holds true. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Scientific Notation

b) $2 \times 10^7$
[Solution Description]
Scientific notation expresses numbers in the form $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. Since $2$ crores is $20,000,000$, it can be written as: $2 \times 10^7$

Key Concept: Exponential Growth, Folding Paper

b) 1.024 cm
[Solution Description]
The thickness of the paper doubles with each fold. For 10 folds, the thickness increases by $2^{10} = 1024$ times.
Initial thickness: $0.001$ cm
Final thickness after 10 folds: $0.001 \times 2^{10} = 0.001 \times 1024 = 1.024$ cm

Key Concept: Exponential Growth, Comparison of Folds

b) 1024
[Solution Description]
The thickness after 10 folds is $0.001 \times 2^{10} = 1.024$ cm.
The thickness after 20 folds is $0.001 \times 2^{20} = 1048.576$ cm.
The ratio of thickness after 20 folds to 10 folds is:
$\frac{1048.576}{1.024} = 1024$

Key Concept: Exponents, Combined Growth

b) $6^4$
[Solution Description]
Using the exponent rule $m^a \times n^a = (mn)^a$, we can combine the exponents:
$3^4 \times 2^4 = (3 \times 2)^4 = 6^4$

Key Concept: Exponents

b) $10^5$
[Solution Description]
Using the exponent rule $m^a \times n^a = (m \times n)^a$, we can rewrite $2^5 \times 5^5$ as $(2 \times 5)^5 = 10^5$.

Key Concept: Exponential Growth

b) 29th day
[Solution Description]
Since the number of lotuses doubles every day, the pond must have been half-covered one day before it was fully covered. Therefore, on the 29th day, the pond was half-covered.

Key Concept: Exponential Growth, Magical Pond

c) Day 9
[Solution Description]
Since the lotuses triple daily, the pond would be one-third full one day before becoming full. Therefore, if full on day 10, it was one-third full on day 9.

Key Concept: Exponential Growth

c) 14th day
[Solution Description]
Since the number of lotuses doubles every day, the pond must have been half-covered one day before it was fully covered.
$\text{Half-covered day} = \text{Full day} - 1 = 15 - 1 = 14.$

Key Concept: Negative Exponents

c) $\frac{1}{9}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this to $3^{-2}$, we get $\frac{1}{3^2} = \frac{1}{9}$.

Key Concept: Negative Exponents and Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the given assertion, first simplify $\left(\frac{1}{5}\right)^{-3}$ using the property of negative exponents:
$\left(\frac{1}{5}\right)^{-3} = 5^3 = 125.$
Now multiply it by $10^4$:
$125 \times 10^4 = 1.25 \times 10^6.$
Thus, Assertion is true.
Reason states the definition of negative exponents and scientific notation, which is also correct.
Furthermore, Reason correctly explains how the simplification in the Assertion was achieved.

Key Concept: Prime Factorization, Exponential Form

a) $2^5 \times 3^3 \times 5^2$
[Solution Description]
To express 21600 in prime factorization exponential form:
Step 1: Break down 21600 into its prime factors.
21600 ÷ 2 = 10800 ÷ 2 = 5400 ÷ 2 = 2700 ÷ 2 = 1350 ÷ 2 = 675 ÷ 3 = 225 ÷ 3 = 75 ÷ 3 = 25 ÷ 5 = 5 ÷ 5 = 1.
So, 21600 = $2^5 \times 3^3 \times 5^2$.

Key Concept: Exponential Notation and Operations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $2^3 \times 2^4$, we use the exponent rule $n^a \times n^b = n^{a+b}$. Here, the base is 2 for both terms. So, $2^3 \times 2^4 = 2^{3+4} = 2^7$. Both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Scientific Notation, Exponent Operations

c) $3.081 \times 10^8$
[Solution Description]
Scientific notation represents a number as $a \times 10^b$, where $1 \leq |a| < 10$ and $b$ is an integer. For $308100000$, we move the decimal point to after the first digit: $3.081 \times 10^8$

Key Concept: Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial thickness of the paper is $0.001 \, \text{cm}$. After 1 fold, it becomes $0.001 \times 2 = 0.002 \, \text{cm}$. After 2 folds, it becomes $0.002 \times 2 = 0.004 \, \text{cm}$, and so on. After 5 folds, the thickness will be $0.001 \times 2^5 = 0.032 \, \text{cm}$. Thus, the Assertion is true. The Reason states that the thickness doubles after each fold, which is also true and correctly explains the Assertion because the exponential growth model ($2^n$) is applied here.

Key Concept: Exponential vs. Linear Growth

c) Exponential growth is faster than linear growth
[Solution Description]
Exponential growth gives:
$0.001 \text{ cm} \times 2^{10} = 1.024 \text{ cm}.$
In linear growth, for 10 steps adding $0.001 \text{ cm}$ each time, the total would be:
$0.001 \text{ cm} \times 10 = 0.01 \text{ cm}.$
Exponential growth results in much larger values compared to linear growth.

Key Concept: Scientific Notation Conversion

c) $7.5 \times 10^6$
[Solution Description]
Scientific notation is written as $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. To convert 7,500,000 to scientific notation: Move the decimal point 6 places to the left to get $7.5$. Since the decimal was moved 6 places, $y = 6$. Thus, the scientific notation is: $7.5 \times 10^6$

Key Concept: Negative Exponents

b) $\frac{1}{5^4}$
[Solution Description]
A negative exponent represents the reciprocal of the base raised to the positive exponent. Thus:
$5^{-4} = \frac{1}{5^4}$
The correct equivalent form is $\frac{1}{5^4}$.

Key Concept: Power of a Power, Zero Exponent

b) $64$
[Solution Description]
Any non-zero number raised to the power of 0 is 1, so $(5^2)^0 = 1$. For the second part, use the rule $(n^a)^b = n^{a \times b}$, so $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$. Multiplying these results together gives $1 \times 64 = 64$.

Key Concept: Exponent Rules

b) $2^{12}$
[Solution Description]
Using the power of a power rule, $(n^a)^b = n^{a \times b}$. Applying this rule to $(2^3)^4$:
$(2^3)^4 = 2^{3 \times 4} = 2^{12}$
The simplified form is $2^{12}$.

Key Concept: Power Notation, Negative Exponents

c) $4$
[Solution Description]
We know that $81$ can be written as $3^4$. So, $\frac{1}{81} = 3^{-4}$. Therefore, $3^{-x} = 3^{-4}$. Since the bases are equal, the exponents must also be equal. Hence, $-x = -4$, which gives $x = 4$.

Key Concept: Division of exponents with the same base

b) $7^{4}$
[Solution Description]
To solve $\frac{7^6}{7^2}$, use the rule $n^a \div n^b = n^{a - b}$. Here, the base $n = 7$, exponent $a = 6$, and exponent $b = 2$. So, $\frac{7^6}{7^2} = 7^{6 - 2} = 7^4$.

Key Concept: Multiplication of Exponents

b) $5^5$
[Solution Description]
To simplify $5^4 \times 5^{-2} \times 5^3$, we use the rule for multiplying exponents with the same base: $n^a \times n^b = n^{a+b}$.
Here, the exponents are added together: $4 + (-2) + 3 = 5$.
So, the simplified form is $5^5$.

Key Concept: Division of Exponents

b) $3^1$
[Solution Description]
First, multiply the exponents in the numerator: $3^7 \times 3^{-4} = 3^{7 + (-4)} = 3^3$.
Now, divide by the denominator using the rule $n^a \div n^b = n^{a-b}$: $3^3 \div 3^2 = 3^{3-2} = 3^1$.
The simplified form is $3^1$.

Key Concept: Negative exponents

b) $2^{2}$
[Solution Description]
To solve $2^{-3} \times 2^5$, use the rule $n^a \times n^b = n^{a + b}$. Here, the base $n = 2$, exponent $a = -3$, and exponent $b = 5$. So, $2^{-3} \times 2^5 = 2^{-3 + 5} = 2^2$.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $3^5 \times 3^{-2}$, we use the exponent rule for multiplication:
$n^a \times n^b = n^{a + b}$.
Applying this:
$3^5 \times 3^{-2} = 3^{5 + (-2)} = 3^3$.
Thus, the Assertion is true.
The Reason correctly states the rule used to solve it, so it is also true and explains the Assertion.

Key Concept: Prime factorization in exponential form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, perform the prime factorization of 540:
$540 = 2 \times 270 = 2 \times 2 \times 135 = 2 \times 2 \times 3 \times 45 = 2 \times 2 \times 3 \times 3 \times 15 = 2 \times 2 \times 3 \times 3 \times 3 \times 5$
Expressing this in exponential form gives $2^2 \times 3^3 \times 5^1$. Thus, the Assertion is true.
The Reason states that prime factorization breaks down a number into the product of prime numbers raised to their respective powers, which is correct and indeed explains why the Assertion holds true. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Exponential Form, Prime Factorization

b) 1
[Solution Description]
First, find prime factors of 504:
504 ÷ 2 = 252
252 ÷ 2 = 126
126 ÷ 2 = 63
63 ÷ 3 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
So, 504 = 2^3 × 3^2 × 7^1
Given equation: a + b - c = 4
From prime factors, we have:
a = 3, b = 2, c = 1
Verify equation:
3 + 2 - 1 = 4 which satisfies given condition.
Thus, value of c is 1.

Key Concept: Prime factorization in exponential form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- Step 1: Prime factorize 36. We know that 36 = 6 × 6.
- Step 2: Further factorize 6 into its prime factors: 6 = 2 × 3.
- Step 3: Substitute back into the equation: 36 = (2 × 3) × (2 × 3).
- Step 4: Regroup the terms to express them in exponential form: 36 = $2^2 \times 3^2$.
- Conclusion: Both Assertion (A) and Reason (R) are true, and R correctly explains A.

Key Concept: Numerical Evaluation

c) 392
[Solution Description]
To evaluate $7^2 \times 2^3$, first calculate the exponents:
$7^2 = 49$
$2^3 = 8$
Multiply them together: $49 \times 8 = 392$.

Key Concept: Product of Powers with the Same Base

a) $3^7$
[Solution Description]
Using the property of exponents that states $n^a \times n^b = n^{a+b}$, we can simplify $3^5 \times 3^2$ as follows:
$3^5 \times 3^2 = 3^{5+2} = 3^7$

Key Concept: Power of a Power

b) $5^6$
[Solution Description]
To simplify $(5^2)^3$, we use the power of a power property: $(n^a)^b = n^{a \times b}$. Here, the base is 5, and the exponents are 2 and 3:
$(5^2)^3 = 5^{2 \times 3} = 5^6$
Therefore, the simplified form of the expression is $5^6$.

Key Concept: Exponents, Negative exponents, Power of a power

b) $3^{-6}$
[Solution Description]
Step 1: Simplify the numerator inside the bracket using the property $n^a \times n^b = n^{a+b}$:
$3^4 \times 3^{-2} = 3^{4 + (-2)} = 3^2$
Step 2: Now divide the simplified numerator by the denominator using the property $n^a \div n^b = n^{a-b}$:
$\frac{3^2}{3^5} = 3^{2-5} = 3^{-3}$
Step 3: Apply the power of a power property $(n^a)^b = n^{ab}$ to the entire expression:
$\left(3^{-3}\right)^2 = 3^{-6}$
The simplified form is $3^{-6}$.

Key Concept: Properties of Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $5^3 \times 5^{-1} = 5^2$. Let's verify this using the product of powers property mentioned in the reason.
Step 1: Apply the product of powers property, which states $n^a \times n^b = n^{a+b}$. Here, $n = 5$, $a = 3$, and $b = -1$.
Step 2: Calculate the exponents: $3 + (-1) = 2$.
Step 3: Thus, $5^3 \times 5^{-1} = 5^{3 + (-1)} = 5^2$.
Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Multiplication of Powers with the Same Base

d) $3^7$
[Solution Description]
To simplify $3^5 \times 3^2$, we use the property of exponents: $n^a \times n^b = n^{a + b}$. Here, the base is the same (3), so we add the exponents:
$3^5 \times 3^2 = 3^{5+2} = 3^7$
Therefore, the simplified form of the expression is $3^7$.

Key Concept: Negative exponents, Exponential operations

a) $\frac{1}{9}$
[Solution Description]
First, simplify the numerator using $n^a \times n^b = n^{a+b}$: $3^{2-5} = 3^{-3}$.
Now, divide by $3^{-1}$ using $n^a \div n^b = n^{a-b}$: $3^{-3-(-1)} = 3^{-2}$.
The simplified form is $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

Key Concept: Zero exponent, Negative exponents

d) Any non-zero real number
[Solution Description]
Substitute $y = 4$ into the equation: $(x^0 + 4^{-1})^{-1} = 2$.
Simplify $x^0 = 1$ and $4^{-1} = \frac{1}{4}$: $(1 + \frac{1}{4})^{-1} = (\frac{5}{4})^{-1} = \frac{4}{5}$.
However, $\frac{4}{5} \neq 2$, so there is no solution. But since $x^0 = 1$ for any $x \neq 0$, $x$ can be any non-zero real number. The question may have an error or missing condition.

Key Concept: Negative Exponent

d) $\frac{1}{9}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this rule:
$3^{-2} = \frac{1}{3^2} = \frac{1}{9}.$

Key Concept: Exponent Operations

c) $\frac{1}{49}$
[Solution Description]
Using the exponent division rule: $n^a ÷ n^b = n^{a-b}$. Here, $7^4 ÷ 7^6 = 7^{4-6} = 7^{-2}$.
By the negative exponent rule, $7^{-2} = \frac{1}{7^2} = \frac{1}{49}$.

Key Concept: Combination Calculation

b) $26^{6}$
[Solution Description]
Each slot has 26 choices (A-Z). For 6 slots, total combinations are $26 \times 26 \times 26 \times 26 \times 26 \times 26 = 26^6$.

Key Concept: Negative Exponents

b) $\frac{1}{10^5}$
[Solution Description]
The general form for negative exponents is $n^{-a} = \frac{1}{n^a}$. Applying this to $10^{-5}$, we get $\frac{1}{10^{+5}}$.

Key Concept: Exponent Rules, Negative Exponents

c) $3^7$
[Solution Description]
To solve this, we first simplify the numerator and denominator separately using exponent rules. The numerator $3^5 \times 3^{-2} = 3^{5-2} = 3^3$. The denominator $3^0 \times 3^{-4} = 1 \times 3^{-4} = 3^{-4}$ because any number to the power of 0 is 1. Now, dividing numerator by denominator: $\frac{3^3}{3^{-4}} = 3^{3-(-4)} = 3^{7}$. Therefore, the simplified form is $3^7$.

Key Concept: Fractional Exponents, Power Division

b) 5
[Solution Description]
First, express 4 as a power of 2: $4 = 2^2$. So, $4^{10} = (2^2)^{10} = 2^{20}$. Now the equation becomes $\frac{2^{20}}{2^{15}} = 2^x$. Using the division rule for exponents: $2^{20-15} = 2^x$ which simplifies to $2^5 = 2^x$. Therefore, x must be 5.

Key Concept: Exponent subtraction rule and negative exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $5^{-3} \div 5^2$, we apply the exponent subtraction rule: $5^{-3} \div 5^2 = 5^{-3 - 2} = 5^{-5}$. The Reason correctly states the rule for dividing exponents with the same base. Therefore, both Assertion and Reason are true, and Reason explains Assertion.

Key Concept: Exponent subtraction rule, Negative exponents

a) $7^3$
[Solution Description]
First, multiply the terms in the numerator using the exponent addition rule: $7^8 \times 7^{-3} = 7^{8 + (-3)} = 7^5$. Next, divide the result by the denominator using the exponent subtraction rule: $\frac{7^5}{7^2} = 7^{5 - 2} = 7^3$. Therefore, the simplified form is $7^3$.

Key Concept: Exponent subtraction rule

b) $2^2$
[Solution Description]
To simplify $2^{5} \div 2^{3}$, we use the exponent subtraction rule: $n^a \div n^b = n^{a - b}$. Here, $n = 2$, $a = 5$, and $b = 3$. Applying the rule:
$2^{5} \div 2^{3} = 2^{5 - 3} = 2^2$
Calculating $2^2$ gives $4$.

Key Concept: Combining exponents

c) $3^5$
[Solution Description]
Using the exponent addition rule: $n^a \times n^b = n^{a + b}$. For $3^{4} \times 3^{-1} \times 3^{2}$:
$3^{4} \times 3^{-1} \times 3^{2} = 3^{4 - 1 + 2} = 3^5$
The simplified form is $3^5$.

Key Concept: Negative Exponent Simplification

c) $\frac{1}{125}$
[Solution Description]
Using the rule for negative exponents, $n^{-a} = \frac{1}{n^a}$. Here, $n = 5$ and $a = 3$. Thus,
$5^{-3} = \frac{1}{5^3} = \frac{1}{125}.$

Key Concept: Zero Exponent

b) 1
[Solution Description]
For any non-zero number or variable, raising it to the power of zero gives 1. Thus:
$(5x)^0 = 1$
Therefore, the correct answer is 1.

Key Concept: Exponent Operations

c) $\frac{1}{4}$
[Solution Description]
Using the exponent rule for multiplication with the same base:
$2^4 \times 2^{-6} = 2^{4 + (-6)} = 2^{-2} = \frac{1}{2^{2}} = \frac{1}{4}$
The correct answer is $\frac{1}{4}$.

Key Concept: Operations with Negative Exponents, Multiplication

b) $125$
[Solution Description]
To simplify the expression, we will use the rules for multiplying and dividing exponents with the same base.
Step 1: Rewrite the division as multiplication by the reciprocal:
$5^{-3} \times 5^{2} \times 5^{4}$
Step 2: Apply the product of powers rule $a^{m} \times a^{n} = a^{m+n}$:
$5^{-3 + 2 + 4} = 5^{3}$
Step 3: Calculate $5^{3}$:
$5^{3} = 125$
The simplified form of the expression is $125$.

Key Concept: Powers of 10 - Basic Conversion

b) 5074
[Solution Description]
To solve this, multiply each term by its respective power of 10 and add them together:
$5 \times 10^3 = 5000$
$7 \times 10^1 = 70$
$4 \times 10^0 = 4$
Adding them:
$5000 + 70 + 4 = 5074$
The standard form is 5074.

Key Concept: Powers of 10, Expanded Form

b) $(8 \times 10^3) + (4 \times 10^2) + (9 \times 10^1) + (3 \times 10^0)$
[Solution Description]
To express 8493 using powers of 10, we break it down into its place values:
$8493 = (8 \times 1000) + (4 \times 100) + (9 \times 10) + 3$
Converting each term to powers of 10:
$8493 = (8 \times 10^3) + (4 \times 10^2) + (9 \times 10^1) + (3 \times 10^0)$

Key Concept: Powers of 10

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $10^3$ can be written as $\frac{1}{10^{-3}}$. From the given syllabus, we know that for any non-zero number $n$, the relation $n^a = \frac{1}{n^{-a}}$ holds true. Substituting $n = 10$ and $a = 3$ in the general form gives us $10^3 = \frac{1}{10^{-3}}$. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Large Numbers - Naming Conventions

a) Arab
[Solution Description]
According to the Indian numbering system:
$10^9 = 100 \times 10^7 = \text{arab}$

Key Concept: Scientific Notation, Large Numbers

c) $7.89 \times 10^6$
[Solution Description]
To convert 7,89,00,000 to scientific notation, we first write it as 7.89 multiplied by a power of 10. Counting the digits after the first digit (7), we have 7 digits (including the 8 and 9). Therefore, the exponent is 6 (since we move the decimal point 6 places to the left from its original position). Hence, the scientific notation is $7.89 \times 10^6$.

Key Concept: Scientific Notation, Conversion

c) $3.43 \times 10^6$
[Solution Description]
To convert 34,30,000 to scientific notation, identify the coefficient between 1 and 10. Moving the decimal point 6 places to the left gives 3.43. Thus, the number becomes $3.43 \times 10^6$.

Key Concept: Scientific Notation

c) $4.5 \times 10^3$
[Solution Description]
To convert 4,500 to scientific notation:
1. Move the decimal point after the first non-zero digit: 4.5
2. Count how many places the decimal moved from its original position (from after the last zero to after 4): 3 places.
3. Thus, $4,500 = 4.5 \times 10^3$.

Key Concept: Scientific Notation, Exponent Importance

d) Assertion is false, but Reason is true.
[Solution Description]
To compare $3.6 \times 10^5$ and $36 \times 10^4$, we first convert both to the same form.
$3.6 \times 10^5 = 360,000$ and $36 \times 10^4 = 360,000$. Thus, both numbers are equal.
Assertion (A) is false because the two numbers are equal, not greater or smaller.
Reason (R) is also false because the magnitude depends on both the coefficient and the exponent, not just the exponent.
Therefore, the correct answer is (d).

Key Concept: Scientific Notation

c) $3.05 \times 10^4$
[Solution Description]
To convert 30,500 to scientific notation:
1. Identify the coefficient: Move the decimal point after the first non-zero digit (3.05).
2. Count the number of places moved (4 places to the left).
3. Write in the form $x \times 10^y$: $3.05 \times 10^4$.

Key Concept: Scientific Notation, Mass Conversion

a) $5.976 \times 10^{24}$ kg
[Solution Description]
The given mass of the Earth is 59,76,00,00,00,00,00,00,00,00,00,000 kg.
Step 1: Rewrite the number without commas: 5976000000000000000000000 kg.
Step 2: Count the number of digits after the first digit (5). Total digits = 25.
Step 3: Move the decimal point to after the first non-zero digit (5.976).
Step 4: The exponent is the number of places moved: 24.
So, the scientific notation is $5.976 \times 10^{24}$ kg.

Key Concept: Exponent Rules

b) $5^5$
[Solution Description]
Using exponent rules:
1. Multiply the exponents for the numerator: $5^3 \times 5^4 = 5^{3+4} = 5^7$.
2. Divide by the denominator: $5^7 \div 5^2 = 5^{7-2} = 5^5$.

Key Concept: Scientific Notation

b) $7.004 \times 10^{10}$
[Solution Description]
To express $70,04,00,00,000$ in scientific notation:
1. Identify the coefficient: Move the decimal point to the left until it is after the first non-zero digit: $7.004$.
2. Count the number of places moved: The decimal moves 10 places to the left.
3. Write in scientific notation: $7.004 \times 10^{10}$.

Key Concept: Scientific Notation

b) $2.5 \times 10^6$
[Solution Description]
To find how many times more ants there are compared to humans, divide the number of ants by the number of humans:
$\frac{2 \times 10^{16}}{8 \times 10^9} = \frac{2}{8} \times 10^{16-9} = 0.25 \times 10^7 = 2.5 \times 10^6$
Therefore, there are $2.5 \times 10^6$ (2.5 million) times more ants than humans.

Key Concept: Historical Context, Powers of 10

b) $10^{11}$
[Solution Description]
According to historical context from the syllabus, the term "niyuta" was used in ancient Indian texts like the *Lalitavistara* to refer to $10^{11}$.

Key Concept: Exponential vs. Linear Growth, Scientific Notation

a) $6.4 \times 10^9$ m
[Solution Description]
Each fold doubles the thickness, so the thickness after $n$ folds is given by $T = T_0 \times 2^n$, where $T_0$ is the initial thickness.
Initial thickness $T_0 = 0.1$ mm = $0.1 \times 10^{-3}$ m = $1 \times 10^{-4}$ m
Thickness after 46 folds:
$T = 1 \times 10^{-4} \times 2^{46}$
Calculate $2^{10} = 1024 \approx 10^3$
So, $2^{40} = (2^{10})^4 \approx (10^3)^4 = 10^{12}$
And $2^{46} = 2^{40} \times 2^6 = 10^{12} \times 64 = 6.4 \times 10^{13}$
Now, $T = 1 \times 10^{-4} \times 6.4 \times 10^{13} = 6.4 \times 10^9$ m
The distance to the Moon is $3.84 \times 10^8$ m, which is less than $6.4 \times 10^9$ m, confirming the claim.

Key Concept: Scientific Notation, Real-World Comparisons

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the ratio of stars to ants using their given values:
$\frac{2 \times 10^{23}}{2 \times 10^{16}} = 10^{23-16} = 10^7$
This confirms that the assertion is correct. The reason states that the ratio can be found by subtracting the exponents, which is a valid rule for division in scientific notation. Thus, the reason not only supports but also correctly explains the assertion.

Key Concept: Scientific Notation, Applications

c) $1 \times 10^{10}$
[Solution Description]
Step 1: Multiply the rate of calculations by the time to get the total calculations.
$1.25 \times 10^{15} \times 8 \times 10^{-6} = (1.25 \times 8) \times 10^{15-6} = 10 \times 10^9 = 1 \times 10^{10}$
Step 2: Answer is $1 \times 10^{10}$ calculations.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Scientific notation requires the coefficient $x$ to satisfy $1 \leq x < 10$ and the exponent $y$ to be an integer. Here, $5.9$ is between 1 and 10, and $3$ is an integer. Thus, both Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Scientific Notation, Expanded Form

a) $7.2381 \times 10^6$
[Solution Description]
Step 1: Add all the terms in the expanded form to get the standard number.
$7 \times 10^6 = 7,000,000$
$2 \times 10^5 = 200,000$
$3 \times 10^4 = 30,000$
$8 \times 10^3 = 8,000$
$1 \times 10^2 = 100$
Step 2: Sum all the values: $7,000,000 + 200,000 = 7,200,000$; $7,200,000 + 30,000 = 7,230,000$; $7,230,000 + 8,000 = 7,238,000$; $7,238,000 + 100 = 7,238,100$.
Step 3: Convert $7,238,100$ to scientific notation by writing it as $7.2381 \times 10^6$.

Key Concept: Scientific Notation, Magnitude Comparison

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare $3.5 \times 10^7$ and $4.8 \times 10^6$, we examine their exponents. The exponent of the first number is 7, while the exponent of the second number is 6. Since 7 > 6, $3.5 \times 10^7$ is indeed greater than $4.8 \times 10^6$, regardless of the coefficients. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the worth of donated jaggery, we multiply Roxie's weight (45 kg) by the cost per kg of jaggery (Rs.70). So, $45 \times 70 = Rs.3150$. The assertion is true as the calculation is correct. The reason is also true because it correctly explains the formula used to calculate the worth of donated goods. Additionally, the reason is the correct explanation for the assertion since it provides the underlying principle for the calculation.

Key Concept: Counting Stars in Scientific Notation

d) $10^{23}$ seconds
[Solution Description]
If there are $10^{23}$ stars and counting one star per second, the time required is $10^{23}$ seconds. This is already in scientific notation.

Key Concept: Worth of Donated Items

b) Rs.3150
[Solution Description]
The worth of donated jaggery can be calculated using the formula: $\text{Worth of jaggery} = \text{Roxie's weight} \times \text{Cost of 1 kg jaggery}$. Substituting the given values: $45 \times 70 = Rs.3150$.

Key Concept: Weight estimation, Unit conversion

c) 15,000 coins
[Solution Description]
To find the number of 1-rupee coins that equal Roxie's weight, we first convert her weight to grams and then divide by the weight of one coin.
Step-by-step calculation:
1. Roxie's weight = 45 kg = $45 \times 1000 = 45,000$ grams
2. Weight of one coin = 3 grams
3. Number of coins = $\frac{45,000}{3} = 15,000$ coins

Key Concept: Comparison of Linear and Exponential Growth

c) A bacteria population doubles every day.
[Solution Description]
Exponential growth involves multiplicative increases, unlike linear growth which is additive. Only option c) represents repeated multiplication (doubling every day).

Key Concept: Linear vs Exponential, Real-World Context

b) Exponential becomes more than 50,000x faster by 20 folds
[Solution Description]

For exponential growth:

$T = 0.001 \times 2^n$

For linear growth:

$L = 0.001 + 0.002n$

Here, T represents exponential thickness and L represents linear thickness.

For example, after 20 folds:

$T = 0.001 \times 2^{20}$

$T = 0.001 \times 1,048,576 = 1048.576 \text{ cm}$

Linear growth after 20 folds:

$L = 0.001 + 0.002(20)$

$L = 0.001 + 0.04 = 0.041 \text{ cm}$

Now compare them:

$\frac{1048.576}{0.041} \approx 25,575$

So, after 20 folds, exponential growth gives a thickness about 25,575 times greater than linear growth.

Also, the increase during the $20^{\text{th}}$ fold is:

$0.001 \times 2^{19} = 524.288 \text{ cm}$

Linear growth adds only:

$0.002 \text{ cm}$

$\frac{524.288}{0.002} = 262,144$

So, by the $20^{\text{th}}$ fold, the rate of exponential growth is more than 50,000 times faster than linear growth.

Key Concept: Exponential Growth, Historical Comparisons

a) n is 5 less than dinosaur exponent
[Solution Description]
Chola duration: 900 years $= 3 × 10^{10}$ seconds (given)
Dinosaurs appeared: 200 million years ago $= 2 × 10^8 $years.
Convert to seconds: $2 × 10^8 × 3.15 × 10^7 ≈ 6.3 × 10^{15}$ seconds.
Difference in exponents: 15 - 10 = 5 orders of magnitude.
Comparison: The dinosaur era was about $10^5 (100,000)$ times longer than the Chola reign.

Key Concept: Linear Growth vs. Exponential Growth

b) 1,92,20,00,000 steps
[Solution Description]
To find the number of steps, convert the total distance into centimeters and divide by the step size.
3,84,400 km = 3,84,400 $\times$ 100,000 cm = 38,44,00,00,000 cm.
Number of steps = Total distance / Step size = 38,44,00,00,000 cm / 20 cm = 1,92,20,00,000 steps.

Key Concept: Scientific Notation, Earth's Water Volume

b) $6.93 \times 10^{21}$
[Solution Description]
First, convert 200 ml to liters:
$200 \text{ ml} = 0.2 \text{ liters}$
Now divide the total volume by the volume of one glass:
$\frac{1.386 \times 10^{21}}{0.2} = 6.93 \times 10^{21}$

Key Concept: Scientific notation

c) $3.081 \times 10^8$
[Solution Description]
Scientific notation requires expressing the number in the form $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. For 308,100,000: $308,100,000 = 3.081 \times 10^8$ The decimal moves 8 places to the left.

Key Concept: Scientific Notation Conversion

b) $10^{23}$ seconds
[Solution Description]
To find the time taken to count all the stars, we divide the total number of stars by the counting rate (1 star per second). Given the number of stars as $10^{23}$, the time taken is $10^{23}$ seconds.
Hence, the answer is $10^{23}$ seconds.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 1 crore is equal to $10^7$ in the Indian numbering system. According to the syllabus, a crore is indeed defined as $10^7$, which is equivalent to 100 lakhs since 1 lakh equals $10^5$. Therefore, both the Assertion (A) and Reason (R) are true, and the Reason correctly explains the Assertion.

Key Concept: Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that 5000 can be written as $5 \times 10^3$ in scientific notation. This is correct because moving the decimal point three places to the left gives the coefficient 5 and the exponent 3. The reason states that in scientific notation, the coefficient must be between 1 and 10, which is true and correctly explains why the assertion is valid.

Key Concept: Indian and International number systems, Conversion

b) 500 billion
[Solution Description]
1 kharab in Indian system = $10^{11}$ (100 billion in International).
Therefore, 5 kharab = $5 \times 10^{11} = 500,000,000,000$.
To convert to billions: Divide by $10^9$ (since 1 billion = $10^9$),
$\frac{5 \times 10^{11}}{10^9} = 5 \times 10^2 = 500$ billion.

Key Concept: Standard Form

c) 7
[Solution Description]
To express 2 crores in standard form:
$2 \, \text{crores} = 2 \times 1,00,00,000 = 2 \times 10^7$
Here, the exponent is 7.

Key Concept: Scientific notation, Indian and International number systems

b) $5.976 \times 10^{24}$ kg
[Solution Description]
First, convert the given Indian number to International format by removing commas and grouping digits in sets of three from the right:
$59,76,00,00,00,00,00,00,00,00,00,000$ becomes $5,976,000,000,000,000,000,000,000$ kg.
Now express this in scientific notation: Identify the coefficient (between 1 and 10) and exponent.
Coefficient = $5.976$
Count the number of places the decimal moves from the end to after the first digit: 24 places.
Thus, scientific notation is $5.976 \times 10^{24}$ kg.

Key Concept: Exponential Growth

c) 29th day
[Solution Description]
Since the number of lotuses doubles every day, the pond must have been half-covered one day before it was fully covered.
Therefore, if the pond is fully covered on the 30th day, it was half-covered on the 29th day.

Key Concept: Linear vs Exponential, Paper Folding

a) Folding once exceeds all steps
[Solution Description]
1. Folded paper thickness:
$0.001 \times 2^{42} \text{ cm} = 4.398 \times 10^9 \text{ km}$
2. Earth-Moon steps:
$Each step = 20 cm → Total steps = 384,400 km / 0.0002 km = 1.922 \times 10^9 steps$
Comparison:
$4.398 \times 10^9 \text{ km} \text{ (folded paper)} > 384,400 \text{ km} \text{ (steps)}$

Key Concept: Exponential Growth, Scientific Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial population $P_0 = 100$ cells. The population doubles every hour, so after $n = 10$ hours, the population $P$ can be calculated as:
$$ P = 100 \times 2^{10} $$
We know that $2^{10} = 1024$. Therefore:
$$ P = 100 \times 1024 = 102400 $$
In scientific notation, this is $1.024 \times 10^5$ cells.
The Assertion (A) correctly states the final population, and the Reason (R) provides the correct mathematical model for exponential growth, which explains why the assertion holds true.

Key Concept: Exponent Rules

d) $2^4$
[Solution Description]
Using the exponent rule $\frac{m^a}{n^a} = \left(\frac{m}{n}\right)^a$:
$\frac{10^4}{5^4} = \left(\frac{10}{5}\right)^4 = 2^4$
The simplified exponential form is $2^4$.

Key Concept: Large Number Facts

b) $1 \times 10^{11}$
[Solution Description]
The number of stars in the Milky Way galaxy is approximately $1 \times 10^{11}$, which is 100 billion stars.

Key Concept: Population Counting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the global population is approximately $8 \times 10^9$, which is true as per the given syllabus context. The reason explains that large numbers like $8 \times 10^9$ are used for resource allocation and urban planning, which is also true as mentioned in the syllabus. Moreover, the reason correctly explains why the assertion is significant, as it highlights the practical application of such large numbers in real-world scenarios like governance and planning. Thus, both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Scientific Notation

c) $8 \times 10^6$
[Solution Description]
To express 80,00,000 in scientific notation, we write it as $8 \times 10^6$. Here, the coefficient $x$ is 8, and the exponent $y$ is 6 because we move the decimal point 6 places to the left.

Key Concept: Population and Demographics, Scientific Notation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the world population is approximately $8 \times 10^9$ and highlights its importance in resource planning, which is true based on the syllabus. The Reason explains that scientific notation simplifies the representation and comparison of large numbers, which is also correct. However, the Reason does not directly explain why the population figure is essential for resource planning—it only describes the utility of scientific notation. Thus, both statements are true, but the Reason does not correctly explain the Assertion.

Key Concept: Scientific Notation Application

c) $2 \times 10^7$
[Solution Description]
2 crores = 2,00,00,000.
To write in scientific notation, express it as $x \times 10^y$ where $1 \leq x < 10$. Here, $2,00,00,000 = 2 \times 10^7$.

Key Concept: Indian and International Naming Conventions

b) Indian: neel, International: trillion
[Solution Description]
According to the syllabus:
- In the Indian system, $10^{13}$ is called "neel".
- In the International system, $10^{13}$ is not a standard named power; the closest is trillion ($10^{12}$) or ten trillion ($10^{13}$). However, the exact match for $10^{13}$ is not explicitly mentioned, but in the Indian system, it is "neel".
Thus, $10^{13}$ corresponds to "neel" in the Indian system, while it is not directly named in the International system beyond trillion ($10^{12}$).

Key Concept: Scientific Notation Basics

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that $6.02 \times 10^{23}$ is written in scientific notation. This is correct because the coefficient (6.02) lies between 1 and 10, and the exponent (23) is an integer, satisfying the definition of scientific notation.
The Reason correctly defines the conditions for a number to be in scientific notation: the coefficient must be between 1 and 10, and the exponent must be an integer.
Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Exponent Comparison

c) Distance between Sun and Earth
[Solution Description]
Compare the exponents first:
- Sun-Saturn: $1.4335 \times 10^{12}$ m ($y = 12$)
- Saturn-Uranus: $1.439 \times 10^{12}$ m ($y = 12$)
- Sun-Earth: $1.496 \times 10^{11}$ m ($y = 11$)
The exponent $11$ is smaller than $12$, so Sun-Earth distance is the smallest.