Key Concept: Exponential Growth, Folding Paper

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

Key Concept: Exponential Growth

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

Key Concept: Scientific Notation

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

Key Concept: Exponential Growth, Comparison of Folds

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

Key Concept: 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: 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 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: Exponential Growth

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

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: 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: Operations with Exponents

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

Key Concept: Negative Exponents

c) $\frac{1}{25}$
[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: 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 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: 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: 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: Exponent Properties, Exponential Growth

c) 320
[Solution Description]
The number of bacteria doubles every hour, so after $t$ hours, the count becomes $10 \times 2^t$. For $t = 5$, the calculation is:
$10 \times 2^5 = 10 \times 32 = 320$

Key Concept: Exponential Notation and Operations: Power Notation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's analyze both the Assertion (A) and Reason (R):
1. Assertion (A): Evaluate $(-3)^4$.
- $(-3)^4 = (-3) \times (-3) \times (-3) \times (-3)$.
- Multiplying two negatives gives a positive: $(-3) \times (-3) = 9$.
- Again multiplying two positives: $9 \times 9 = 81$.
- So, $(-3)^4 = 81$, which is indeed positive. Thus, Assertion (A) is true.
2. Reason (R): The reason states that any negative base raised to an even exponent results in a positive number.
- This is a fundamental property of exponents:
For any non-zero real number $n$ and even integer $a$, $(-n)^a = n^a > 0$.
- Example: $(-2)^6 = 64$ (positive).
- Hence, Reason (R) is true and correctly explains Assertion (A).
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that $5^3 = 125$. This is correct because $5^3 = 5 \times 5 \times 5 = 125$. The reason explains the general definition of exponential notation, where $n^a$ means $n$ multiplied by itself $a$ times. Since the assertion is a specific example of this rule and correctly applies it, the reason is the correct explanation of the assertion.

Key Concept: 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: Division of Exponents

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

Key Concept: 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: 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: Division of exponents with the same base

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

Key Concept: Multiplication of exponents 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: 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: 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, 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: 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: Power of a Power

a) $2^{12}$
[Solution Description]
Using the power of a power property $(n^a)^b = n^{a \times b}$, we can simplify $(2^3)^4$ as follows:
$(2^3)^4 = 2^{3 \times 4} = 2^{12}$

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: 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: 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: Zero Exponent

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

Key Concept: Combining Exponents

b) 16
[Solution Description]
First, simplify the division inside the parentheses using the exponent rule $n^a \div n^b = n^{a-b}$:
$2^{-4} \div 2^{-6} = 2^{-4-(-6)} = 2^{2} = 4$
Now, square the result:
$4^2 = 16$

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: 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: Exponential Growth

b) $2^{29}$
[Solution Description]
Since the lotuses double each day, if the pond is fully covered on the 30th day, it must have been half-covered on the 29th day. Let the full coverage be $2^{30}$, then half coverage is $\frac{2^{30}}{2} = 2^{29}$.

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

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: 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, Negative exponents

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

Key Concept: Exponent subtraction rule

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

Key Concept: 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 and negative exponents

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

Key Concept: Exponent 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: Zero Exponent

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

Key Concept: 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: 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: Large Numbers - Naming Conventions

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

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: 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: 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: Scientific Notation, Exponent Importance

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

Key Concept: Scientific Notation

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

Key Concept: Scientific Notation, 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

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the number 3,00,00,000 can be written as $3 \times 10^6$ in scientific notation. This is correct because:
3,00,00,000 = 3 × 10,00,000 = $3 \times 10^6$.
The reason explains the general rule of scientific notation, which is accurate: a number is expressed as $x \times 10^y$, where $x$ is between 1 and 10, and $y$ is an integer.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

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: Exponent Rules

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

Key Concept: Scientific Notation

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

Key Concept: Scientific Notation

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

Key Concept: Historical Context, Powers of 10

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

Key Concept: 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: Comparing Numbers

c) Ant population is larger
[Solution Description]
Comparing exponents:
1. Ants have exponent 16, humans have exponent 9
2. Larger exponent means bigger number regardless of coefficient
3. Therefore $2 \times 10^{16} > 8 \times 10^9$

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

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: 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: Standard Form

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

Key Concept: Powers of 10

b) $8 \times 10^6$
[Solution Description]
To express 80,00,000 using powers of 10:
Step 1: The coefficient is 8 (a number between 1 and 10).
Step 2: Count the number of zeroes, which is 6.
Hence, the standard form is $8 \times 10^6$.

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: Age in Hours

b) 116160 hours
[Solution Description]
To convert days into hours, multiply the number of days by 24 (since there are 24 hours in a day).
Calculation:
$\text{Hours} = 4840 \times 24 = 116160 \text{ hours}$

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: Powers of 10

c) 7
[Solution Description]
The number $10^7$ has 1 followed by 7 zeros: 1,00,00,000. Thus, it has exactly 7 zeros.

Key Concept: Linear Growth vs. Exponential Growth

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

Key Concept: Linear Growth vs. Exponential Growth

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

Key Concept: Linear 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: Scientific Notation, Earth's Water Volume

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

Key Concept: Comparing large numbers

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

Key Concept: 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: 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

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

Key Concept: Scientific Notation

b) $5.976 \times 10^{24}$
[Solution Description]
To convert the given number into scientific notation, we need to express it in the form $x \times 10^y$ where $1 \leq x < 10$. Starting with the number 59,76,00,00,00,00,00,00,00,00,00,000, we move the decimal point after the first digit (5). Counting the number of places moved gives us $5.976 \times 10^{24}$ because there are 24 digits after the 5.

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: Linear vs Exponential Growth, Scientific Notation

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

Key Concept: Linear vs Exponential 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: Exponent Rules

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

Key Concept: Exponential Growth, Scientific Notation

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

Key Concept: 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: Scientific Notation and Large Numbers

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 Milky Way is $1 \times 10^{11}$, which is correct. The reason explains that scientific notation simplifies large numbers by using a coefficient and a power of 10, which is also true. Furthermore, the reason correctly explains why the assertion uses scientific notation to represent the large number of stars in the Milky Way.

Key Concept: Currency and Large Numbers

c) Hungary
[Solution Description]
In 1946, Hungary issued a currency note with the denomination of 1 sextillion pengő ($10^{21}$ pengő).

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 Application

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

Key Concept: 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 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: Standard Form Conversion

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