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: 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, 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: 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: 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

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: 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: 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: 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: 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: Prime Factorization, Exponential Form

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

Key Concept: Exponential Notation

b) $3^5$
[Solution Description]
The expression $3 \times 3 \times 3 \times 3 \times 3$ has the base 3 multiplied by itself 5 times. Therefore, the exponential form is $3^5$.

Key Concept: Exponential Operations: Power Rules

c) $3^3$
[Solution Description]
Using the exponent rule $n^a \times n^b = n^{a+b}$, first multiply the terms in the numerator:
$3^4 \times 3^2 = 3^{4+2} = 3^6$
Now, divide by the denominator using the rule $n^a \div n^b = n^{a-b}$:
$3^6 \div 3^3 = 3^{6-3} = 3^3$
Therefore, the simplified form is $3^3$.

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: Scientific Notation

b) $1.31 \times 10^2$ cm
[Solution Description]
The thickness is approximately $131 \text{ cm}$. In scientific notation, this is written as:
$1.31 \times 10^2 \text{ cm}.$

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 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: Power Notation

b) $5^4$
[Solution Description]
The expression $5 \times 5 \times 5 \times 5$ means multiplying 5 by itself 4 times. In exponential notation, this is written as $5^4$.

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: 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: Power play, Operations with exponents

c) 128
[Solution Description]
First, simplify the numerator $(2^3 \times 2^5) \div (2^2)^2$. Using $n^a \times n^b = n^{a + b}$, $2^3 \times 2^5 = 2^{3 + 5} = 2^8$. Using $(n^a)^b = n^{a \times b}$, $(2^2)^2 = 2^{2 \times 2} = 2^4$. Now, $2^8 \div 2^4 = 2^{8 - 4} = 2^4$. Next, simplify the denominator $(2^{-1})^3 = 2^{-1 \times 3} = 2^{-3}$. Finally, divide the numerator and denominator: $\frac{2^4}{2^{-3}} = 2^{4 - (-3)} = 2^7 = 128$.

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: 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: 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: Multiplication of exponents with the same base

b) $5^{7}$
[Solution Description]
To solve $5^3 \times 5^4$, use the rule $n^a \times n^b = n^{a + b}$. Here, the base $n = 5$, exponent $a = 3$, and exponent $b = 4$. So, $5^3 \times 5^4 = 5^{3 + 4} = 5^7$.

Key Concept: Prime Factorization and Exponential Representation

b) $2^3 \times 3^4$
[Solution Description]
To find the prime factorization of $648$, divide it by primes:
648 ÷ 2 = 324
324 ÷ 2 = 162
162 ÷ 2 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Prime factors: $2^3 \times 3^4$.

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: 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: Exponentiation, Verification

b) -6912
[Solution Description]
Step 1: Calculate $(-3)^2 = 9$
Step 2: Calculate $4^3 = 64$
Step 3: Multiply results from Step 1 and Step 2:
9 × 64 = 576
Step 4: Simplify denominator:
$2^{-2} = \frac{1}{2^2} = \frac{1}{4}$
$(-3)^{-1} = -\frac{1}{3}$
So denominator is $\frac{1}{4} \times -\frac{1}{3} = -\frac{1}{12}$
Step 5: Divide numerator (576) by denominator (-1/12):
576 ÷ (-1/12) = 576 × (-12) = -6912
Final simplified expression evaluates to $-6912$.

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: 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 Exponents

b) $\frac{1}{25}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this to $5^{-2}$:
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

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: 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: 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: 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: 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

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 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: Exponential Growth

b) 29th day
[Solution Description]
Since the number of lotuses doubles every day, the pond would have been half-covered one day before it was fully covered. Hence, if full coverage occurs on the 30th day, half-coverage occurred on the 29th day.

Key Concept: Fractional Exponents, Power Division

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

Key Concept: Combination Calculation

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

Key Concept: Negative exponents

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, Negative exponents

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

Key Concept: 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: 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 Exponent Simplification

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

Key Concept: 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: Negative Exponents, Zero Exponents

b) $\frac{1}{9}$
[Solution Description]
To solve this problem, we will use the concepts of negative exponents and zero exponents.
Step 1: Evaluate $(-3)^{-2}$ using the negative exponent rule:
$(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$
Step 2: Evaluate $4^0$ using the zero exponent rule:
$4^0 = 1$
Step 3: Multiply the results from Step 1 and Step 2:
$\frac{1}{9} \times 1 = \frac{1}{9}$
The final 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: Large Numbers Notation

c) 7
[Solution Description]
In the Indian numbering system:
$1 \text{ crore} = 1,00,00,000$
Counting the zeros in 1,00,00,000 gives us 7 zeros.

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, Expanded Form

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

Key Concept: Powers of 10 - Basic Conversion

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

Key Concept: Scientific Notation, Large Numbers

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

Key Concept: Scientific Notation

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, Comparison

c) Distance between Sun and Earth
[Solution Description]
To compare the distances, observe the exponents:
1. $1.4335 \times 10^{12}$ has exponent 12,
2. $1.439 \times 10^{12}$ also has exponent 12,
3. $1.496 \times 10^{11}$ has exponent 11.
Since $10^{11} < 10^{12}$, the distance between the Sun and Earth is the smallest. Even if we compare coefficients, the exponent difference outweighs any coefficient comparison.

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, Mass Conversion

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

Key Concept: 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

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, Large Numbers

b) $3.0 \times 10^{22}$ m
[Solution Description]
To convert the given large number to scientific notation, we need to write it in the form $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer. Given number: 30,00,00,00,00,00,00,00,00,000 m Step 1: Count the number of digits after the first digit (3). The number can be written as 3 followed by 22 zeros, so total digits = 23. Step 2: Express the number as $3.0 \times 10^{22}$ m, since moving the decimal point 22 places to the right gives the original number. Therefore, the correct scientific notation is $3.0 \times 10^{22}$ m.

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 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: 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: Scientific Notation, Magnitude Comparison

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

Key Concept: Scientific Notation, Exponent Importance

b) 11.52 times
[Solution Description]
Step 1: Divide the distance to Jupiter by the distance to Mars to find the ratio.
$\frac{6.29 \times 10^8}{5.46 \times 10^7} = \frac{6.29}{5.46} \times 10^{8-7} = 1.152 \times 10^1 = 11.52$
Step 2: Answer is approximately $11.52$ times.

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

d) $4.25 \times 10^4$
[Solution Description]
To convert 42,500 to scientific notation:
Step 1: Rewrite the number as 4.25 (between 1 and 10).
Step 2: Count how many places the decimal moves from its original position to after 4.25, which is 4.
Thus, the standard form is $4.25 \times 10^4$.

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: Age in Hours

b) 116,160 hours
[Solution Description]
To convert days to hours, multiply the number of days by 24: $4840 \times 24$. Calculating step-by-step: First, multiply 4000 × 24 = 96,000. Then, multiply 800 × 24 = 19,200. Finally, multiply 40 × 24 = 960. Add them together: 96,000 + 19,200 = 115,200; 115,200 + 960 = 116,160 hours.

Key Concept: Assertion-Reason

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

Key Concept: 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: Exponential Growth, Time scales

b) 94,500 seconds
[Solution Description]

The bacterial population doubles every hour, so we use the formula:

$P = 100 \times 2^t$

where P is the final population and t is the time in hours.

We need the population to reach 8 billion:

$100 \times 2^t = 8,000,000,000$

$2^t = \frac{8,000,000,000}{100}$

$2^t = 80,000,000$

Now,

$t = \log_2(80,000,000) \approx 26.25 \text{ hours}$

Convert hours into seconds:

$26.25 \times 3600 = 94,500 \text{ seconds}$

So, it would take approximately 94,500 seconds, or about 26.25 hours, for the bacterial population to reach 8 billion cells.

Key Concept: Linear vs Exponential, Real-World Context

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

For exponential growth:

$T = 0.001 \times 2^n$

For linear growth:

$L = 0.001 + 0.002n$

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

For example, after 20 folds:

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

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

Linear growth after 20 folds:

$L = 0.001 + 0.002(20)$

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

Now compare them:

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

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

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

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

Linear growth adds only:

$0.002 \text{ cm}$

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

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

Key Concept: Exponential Growth

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, Historical Comparisons

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

Key Concept: Indian and International Number Systems, Powers of 10

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this, we first verify the assertion:
1. According to the Indian system, $1 \text{ kharab} = 100 \times 1 \text{ arab} = 100 \times 1,00,00,00,000 = 1,00,00,00,00,000$ or $10^{11}$.
2. In the international system, $100 \text{ billion} = 100 \times 1,000,000,000 = 100,000,000,000$ or $10^{11}$.
Thus, the Assertion is true.
Next, we examine the Reason:
- In the Indian system, each term is multiplied by $100$ (e.g., $1 \text{ lakh} = 100 \times 10^3$, $1 \text{ crore} = 100 \times 10^5$).
- In the international system, each term is multiplied by $1000$ (e.g., $1 \text{ million} = 1000 \times 10^3$, $1 \text{ billion} = 1000 \times 10^6$).
Therefore, the Reason is also true. However, while the numerical value equivalence in the assertion is correct, the reason does not directly explain why $1 \text{ kharab}$ equals $100 \text{ billion}$ as it only describes the multiplicative steps in each system. Hence, the Reason does not correctly explain the Assertion.

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: 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: 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: 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: 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: 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: 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

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: 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: 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: Scientific Notation

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

Key Concept: Population 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: 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: 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: 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.

Key Concept: Indian and International Naming Conventions

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

Key Concept: Scientific Notation

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, $70,04,00,00,000 = 7.004 \times 10^{10}$.