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, Power of 2

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The thickness of the paper after each fold doubles the previous thickness. So, the formula for thickness after $n$ folds is $T = 0.001 \times 2^n$ cm. For $n = 30$, we get:
$T = 0.001 \times 2^{30} \text{ cm} = 0.001 \times 1,\!073,\!741,\!824 \text{ cm} = 1,\!073,\!741.824 \text{ cm} = 10.737 \text{ km}.$
Thus, the assertion is true. The reason provides the correct formula for calculating the thickness, making it true and the correct explanation of the assertion.

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 $0.001$ cm. After each fold, the thickness doubles. So after $n$ folds, the thickness is given by $0.001 \times 2^n$ cm. For 46 folds:
$0.001 \times 2^{46} = 0.001 \times 7.03687 \times 10^{13}$ cm ≈ $7.03687 \times 10^8$ km ≈ $703,687$ km.
Hence, the assertion is true. The reason correctly explains the assertion as it describes the exponential growth mechanism where each fold doubles the thickness.

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: 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, 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 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

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 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 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: 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: Scientific Notation

c) $4.5 \times 10^5$
[Solution Description]
Scientific notation is expressed as $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. For $450,000$, we move the decimal point 5 places to the left to get $4.5$, so the exponent $y = 5$. Thus, $450,000 = 4.5 \times 10^5$.

Key Concept: Exponential Notation

a) $2^8$
[Solution Description]
To solve $2^3 \times 2^5$, we use the exponent rule $n^a \times n^b = n^{a + b}$. Here, the base ($n$) is the same (which is 2), so we can directly add the exponents: $3 + 5 = 8$. Thus, $2^3 \times 2^5 = 2^{8}$.

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

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

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 Growth: Doubling Effect

b) 4.096 cm
[Solution Description]
The thickness doubles after each fold. For 12 folds, the total increase in thickness can be calculated using the formula:
$\text{Final Thickness} = \text{Initial Thickness} \times 2^n$
Here, $n = 12$ and Initial Thickness = 0.001 cm.
Substituting the values:
$\text{Final Thickness} = 0.001 \, \text{cm} \times 2^{12}$
Calculate $2^{12} = 4096$:
$0.001 \, \text{cm} \times 4096 = 4.096 \, \text{cm}$
Therefore, the thickness after 12 folds is 4.096 cm.

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: 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 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: 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: 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: Multiplying exponents, Dividing exponents

c) 1
[Solution Description]
First, simplify the numerator $5^6 \times 5^{-2}$ using the rule $n^a \times n^b = n^{a + b}$. So, $5^6 \times 5^{-2} = 5^{6 + (-2)} = 5^4$. Next, simplify the denominator $5^3 \div 5^{-1}$ using the rule $n^a \div n^b = n^{a - b}$. So, $5^3 \div 5^{-1} = 5^{3 - (-1)} = 5^4$. Now, the expression becomes $\left(\frac{5^4}{5^4}\right)^2 = \left(5^{4 - 4}\right)^2 = \left(5^0\right)^2 = 1^2 = 1$.

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: 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: 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: Prime Factorization, Exponentiation

b) $2^3 \times 3^5 \times 5$
[Solution Description]
First, perform prime factorization of 9720:
Divide by 2:
9720 ÷ 2 = 4860
4860 ÷ 2 = 2430
2430 ÷ 2 = 1215
1215 is divisible by 3:
1215 ÷ 3 = 405
405 ÷ 3 = 135
135 ÷ 3 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
Finally, 5 ÷ 5 = 1
So, the prime factors are:
2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5
In exponential form:
2^3 × 3^5 × 5^1
Therefore, the correct answer is $2^3 \times 3^5 \times 5$.

Key Concept: Exponential form representation

d) $5^2 \times 7^3$
[Solution Description]
The expression $5 \times 5 \times 7 \times 7 \times 7$ represents two 5's and three 7's multiplied together.
In exponential form, this becomes $5^2 \times 7^3$.

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: 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: 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: 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 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: 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: 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: Negative Exponents

c) 5
[Solution Description]
To simplify $5^{-3} \times 5^4$, we use the exponent rule $n^a \times n^b = n^{a+b}$. Here, $a = -3$ and $b = 4$.
$5^{-3} \times 5^4 = 5^{-3+4} = 5^1 = 5$

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 $5^0 = 1$. According to the exponent rules, any non-zero number raised to the power of zero is indeed equal to one. This is derived from the property $x^a ÷ x^a = x^{a-a} = x^0 = 1$, where $x ≠ 0$. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Exponents and Division

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $2^{10} \div 2^4$, we use the exponent division rule: $\frac{a^m}{a^n} = a^{m-n}$ where $a \neq 0$. Here, the base is 2, so:
$2^{10} \div 2^4 = 2^{10-4} = 2^6$.
The assertion correctly applies the exponent division rule, and the reason accurately states the rule used. Therefore, both are true, and the reason explains 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: 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: Exponent subtraction rule, Negative exponents

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve $\frac{n^5 \times n^{-2}}{n^{-3}}$, first multiply the numerator terms by adding their exponents: $n^5 \times n^{-2} = n^{5 + (-2)} = n^3$. Then, divide by the denominator using the exponent subtraction rule: $\frac{n^3}{n^{-3}} = n^{3 - (-3)} = n^{6}$. The assertion is correct as it simplifies to $n^6$. The reason correctly states the rules applied in the simplification, explaining why the assertion holds true.

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: 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) $4$
[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: Exponent Property Application

b) $-4$
[Solution Description]
Rewrite $\frac{1}{16}$ as a power of 2:
$\frac{1}{16} = 2^{-4} \text{ because } 16 = 2^4.$
So,
$2^x = 2^{-4} \implies x = -4.$

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: 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: 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: Negative Exponents, Powers of 10

b) 5
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Here, $n = 10$ and $-a = -5$, so $a = 5$.

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

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, 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: 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, 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, 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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To express 7,00,00,000 in scientific notation, we first identify the coefficient and exponent. The coefficient must be a number between 1 and 10, which for this number is 7. Now, count the number of digits after the first digit (7): there are 6 zeros, so the exponent is 6. Therefore, the number can be written as $7 \times 10^6$. The reason correctly explains that the coefficient must be between 1 and 10 and how the exponent is determined by counting the digits after the first digit. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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) $4.5 \times 10^5$
[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, 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) $3.05 \times 10^4$
[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

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]
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: Historical Context

b) 100
[Solution Description]
To find how many ayutas make up one niyuta, divide the value of a niyuta by the value of an ayuta:
$\frac{10^{11}}{10^9} = 10^{11-9} = 10^2 = 100$
Therefore, 100 ayutas make up one niyuta.

Key Concept: Scientific Notation, Real-World Comparisons

a) $2 \times 10^{12}$
[Solution Description]
To find how many times more stars are in the observable universe compared to the Milky Way, divide the number of stars in the universe by the number of stars in the Milky Way.
Number of stars in the universe = $2 \times 10^{23}$
Number of stars in the Milky Way = $1 \times 10^{11}$
Ratio = $\frac{2 \times 10^{23}}{1 \times 10^{11}} = 2 \times 10^{(23-11)} = 2 \times 10^{12}$
So, the observable universe has $2 \times 10^{12}$ times more stars than the Milky Way.

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

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: 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: 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: Coin Weight Calculation

b) 6429 coins
[Solution Description]
First, convert Roxie's weight from kilograms to grams:
$45 \text{ kg} = 45 \times 1000 = 45000 \text{ grams}$
Next, divide the total weight in grams by the weight of one coin:
$\text{Number of coins} = \frac{45000}{7} \approx 6428.57$
Since we can't have a fraction of a coin, we round up to the nearest whole number: 6429 coins.

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: 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 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: 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

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: 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: 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: Large Number Naming Systems

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 number system, which is correct. The reason explains that a crore is 100 lakhs, and since 1 lakh is $10^5$, multiplying them gives $10^7$. This correctly justifies the assertion.

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: 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: Scientific Notation

b) $3 \times 10^{20}$
[Solution Description]
To convert the given number to scientific notation:
$30,00,00,00,00,00,00,00,00,000 = 3 \times 10^{20} \, \text{meters}$
We count the number of zeroes after the digit '3', which is 20, and write it in the form $x \times 10^y$.

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: 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, Pond Example

c) Day 27
[Solution Description]
Since coverage doubles daily:
Day 30 = 100% → Day 29 = 50% → Day 28 = 25% → Day 27 = 12.5%
Using exponential formula:
$\text{Coverage} = 100\% \times 2^{-(30-n)} = 12.5\%$
Solving: $2^{-(30-n)} = 0.125 = 2^{-3}$ ⇒ n = 27

Key Concept: Linear vs Exponential Comparison

d) Number of bacteria doubling every hour in a lab culture
[Solution Description]
Exponential growth involves repeated multiplication, such as doubling or tripling. Among the options: folding paper (thickness doubles with each fold), steps to cover distance (linear addition), and monthly salary increment (fixed addition) are not exponential. The number of bacteria doubling every hour is an example of exponential growth because it involves multiplication.

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: Exponential Growth Calculation

c) 29th day[Solution Description]Exponential growth involves multiplication, where the quantity doubles each day. If the pond is fully covered on the $30^{th}$ day, it means the coverage doubled from the previous day. Hence, on the $29^{th}$ day, the pond must have been half-covered because doubling the coverage on the $29^{th}$ day results in full coverage on the $30^{th}$ 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: 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: Indian System, Exponents

c) 250,000 crore
[Solution Description]
From the Indian system, 1 kharab $= 10^{11}$ rupees and 1 crore $= 10^7$ rupees. To convert kharab to crore, divide:
$\frac{25 \times 10^{11}}{10^7} = 25 \times 10^4 = 250000 \text{ crore}$
Thus, 25 kharab rupees equals 250,000 crore rupees.

Key Concept: Scientific Notation Conversion

b) 1,43,35,00,00,000 m
[Solution Description]
To convert $1.4335 \times 10^{12}$ to the Indian number system:
First, calculate $10^{12} = 1$ lakh crore.
Then, multiply by the coefficient: $1.4335 \times 1$ lakh crore $= 1.4335$ lakh crore.
This equals approximately 1,43,35,00,00,000 m.

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.

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, Real-World Application

c) $3.17 \times 10^{15}$ years
[Solution Description]
Step 1: Calculate the total time required in seconds.
Total stars = $10^{23}$
Time per star = 1 second
Total time = $10^{23}$ seconds
Step 2: Convert seconds to years.
Seconds in a year = $3.15 \times 10^7$
Total years = $\frac{10^{23}}{3.15 \times 10^7} = \frac{1}{3.15} \times 10^{23-7} = 0.317 \times 10^{16}$
Step 3: Adjust the coefficient to be between 1 and 10.
$0.317 \times 10^{16} = 3.17 \times 10^{15}$ years
It would take approximately $3.17 \times 10^{15}$ years.