Key Concept: Exponential Growth

d) 1024 times
[Solution Description]
The thickness increases by a factor of $2^n$, where $n$ is the number of folds. For 10 folds:
$2^{10} = 1024$
So, the thickness increases 1024 times.

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: Power Operations

b) $3^3$
[Solution Description]
Using the rule for division of exponents with the same base, $n^a \div n^b = n^{a - b}$. Applying this rule:
$3^5 \div 3^2 = 3^{5 - 2} = 3^3$

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, 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: 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, Folding Paper Experiment

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we calculate the thickness after 46 folds using the formula for exponential growth: $T = T_0 \times 2^n$, where $T_0$ is the initial thickness (0.001 cm) and $n$ is the number of folds (46).
Step 1: Calculate $2^{46}$. Using logarithmic approximations or computational tools, $2^{46} \approx 7.04 \times 10^{13}$.
Step 2: Multiply the result by the initial thickness. $T = 0.001 \, \text{cm} \times 7.04 \times 10^{13} = 7.04 \times 10^{10} \, \text{cm}$.
Step 3: Convert centimeters to kilometers. Since $1 \, \text{km} = 10^5 \, \text{cm}$, $T = \frac{7.04 \times 10^{10}}{10^5} \, \text{km} = 7.04 \times 10^5 \, \text{km} \approx 7,00,000 \, \text{km}$.
Both the assertion and reason are true, and the reason correctly explains the assertion as it provides the mathematical basis for the exponential growth calculation.

Key Concept: Exponential Growth, Folding Paper

b) 32.768 cm
[Solution Description]
The thickness doubles with each fold. So after 15 folds, thickness = $0.001 \times 2^{15}$ cm. Calculating step by step:
$2^1 = 2$
$2^2 = 4$
...
$2^{10} = 1024$
$2^{15} = 32768$
Final thickness = $0.001 \times 32768 = 32.768$ cm

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: Operations with Exponents

b) $7^7$
[Solution Description]
Using the exponent rule $n^a \times n^b = n^{a+b}$, we can simplify $7^3 \times 7^4$ as $7^{3+4} = 7^7$.

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: Exponential Notation and Operations

a) $3^3 \times 7^2 \times 2^{-2}$
[Solution Description]
To simplify $\frac{3^5 \times 7^3 \times 2^4}{3^2 \times 7 \times 2^6}$, we use the rule $n^a \div n^b = n^{a-b}$ for each common base.
Step 1: Simplify the exponents of 3.
$3^5 \div 3^2 = 3^{5-2} = 3^3$.
Step 2: Simplify the exponents of 7.
$7^3 \div 7^1 = 7^{3-1} = 7^2$.
Step 3: Simplify the exponents of 2.
$2^4 \div 2^6 = 2^{4-6} = 2^{-2}$.
Final simplified form: $3^3 \times 7^2 \times 2^{-2}$.

Key Concept: Exponential Growth, Doubling Effect

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that the thickness of a paper folded 20 times would exceed the height of Mount Everest, while the Reason provides the formula for calculating the thickness after $n$ folds and calculates the thickness for 20 folds as 10.48576 m, which is less than the height of Mount Everest (8848 m). Here, the Assertion is false because the calculated thickness (10.48576 m) is significantly less than the height of Mount Everest (8848 m). However, the Reason correctly explains the calculation using the exponential growth formula and arrives at the correct numerical result. Therefore, the Assertion is false, but the Reason is true.

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 Growth, Scientific Notation

c) 32.768 cm
[Solution Description]
The thickness doubles with each fold, so after $n$ folds, the thickness is given by $0.001 \times 2^n$ cm. For $n = 15$, the calculation is:
$0.001 \times 2^{15} = 0.001 \times 32768 = 32.768 \text{ cm}$

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 Notation and Power of a Power

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(3^4)^5$ simplifies to $3^{20}$. This is true because applying the power of a power rule $(n^a)^b = n^{a \times b}$, we get $(3^4)^5 = 3^{4 \times 5} = 3^{20}$. The reason correctly explains this simplification using the power of a power rule. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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 $5^3 = 125$. This is correct because $5^3 = 5 \times 5 \times 5 = 125$. The reason explains the general definition of exponential notation, where $n^a$ means $n$ multiplied by itself $a$ times. Since the assertion is a specific example of this rule and correctly applies it, the reason is the correct explanation of the assertion.

Key Concept: Zero Exponent

c) $1$
[Solution Description]
Any non-zero number raised to the power of zero is equal to 1. Therefore, $7^0 = 1$.

Key Concept: Product of Powers, Quotient of Powers

b) $\frac{1}{5}$
[Solution Description]
First, simplify the numerator and denominator separately using the product of powers rule ($n^a \times n^b = n^{a+b}$). Numerator: $5^4 \times 5^{-2} = 5^{4 + (-2)} = 5^2$. Denominator: $5^0 \times 5^3 = 1 \times 5^3 = 5^3$. Now, apply the quotient of powers rule ($n^a \div n^b = n^{a-b}$): $\frac{5^2}{5^3} = 5^{2-3} = 5^{-1} = \frac{1}{5}$.

Key Concept: Multiplication of Exponents with Same Base

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the rule for multiplication of exponents with the same base:
$n^a \times n^b = n^{a + b}$.
Given the assertion: $3^5 \times 3^{-2}$.
Applying the rule: $3^{5 + (-2)} = 3^{3}$.
The reason correctly explains that exponents are added when multiplying with the same base.

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: 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: Multiplying exponents, Generalized forms

a) $-1$
[Solution Description]
Using the rule $n^a \times n^b = n^{a + b}$, we can combine all the exponents: $x^{-4} \times x^5 \times x^{-2} = x^{-4 + 5 + (-2)} = x^{-1}$. Therefore, the value of $k$ is $-1$.

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) $2^3 \times 3^4$
[Solution Description]
To find the prime factorization of 648, start dividing by the smallest prime number, which is 2:
$648 \div 2 = 324 \\ \\ 324 \div 2 = 162 \\ \\ 162 \div 2 = 81$
Now, 81 is not divisible by 2, so move to the next prime number, 3:
$81 \div 3 = 27 \\ \\ 27 \div 3 = 9 \\ \\ 9 \div 3 = 3 \\ \\ 3 \div 3 = 1$
The prime factors are $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$. In exponential form: $2^3 \times 3^4$.

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: Expressing Numbers in Exponential Form

b) $5^2 \times 7^3$
[Solution Description]
To express $5 \times 5 \times 7 \times 7 \times 7$ in exponential form, count the number of times each base is multiplied by itself.
5 is multiplied by itself 2 times: $5^2$.
7 is multiplied by itself 3 times: $7^3$.
The exponential form is $5^2 \times 7^3$.

Key Concept: Prime factorization, Exponential form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's perform the prime factorization of 3600:
1. Divide 3600 by 2 (smallest prime): 3600 ÷ 2 = 1800
2. Continue dividing by 2: 1800 ÷ 2 = 900 → 900 ÷ 2 = 450 → 450 ÷ 2 = 225 (total four divisions by 2)
3. Next divide by 3 (next prime): 225 ÷ 3 = 75 → 75 ÷ 3 = 25 (total two divisions by 3)
4. Finally divide by 5: 25 ÷ 5 = 5 → 5 ÷ 5 = 1 (total two divisions by 5)
Thus, 3600 = $2^4 \times 3^2 \times 5^2$, which matches the assertion. The reason correctly describes the process of prime factorization. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Prime factorization, Exponents

a) $2 \times 3^2 \times 5^{-1}$
[Solution Description]
Step 1: Simplify the expression $\frac{x^2 \times y}{x \times y^2}$ to $\frac{x}{y}$:
$\frac{x^2 \times y}{x \times y^2} = \frac{x^{2-1}}{y^{2-1}} = \frac{x}{y}$
Step 2: Substitute the given values for $x$ and $y$:
$\frac{x}{y} = \frac{2^3 \times 3^4 \times 5^2}{2^2 \times 3^2 \times 5^3}$
Step 3: Simplify using the property $n^a \div n^b = n^{a-b}$ for each prime factor:
$2^{3-2} \times 3^{4-2} \times 5^{2-3} = 2^1 \times 3^2 \times 5^{-1}$
The simplified form is $2 \times 3^2 \times 5^{-1}$.

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: 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: Properties of Powers, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's evaluate the assertion step-by-step using the properties of exponents.
1. Given expression: $5^{-3} \times 5^2$
2. Using the product of powers property:
$n^a \times n^b = n^{a+b}$
So, $5^{-3} \times 5^2 = 5^{-3+2} = 5^{-1}$
3. Using the negative exponent property:
$n^{-a} = \frac{1}{n^a}$
Therefore, $5^{-1} = \frac{1}{5^1} = \frac{1}{5}$
The assertion is true as it correctly simplifies to $\frac{1}{5}$.
The reason states the correct properties of exponents ($n^{-a} = \frac{1}{n^a}$ and $n^a \times n^b = n^{a+b}$), which are used to derive the assertion. Thus, the reason is the correct explanation for the assertion.

Key Concept: Properties of Powers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The property of exponents states that when two powers with the same base are multiplied, their exponents must be added. For example, $3^2 \times 3^4 = 3^{2+4} = 3^6$. The assertion correctly states this property, and the reason provides the correct explanation for the assertion.

Key Concept: Zero Exponent

b) 1
[Solution Description]
According to the zero exponent rule, any non-zero number raised to the power of 0 is equal to 1. Therefore, $5^0 = 1$.

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

b) 8.5
[Solution Description]
First, evaluate $7^0$ and $4^{-2}$ separately:
$7^0 = 1 \quad \text{(Any non-zero number to the power of 0 is 1)}$
$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
Now add these results and multiply by 8:
$(1 + \frac{1}{16}) \times 8 = \left(\frac{17}{16}\right) \times 8 = \frac{17 \times 8}{16} = \frac{136}{16} = 8.5$

Key Concept: Negative and Zero Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, the assertion states that $5^0 = 1$. According to the properties of exponents, any non-zero number raised to the power of 0 equals 1. Therefore, the assertion is true.
The reason given is that for any non-zero number $n$, $n^0 = 1$. This is a fundamental property of exponents and correctly explains why $5^0 = 1$. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Exponents, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $\frac{1}{10^{-5}} = 10^5$. Using the property of negative exponents, we know that $n^{-a} = \frac{1}{n^a}$. Therefore, $\frac{1}{10^{-5}}$ can be rewritten as $10^5$ because $\frac{1}{\frac{1}{10^5}} = 10^5$. The reason provides the general form of the negative exponent rule, which correctly explains why the assertion is true. Hence, both the assertion and reason are true, and the reason is the correct explanation for the assertion.

Key Concept: Negative Exponents, Multiplicative Inverse

d) $5^4$
[Solution Description]
First, convert all terms to base 5: $25 = 5^2$ and $125 = 5^3$. Rewrite the expression: $5^{-3} \times (5^2)^2 \div (5^3)^{-1} = 5^{-3} \times 5^4 \div 5^{-3}$. When dividing exponents with the same base, subtract them: $5^{-3+4-(-3)} = 5^{1+3} = 5^4$.

Key Concept: Exponent Division

b) $2^{75}$
[Solution Description]
Using the exponent division rule, $n^a \div n^b = n^{a-b}$. Here, $n = 2$, $a = 100$, and $b = 25$. So, $2^{100} \div 2^{25} = 2^{100-25} = 2^{75}$.

Key Concept: Exponent Rules

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that $2^5 \div 2^3 = 4$. Let us verify this:
Step 1: Calculate $2^5$ which is $2 \times 2 \times 2 \times 2 \times 2 = 32$.
Step 2: Calculate $2^3$ which is $2 \times 2 \times 2 = 8$.
Step 3: Divide $32 \div 8 = 4$. So, the Assertion is true.
The Reason states the exponent rule $n^a \div n^b = n^{a-b}$. This is a valid rule as per the syllabus.
Applying this rule to the Assertion:
$2^5 \div 2^3 = 2^{5-3} = 2^2 = 4$, which matches the Assertion. Thus, the Reason correctly explains the Assertion.

Key Concept: Exponent Subtraction Rule

b) $5^{5}$
[Solution Description]
Using the exponent subtraction rule, when dividing powers with the same base, subtract the exponents:
$5^{3} \div 5^{-2} = 5^{3 - (-2)} = 5^{5}$

Key Concept: Combining Exponents

c) $4$
[Solution Description]
When multiplying powers with the same base, add the exponents:
$2^{-3} \times 2^{5} = 2^{-3 + 5} = 2^{2} = 4$

Key Concept: Negative exponents, Division of powers

a) $1$
[Solution Description]
First, simplify the expression inside the parentheses: Multiply $5^3 \times 5^{-5} = 5^{3 + (-5)} = 5^{-2}$. Then, divide by $5^{-2}$: $\frac{5^{-2}}{5^{-2}} = 5^{-2 - (-2)} = 5^0 = 1$. Finally, take the reciprocal as per the negative exponent outside the parentheses: $1^{-1} = 1$.

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: 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: Negative Exponents

c) $\frac{1}{9}$
[Solution Description]
To find the value of $3^{-2}$, we use the property of negative exponents:
$3^{-2} = \frac{1}{3^{2}} = \frac{1}{9}$
Therefore, the correct answer is $\frac{1}{9}$.

Key Concept: Negative exponents, Zero exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that $x^0 + x^{-1} = \frac{x + 1}{x}$. According to the rules of exponents, $x^0 = 1$ for any non-zero $x$ (given in Reason), and $x^{-1} = \frac{1}{x}$ (also given in Reason). Substituting these into the Assertion: $x^0 + x^{-1} = 1 + \frac{1}{x} = \frac{x}{x} + \frac{1}{x} = \frac{x + 1}{x}$. Thus, both Assertion and Reason are true, and the Reason correctly explains the simplification.

Key Concept: Zero Exponent Evaluation

c) $\frac{10}{9}$
[Solution Description]
Break it into two parts:
$(7)^0 = 1 \text{ (any non-zero number to the power 0 is 1)}$
$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
Now add them together:
$1 + \frac{1}{9} = \frac{10}{9}.$

Key Concept: Powers of 10

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's verify the assertion using the reason:
Step 1: According to the negative exponent rule, $n^a = \frac{1}{n^{-a}}$.
Step 2: Substitute $n = 10$ and $a = 3$ in the above formula.
Step 3: We get $10^3 = \frac{1}{10^{-3}}$, which matches the assertion.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Negative Exponents

c) $10^2$
[Solution Description]
Using the rule for negative exponents:
$\frac{1}{10^{-a}} = 10^{a}$
Here, $a = 2$, so:
$\frac{1}{10^{-2}} = 10^{2}$
Hence, the answer is $10^2$.

Key Concept: Negative Exponents - Simplification

b) $\frac{1}{25}$
[Solution Description]
Using the rule for negative exponents, $n^{-a} = \frac{1}{n^a}$. Therefore:
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Key Concept: Magnitude Comparison

b) $3.2 \times 10^8$
[Solution Description]
To compare magnitudes, focus on the exponent $y$ in $x \times 10^y$. A higher exponent indicates a larger magnitude regardless of the coefficient. Here, the exponents are:
- $3.2 \times 10^8$ has exponent 8,
- $5.6 \times 10^6$ has exponent 6.
Since $8 > 6$, $3.2 \times 10^8$ has a larger magnitude.

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

b) $4.75 \times 10^6$
[Solution Description]
To express 4,750,000 in scientific notation, we need to write it as $x \times 10^y$ where $1 \leq x < 10$. First, identify the coefficient by moving the decimal point to the left until there is only one non-zero digit before the decimal: $4.75$. The exponent $y$ is the number of places the decimal was moved, which is 6. Thus, $4,750,000 = 4.75 \times 10^6$.

Key Concept: Scientific Notation

c) $7.2 \times 10^4$
[Solution Description]
To convert 72,000 to scientific notation:
1. Move the decimal point after the first non-zero digit: 7.2
2. Count how many places the decimal moved from its original position (from after the last zero to after 7): 4 places.
3. Thus, $72,000 = 7.2 \times 10^4$.

Key Concept: Scientific Notation, Magnitude and Precision

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To convert $7,200,000,000,000$ to scientific notation:
Step 1: Move the decimal to the left until it is after the first non-zero digit (here, 7). This requires moving the decimal 12 places to the left.
Step 2: The coefficient $x$ must satisfy $1 \leq x < 10$. Here, $x = 7.2$. Step 3: The exponent $y$ is the number of places the decimal was moved, so $y = 12$. Thus, $7,200,000,000,000 = 7.2 \times 10^{12}$. The Reason states that the exponent must match the number of decimal places moved, which is correct and explains why the Assertion is true. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

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

d) $7.2 \times 10^6$
[Solution Description]
To convert 7,200,000 to scientific notation:
1. Identify the coefficient: Move the decimal point after the first non-zero digit (7.2).
2. Count the number of places moved (6 places to the left).
3. Write in the form $x \times 10^y$: $7.2 \times 10^6$.

Key Concept: Scientific Notation

c) $4.5 \times 10^5$
[Solution Description]
To convert 450,000 to scientific notation:
1. Identify the coefficient: Move the decimal point after the first non-zero digit (4.5).
2. Count the number of places moved (5 places to the left).
3. Write in the form $x \times 10^y$: $4.5 \times 10^5$.

Key Concept: Scientific Notation and Comparison of Large Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To compare the numbers, we look at their exponents since they are both in scientific notation. The number of ants is $2 \times 10^{16}$ and the number of trees is $3 \times 10^{12}$. The exponent for ants ($16$) is larger than the exponent for trees ($12$), so $2 \times 10^{16} > 3 \times 10^{12}$, regardless of the coefficients. Hence, the assertion is true. The reason states that the exponent determines the magnitude and correctly explains why $10^{16} > 10^{12}$.

Key Concept: Exponent Comparison

b) Stars in universe ($2 \times 10^{23}$)
[Solution Description]
Compare exponents of given quantities:
1. Milky Way stars ($1 \times 10^{11}$)
2. Stars in universe ($2 \times 10^{23}$)
3. Grains of sand ($1 \times 10^{21}$)
4. Highest exponent is 23 for stars in universe

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: 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

b) $7 \times 10^6$
[Solution Description]
To convert 7,000,000 to scientific notation:
Step 1: Identify the coefficient. The number between 1 and 10 here is 7.
Step 2: Count the number of digits after 7, which is 6.
So, the exponent is 6.
Therefore, the standard form is $7 \times 10^6$.

Key Concept: Standard Form

c) $1.439 \times 10^{12}$ m
[Solution Description]
The given number is $1,439,000,000,000$ m.
Move the decimal point to the left until it is just after the first non-zero digit, which gives $1.439$.
Count the number of places moved (12 places). Therefore, the scientific notation is $1.439 \times 10^{12}$ m.

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

c) $5.9853 \times 10^4$
[Solution Description]
To express $59,853$ in scientific notation:
First, move the decimal point to the left until it is just after the first non-zero digit. Here, we get $5.9853$.
Count the number of places moved (4 places), which gives the exponent for 10. Thus, $59,853 = 5.9853 \times 10^4$.

Key Concept: Scientific notation, Large numbers

c) $3.17 \times 10^{15}$ years
[Solution Description]
The time taken to count all the stars can be found by dividing the total number of stars by the counting rate (1 star per second).
Step-by-step calculation:
1. Total stars = $10^{23}$
2. Counting rate = 1 star/second
3. Time in seconds = $10^{23}$ seconds
Convert seconds to years for better understanding:
1 minute = 60 seconds
1 hour = 3600 seconds
1 day = 86,400 seconds
1 year ≈ 31,536,000 seconds (assuming 365.25 days/year)
Total years = $\frac{10^{23}}{3.1536 \times 10^7} \approx 3.17 \times 10^{15}$ years.

Key Concept: Weight-Based Donation Calculations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The worth of donated jaggery is calculated using the formula:
$\text{Worth of jaggery} = \text{Roxie’s weight} \times \text{cost of 1 kg jaggery}$.
Given Roxie's weight is 45 kg and cost of 1 kg jaggery is Rs.70, the calculation is $45 \times 70 = Rs.3150$.
The reason correctly describes the general method for calculating the worth of any donated item based on weight and unit cost. Therefore, both Assertion and Reason are true, and Reason explains the Assertion correctly.

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: Weight-Donation Cost Calculation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the cost of jaggery donated by Nanjundappa depends on Roxie’s weight and the price per kg of jaggery, which is mathematically represented as $\text{Worth of jaggery} = \text{Roxie’s weight} \times \text{Cost of 1 kg jaggery}$. This is correct because the total cost is calculated by multiplying the weight by the unit price. The reason explains the general principle behind Tulābhāra, where the worth of donated goods depends on the individual’s weight and the commodity's unit price. Since the reason correctly explains the assertion, both are true and related.

Key Concept: Linear Growth vs. Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that linear growth (adding 100 individuals per year) takes longer to double the population size than exponential growth (fixed growth rate). This is true because linear growth increases by a fixed amount each time, whereas exponential growth multiplies the current population by a factor, accelerating the growth rate. The reason correctly explains the difference between additive (linear) and multiplicative (exponential) growth patterns, justifying why exponential growth leads to faster doubling. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Exponential Growth

c) 1.024 cm
[Solution Description]

When a paper is folded once, its thickness becomes double. So, after every fold, the thickness keeps multiplying by 2. The formula is:

$\text{Final Thickness} = \text{Initial Thickness} \times 2^n$

Here, initial thickness $= 0.001\ \text{cm}$ and number of folds $n = 10.$

$0.001 \times 2^{10} = 0.001 \times 1024 = 1.024\ \text{cm}$

So, after folding the paper 10 times, its thickness will be $1.024\ \text{cm}.$

Key Concept: Linear Growth vs. Exponential Growth

c) Covering distance between Earth and Moon with steps of 20 cm each
[Solution Description]
Linear growth involves adding a fixed amount repeatedly, such as walking steps of equal length. Among the options, covering distance with equal-sized steps $(20 + 20 + \ldots)$ is linear growth.

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: Comparing large numbers

c) 1000
[Solution Description]
1 billion is $10^9$ and 1 million is $10^6$. To find how many millions make a billion, divide 1 billion by 1 million:
$\frac{10^9}{10^6} = 10^{9-6} = 10^3 = 1000$
So, 1000 million make 1 billion.

Key Concept: Exponential Notation, Universe Age

c) $4.35 \times 10^{17}$ seconds
[Solution Description]
First, convert 13.8 billion years to seconds:
$13.8 \times 10^9 \text{ years} \times 3.154 \times 10^7 \text{ seconds/year} = 13.8 \times 3.154 \times 10^{16}$
$= 43.5252 \times 10^{16} \approx 4.35 \times 10^{17} \text{ seconds}$

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 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: 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: Power Play

d) 9 zeros
[Solution Description]
In the Indian numbering system:
$1 \, \text{arab} = 100 \, \text{crores} = 100 \times 1,00,00,000 = 1,00,00,00,000$
Counting the zeros gives 9 zeros.

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

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: 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 Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if a pond is fully covered with lotuses on the 30th day and the lotuses double every day, then the pond was half-covered on the 29th day. This is correct because exponential growth involves doubling (multiplying by 2), so one day before full coverage (day 30) would be half coverage (day 29). The reason explains that exponential growth is multiplicative, which correctly justifies the assertion since doubling is an example of multiplicative growth.

Key Concept: Linear vs Exponential Growth, Scientific Notation

c) After 21 hours
[Solution Description]
First, model both populations:
1. Exponential growth: $P_e(t) = 100 \times 2^{t/3}$
2. Linear growth: $P_l(t) = 100 + 50t$
We need to solve:
$100 \times 2^{t/3} - (100 + 50t) > 10,000$
Test option values:
a) t = 24 hours:
$P_e(24) = 100 \times 2^8 = 25,600 \\ P_l(24) = 100 + 50 \times 24 = 1,300 \\ Difference = 24,300 > 10,000$
b) t = 18 hours:
$P_e(18) = 100 \times 2^6 = 6,400 \\ P_l(18) = 100 + 50 \times 18 = 1,000 \\ Difference = 5,400 < 10,000$ c) t = 15 hours: $P_e(15) = 100 \times 2^5 = 3,200 \\ P_l(15) = 100 + 50 \times 15 = 850 \\ Difference = 2,350 10,000$ The smallest t satisfying the condition is 21 hours.

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: Real-World Applications

c) Jupiter is about 318 times heavier than Earth.
[Solution Description]
To compare the masses of Jupiter and Earth, divide Jupiter's mass by Earth's mass:
$\frac{1.898 \times 10^{27}}{5.976 \times 10^{24}} = \frac{1.898}{5.976} \times 10^{27-24} = 0.3176 \times 10^3 \approx 317.6$
Therefore, Jupiter's mass is approximately 318 times that of Earth's mass.

Key Concept: Exponents, Thought Experiments

c) $2.4 \times 10^{11}$
[Solution Description]
Total population = $8 \times 10^9$. Each person has 30 pieces of clothing.
Calculation:
$8 \times 10^9 \times 30 = 240 \times 10^9 = 2.4 \times 10^{11}$
The total number of clothing pieces is $2.4 \times 10^{11}$.

Key Concept: Currency and Large Numbers

c) $1 \times 10^{14}$
[Solution Description]
100 trillion can be expressed numerically as 100,000,000,000,000.
To write this in scientific notation, move the decimal point to just after the first digit (1), giving 1.00.
Count the number of places the decimal moved: 14 places to the left. Hence, the exponent is 14.
Therefore, 100 trillion is written as $1 \times 10^{14}$ in scientific notation.

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: Standard Form Conversion

c) $3.43 \times 10^6$
[Solution Description]
To convert 34,30,000 to standard form:
Step 1: Identify the coefficient by placing the decimal after the first non-zero digit: $3.43$.
Step 2: Count the number of places the decimal has moved from its original position (6 places).
Thus, $34,30,000 = 3.43 \times 10^6$.

Key Concept: Scientific Notation, Exponent Significance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number of stars in the observable universe ($10^{23}$) is much larger than the number of grains of sand on Earth's beaches ($7.5 \times 10^{18}$). This is true as $10^{23} > 7.5 \times 10^{18}$. The reason states that the exponent in scientific notation indicates the order of magnitude, which is correct and explains why the comparison is straightforward. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Scientific Notation

b) $7.004 \times 10^{10}$
[Solution Description]
To convert 70,04,00,00,000 to scientific notation:
Step 1: Count the number of digits after the first digit (7). Here, there are 10 digits after 7.
Step 2: Write the coefficient as a number between 1 and 10. Move the decimal point after the first digit: $7.004$.
Step 3: Multiply by $10$ raised to the power equal to the number of places the decimal moved (10).
Thus, $70,04,00,00,000 = 7.004 \times 10^{10}$.

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.