Key Concept: Exponential Growth

b) 0.032 cm
[Solution Description]
The thickness doubles with each fold. After 5 folds, the thickness can be calculated as:
$0.001 \times 2^5 = 0.001 \times 32 = 0.032 \, \text{cm}$

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: Exponential Growth

c) 0.128 cm
[Solution Description]
Using the formula for exponential growth:
$0.001 \times 2^7 = 0.001 \times 128 = 0.128 \, \text{cm}$

Key Concept: Exponential Growth

b) 1.024 cm
[Solution Description]
The initial thickness of the paper is $0.001$ cm. Each fold doubles the thickness. After $n$ folds, the thickness becomes $0.001 \times 2^n$ cm. For $n = 10$, the thickness is:
$0.001 \times 2^{10} = 0.001 \times 1024 = 1.024 \, \text{cm}$

Key Concept: Exponentiation

b) $2^4$
[Solution Description]
Using the property $\frac{m^a}{n^a} = \left(\frac{m}{n}\right)^a$,
$\frac{10^4}{5^4} = \left( \frac{10}{5} \right)^4 = 2^4.$

Key Concept: Exponential Growth

c) $0.008$ cm
[Solution Description]
The thickness doubles with each fold. After the first fold, it becomes $0.001 \times 2 = 0.002$ cm. After the second fold, it becomes $0.002 \times 2 = 0.004$ cm. After the third fold, it becomes $0.004 \times 2 = 0.008$ cm or $0.001 \times 2^3 = 0.008$ cm.

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

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: Exponential Growth in Paper Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given table, the initial thickness of the paper is $0.001 cm$. After each fold, the thickness doubles. Using the formula for exponential growth:
Thickness after $n$ folds = Initial thickness $\times 2^n$.
For $n = 7$, the thickness becomes $0.001 cm \times 2^7 = 0.128 cm$, which matches the assertion. The reason correctly explains why the thickness is $0.128 cm$ after 7 folds because it describes the doubling mechanism.

Key Concept: Negative Exponents

c) $\frac{1}{25}$
[Solution Description]
Using the negative exponent rule $n^{-a} = \frac{1}{n^a}$, we have $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$.

Key Concept: Operations with Exponents, Negative Exponents

a) $\frac{2}{625}$
[Solution Description]
To evaluate $\left(\frac{2^{-3} \times 5^2}{10^{-2}}\right)^{-1}$:
Step 1: Rewrite the denominator $10^{-2}$ as $\frac{1}{10^2}$.
The expression becomes $\left(2^{-3} \times 5^2 \times 10^2\right)^{-1}$.
Step 2: Simplify $10^2$ to $(2 \times 5)^2 = 2^2 \times 5^2$.
The expression now is $\left(2^{-3} \times 5^2 \times 2^2 \times 5^2\right)^{-1}$.
Step 3: Combine like terms using $n^a \times n^b = n^{a+b}$.
For 2: $-3 + 2 = -1$. For 5: $2 + 2 = 4$.
So, $\left(2^{-1} \times 5^4\right)^{-1}$.
Step 4: Apply the exponent $-1$ to each term inside the parentheses.
$(2^{-1})^{-1} = 2^1$, $(5^4)^{-1} = 5^{-4}$.
Final result: $2^1 \times 5^{-4} = \frac{2}{5^4}$.
Simplified numerical value: $\frac{2}{625}$.

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: Exponential Notation and Operations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $(-3)^4 \times (-3)^5 = (-3)^9$. According to the rule of exponents for multiplication with the same base, $n^a \times n^b = n^{a+b}$. Here, the base is $-3$, and the exponents are $4$ and $5$. Adding them gives $4 + 5 = 9$. Therefore, $(-3)^4 \times (-3)^5 = (-3)^{9}$ is correct. The reason correctly explains the assertion by stating the exponent addition rule for multiplication with the same base. Both the assertion and reason are true, and the reason justifies the assertion.

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: 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: 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: 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: 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: 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: Exponential Form

c) $(-3)^3 \times 2^2$
[Solution Description]
To convert the product into exponential form, count how many times each base is multiplied by itself. Here, $-3$ is multiplied three times, and $2$ is multiplied twice. Therefore:
$(-3) \times (-3) \times (-3) \times 2 \times 2 = (-3)^3 \times 2^2$
The correct exponential form is $(-3)^3 \times 2^2$.

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: 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: 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: Power of a Power

b) $2^{12}$
[Solution Description]
Using the power of a power rule, $(n^a)^b = n^{a \times b}$.
Here, $a = 4$ and $b = 3$, so $(2^4)^3 = 2^{4 \times 3} = 2^{12}$.
Therefore, the answer is $2^{12}$.

Key Concept: Operations with exponents, Zero exponent

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, evaluate the expression inside the parentheses: $3^5 \div 3^{-2}$. Using the division rule of exponents, $n^a \div n^b = n^{a-b}$, we get:
$3^5 \div 3^{-2} = 3^{5-(-2)} = 3^{7}.$
Now, raise this result to the power of zero: $(3^7)^0$. According to the zero exponent rule, any non-zero number raised to the power of zero is equal to $1$. Hence, $(3^7)^0 = 1$.
Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

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: 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: 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, 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: 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: Division of Powers with the Same Base

c) $7^3$
[Solution Description]
To simplify $\frac{7^8}{7^5}$, we use the division of powers property: $n^a \div n^b = n^{a - b}$. Here, the base is the same (7), so we subtract the exponents:
$\frac{7^8}{7^5} = 7^{8-5} = 7^3$
Therefore, the simplified form of the expression is $7^3$.

Key Concept: Zero exponent, Negative exponents

b) $\frac{8}{9}$
[Solution Description]
Step 1: Simplify the exponents inside the brackets. Any number to the power 0 is 1 ($5^0 = 1$, $4^0 = 1$):
$\frac{1 \times 2^{-3}}{3^{-2} \times 1} = \frac{2^{-3}}{3^{-2}}$
Step 2: Rewrite negative exponents using $n^{-a} = \frac{1}{n^a}$:
$\frac{\frac{1}{2^3}}{\frac{1}{3^2}} = \frac{3^2}{2^3} = \frac{9}{8}$
Step 3: Take the reciprocal of the entire expression because of the outer $-1$ exponent:
$\left( \frac{9}{8} \right)^{-1} = \frac{8}{9}$
The final value is $\frac{8}{9}$.

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: 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: 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: 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: Negative exponents and zero exponents

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that for any non-zero number $n$, $n^0 = 1$, which is true because it follows from the exponent division rule: $n^a ÷ n^a = n^{a–a} = n^0$. The reason explains why $0^0$ is undefined, as applying the exponent division rule would involve division by zero, which is not allowed. Both statements are true, but the reason does not explain the assertion; it provides a related but separate explanation about $0^0$.

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: 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: Exponent Division

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

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: 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 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) $3^5$
[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 exponents

c) $\frac{1}{25}$
[Solution Description]
We know that $n^{-k} = \frac{1}{n^k}$. Thus, for $5^{-2}$:
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
The simplified form is $\frac{1}{25}$.

Key Concept: Exponent Subtraction Rule

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this, apply the exponent subtraction rule: $n^a \div n^b = n^{a-b}$. Here, $a = 3$ and $b = -1$.
Calculation: $5^3 \div 5^{-1} = 5^{3 - (-1)} = 5^{4}$.
The Assertion is true because $5^3 \div 5^{-1} = 5^4$.
The Reason is also true as it correctly states the exponent subtraction rule. Moreover, the Reason explains why the Assertion holds true.

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 and Zero Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that $5^0 = 1$, which is true as per the general rule that any non-zero number raised to the power of 0 is 1. The reason provides the correct explanation for this: when you divide $n^a$ by itself, it equals $n^{a-a} = n^0$, and any number divided by itself is 1. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

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: Zero Exponents, Real-world Context

d) $96$ grams
[Solution Description]
To find the original size of the sample, we will use the concept of negative exponents in exponential decay.
Step 1: Given the equation $S \times 2^{-5} = 3$, rewrite $2^{-5}$ as $\frac{1}{2^5}$:
$S \times \frac{1}{32} = 3$
Step 2: Solve for $S$ by multiplying both sides by 32:
$S = 3 \times 32$
Step 3: Perform the multiplication:
$S = 96$
The original size of the sample was $96$ grams.

Key Concept: Powers of 10 - Expanded Form

a) $(3 \times 10^3) + (4 \times 10^2) + (7 \times 10^1) + (5 \times 10^0)$
[Solution Description]
To express $3475$ in expanded form using powers of 10, we break it into each digit multiplied by its place value:
$3475 = (3 \times 10^3) + (4 \times 10^2) + (7 \times 10^1) + (5 \times 10^0)$

Key Concept: Powers of 10, Negative Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $10^{-5} = \frac{1}{10^5}$. This is correct because, by definition, a negative exponent indicates the reciprocal of the base raised to the positive exponent. The reason provides the general form of this property: for any non-zero number $n$ and integer $a$, $n^{-a} = \frac{1}{n^a}$. Since the assertion aligns with the general rule stated in the reason, and the reason correctly explains the assertion, both are true and the reason is the correct explanation of the assertion.

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: 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: 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) $4.5 \times 10^3$
[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

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) $6.03 \times 10^6$
[Solution Description]
To convert 6,030,000 to scientific notation:
1. Move the decimal point after the first non-zero digit: 6.03
2. Count how many places the decimal moved from its original position (from after the last zero to after 6): 6 places.
3. Thus, $6,030,000 = 6.03 \times 10^6$.

Key Concept: Scientific Notation, Population Representation

b) $8.5 \times 10^7$
[Solution Description]
The given population is 8,50,00,000.
Step 1: Rewrite the number without commas: 85000000.
Step 2: Count the number of digits after the first digit (8). Total digits = 8.
Step 3: Move the decimal point to after the first non-zero digit (8.5).
Step 4: The exponent is the number of places moved: 7.
So, the scientific notation is $8.5 \times 10^7$.

Key Concept: Scientific Notation

c) 1,439,000,000,000 m
[Solution Description]
To convert $1.439 \times 10^{12}$ m to standard form:
1. The exponent $10^{12}$ means moving the decimal point 12 places to the right.
2. Starting from $1.439$, moving the decimal 12 places gives $1,439,000,000,000$ m.

Key Concept: Scientific Notation Conversion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion is true because $1,000,000 = 1 \times 10^6$, where the coefficient is 1 (which lies between 1 and 10) and the exponent 6 shows the decimal moved 6 places to the left. Reason is also true because it correctly describes the definition of scientific notation: the coefficient must satisfy $1 \leq x < 10$, and the exponent $y$ represents the number of decimal shifts. Further, the Reason accurately explains why the Assertion is correct.

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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To write a number in scientific notation, we express it as $x \times 10^y$, where $1 \leq x < 10$. Here, $5,000$ is indeed equal to $5 \times 10^3$ because moving the decimal point three places to the left gives $5.000 = 5.0$ and the exponent becomes $3$. The Reason correctly explains that the exponent ($3$ here) represents the number of places the decimal moves, which corresponds to the number of zeros ($3$) after the digit $5$ in $5,000$.

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

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, 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]
To express the number $50,000$ in scientific notation:
Step 1: Identify the coefficient by moving the decimal point to the left until it is between the first and second digit. For $50,000$, the decimal point is after the last zero, so moving it left 4 places gives $5.0$ (which simplifies to $5$).
Step 2: Count the number of places the decimal was moved, which is 4, hence the exponent is $4$.
Therefore, $50,000 = 5 \times 10^4$.
The reason correctly explains that the coefficient must be between $1$ and $10$, and the exponent indicates the number of decimal places moved.

Key Concept: Powers of 10

b) $(1 \times 10^2) + (7 \times 10^1) + (2 \times 10^0)$
[Solution Description]
Break down the number $172$ into its expanded form using powers of 10 as follows:
$172 = (1 \times 100) + (7 \times 10) + (2 \times 1) = (1 \times 10^2) + (7 \times 10^1) + (2 \times 10^0)$.

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: Tulābhāra Donation Cost

b) Rs.3150
[Solution Description]
To find the worth of the donated jaggery, multiply Roxie's weight by the cost per kg of jaggery.
Given:
- Roxie's weight = 45 kg
- Cost of 1 kg jaggery = Rs.70
Calculation:
$\text{Worth of jaggery} = 45 \times 70 = Rs.3150$

Key Concept: Age calculation, Weight estimation

b) 116,160 hours
[Solution Description]
To find Roxie's age in hours, we need to convert her age from days to hours. There are 24 hours in a day. So, we multiply the number of days by 24.
Step-by-step calculation:
1. Number of days = 4840
2. Hours in a day = 24
3. Total hours = $4840 \times 24$
Now, let's compute $4840 \times 24$:
Break it down using distributive property:
$4840 \times 20 = 96,800$
$4840 \times 4 = 19,360$
Add them together:
$96,800 + 19,360 = 116,160$ hours.

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: 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: 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 building a ladder to the Moon with 20 cm steps would require 1,92,20,00,000 steps. This is true because linear growth involves a fixed increase (additive process) with each step. The reason correctly explains that linear growth is additive, as it involves a constant addition (20 cm per step). Therefore, both the assertion and reason are true, and the reason properly explains the assertion.

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: Exponential Growth

b) 0.128 cm
[Solution Description]
The thickness after $n$ folds follows exponential growth as $0.001 \times 2^n$ cm. For 7 folds:
Thickness = $0.001 \times 2^7$ cm = $0.001 \times 128$ cm = 0.128 cm.

Key Concept: Understanding large numbers

c) $10^7$
[Solution Description]
1 crore is defined as $10^7$, which is 1 followed by 7 zeros (1,00,00,000). Let's verify each option:
a) $10^5$ = 1 lakh (1,00,000) – Incorrect.
b) $10^6$ = 1 million (1,000,000) – Incorrect.
c) $10^7$ = 1 crore (1,00,00,000) – Correct.
d) $10^8$ = 10 crore (10,00,00,000) – Incorrect.

Key Concept: Large Number Comparisons, Historical Context

b) niyuta
[Solution Description]
The Lalitavistara mentions that 100 ayutas make a niyuta, and since an ayuta is $10^9$, multiplying by 100 gives:
$100 \times 10^9 = 10^2 \times 10^9 = 10^{11} \text{ (niyuta)}$

Key Concept: Comparison of Large Numbers

b) $2^{100}$
[Solution Description]
We compare $100^2$ and $2^{100}$:
- $100^2 = 10,000$
- $2^{100}$ can be simplified using logarithms or exponentiation:
$2^{10} = 1024 \approx 10^3$
$2^{100} = (2^{10})^{10} \approx (10^3)^{10} = 10^{30}$
Since $10^{30} > 10,000$, $2^{100}$ is larger than $100^2$.

Key Concept: Population Calculation

b) $2 \times 10^9$
[Solution Description]
The total population of sheep and goats is the sum of their individual populations:
$10^9 + 10^9 = 2 \times 10^9$
So, the total population is $2 \times 10^9$.

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 and Magnitude

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Scientific notation is written as $x \times 10^y$, where $x$ is a number between 1 and 10, and $y$ is an integer exponent. The exponent $y$ indicates the power of 10 by which the coefficient $x$ is multiplied, thus representing the order of magnitude of the quantity. For example, $3 \times 10^{20} \, \text{m}$ (distance to Milky Way center) has a much larger magnitude than $5.976 \times 10^{24} \, \text{kg}$ (Earth's mass) despite having a smaller coefficient because $20 < 24$. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Powers of 10, Scientific notation

c) 10,000 times larger
[Solution Description]
To find how many times larger $10^{82}$ is compared to $10^{78}$, we divide them:
$\frac{10^{82}}{10^{78}} = 10^{82-78} = 10^4$.
$10^4$ means 10,000 times larger.

Key Concept: Powers of 10

c) $10^9$
[Solution Description]
In the Indian numbering system, 1 arab is equal to 100 crores. Since 1 crore is $10^7$, then 1 arab would be $100 \times 10^7 = 10^9$.

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: Linear Growth

c) 1,922,000,000 steps
[Solution Description]
To find the number of 20 cm steps required to cover 3,84,400 km:
First, convert 3,84,400 km to cm:
$3,84,400 \text{ km} = 3,84,400 \times 1,00,000 \text{ cm} = 3,84,40,00,00,000 \text{ cm}$
Now, divide by step size (20 cm):
$\frac{3,84,40,00,00,000}{20} = 1,92,20,00,000 \text{ steps}$
Hence, 1,92,20,00,000 steps are needed.

Key Concept: Linear vs Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion (A) correctly defines linear and exponential growth. Linear growth is additive (e.g., $20 + 20 + 20 \ldots$), while exponential growth is multiplicative (e.g., $0.001 \times 2 \times 2 \ldots$). Reason (R) provides a valid example of linear growth, where each step adds a fixed distance (20 cm). Thus, both are true, and (R) explains (A).

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: 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, Large Numbers

b) $5 \times 10^{12}$
[Solution Description]
First, identify the given number of stars: $1 \times 10^{11}$. Multiply this by 50 to find the number of stars in the new galaxy.
Calculation:
$1 \times 10^{11} \times 50 = 50 \times 10^{11}$
To express this in proper scientific notation, adjust the coefficient to be between 1 and 10:
$5 \times 10^{12}$
So, the correct answer is $5 \times 10^{12}$.

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, 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 and Exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine which number is larger, compare their exponents first. Here, $2.5 \times 10^6$ has an exponent of 6, whereas $3.4 \times 10^5$ has an exponent of 5. Since 6 > 5, $2.5 \times 10^6$ is indeed greater than $3.4 \times 10^5$, regardless of the coefficients. Therefore, the Assertion is true. The Reason states that the magnitude of a number in scientific notation is primarily determined by its exponent, which is also true and correctly explains why the Assertion holds. Thus, both are true, and the Reason is the correct explanation for the Assertion.

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: Scientific Notation, Exponent Importance

c) $8.02 \times 10^4$
[Solution Description]
To find how many times farther the exoplanet is compared to the Sun:
Step 1: Write down both distances in scientific notation.
- Exoplanet distance: $1.2 \times 10^{16}$ m
- Sun distance: $1.496 \times 10^{11}$ m
Step 2: Divide the exoplanet distance by the Sun distance.
$\frac{1.2 \times 10^{16}}{1.496 \times 10^{11}} = \frac{1.2}{1.496} \times 10^{16-11} = 0.80214 \times 10^5$
Step 3: Adjust the coefficient to be between 1 and 10.
$0.80214 \times 10^5 = 8.0214 \times 10^4$
The exoplanet is approximately $8.02 \times 10^4$ times farther than the Sun.