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, Real-World Application

b) 10.737 km
[Solution Description]
The thickness after 30 folds is calculated as:
$0.001 \times 2^{30}$ cm
$2^{10} = 1024$, so $2^{30} = (2^{10})^3 = 1024^3 \approx 1.0737 \times 10^9$
Thickness in cm: $0.001 \times 1.0737 \times 10^9 = 1.0737 \times 10^6$ cm
Convert cm to km: $\frac{1.0737 \times 10^6}{100000} = 10.737$ km

Key Concept: Exponential Growth

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

Key Concept: Exponential Growth

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

Key Concept: Exponential Growth, 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: 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 Experiment

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

Key Concept: Exponential Growth in Folding Paper

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the thickness after 10 folds using the formula for exponential growth: $0.001 \text{ cm} \times 2^{10}$.
Step 1: Calculate $2^{10}$ which is $1024$.
Step 2: Multiply by the initial thickness: $0.001 \text{ cm} \times 1024 = 1.024 \text{ cm}$.
The calculation confirms that the assertion is true. The reason provided correctly explains the assertion by stating the exponential growth formula and how it applies to the folding process.

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: Negative Exponents

c) $\frac{1}{9}$
[Solution Description]
The negative exponent rule states that $n^{-a} = \frac{1}{n^a}$. Applying this to $3^{-2}$, we get $\frac{1}{3^2} = \frac{1}{9}$.

Key Concept: Negative Exponents

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

Key Concept: Operations with Exponents

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 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: 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 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: Exponential Growth

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The initial thickness of the paper is $0.001 \, \text{cm}$. After 1 fold, it becomes $0.001 \times 2 = 0.002 \, \text{cm}$. After 2 folds, it becomes $0.002 \times 2 = 0.004 \, \text{cm}$, and so on. After 5 folds, the thickness will be $0.001 \times 2^5 = 0.032 \, \text{cm}$. Thus, the Assertion is true. The Reason states that the thickness doubles after each fold, which is also true and correctly explains the Assertion because the exponential growth model ($2^n$) is applied here.

Key Concept: Exponential Growth: 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: 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: 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: 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: 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: 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, Generalized forms

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

Key Concept: 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: 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: 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: 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 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: 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: 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: 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: 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: Multiplication of Powers with the Same Base

d) $3^7$
[Solution Description]
To simplify $3^5 \times 3^2$, we use the property of exponents: $n^a \times n^b = n^{a + b}$. Here, the base is the same (3), so we add the exponents:
$3^5 \times 3^2 = 3^{5+2} = 3^7$
Therefore, the simplified form of the expression is $3^7$.

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: 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: Exponent Operations

c) $\frac{1}{49}$
[Solution Description]
Using the exponent division rule: $n^a ÷ n^b = n^{a-b}$. Here, $7^4 ÷ 7^6 = 7^{4-6} = 7^{-2}$.
By the negative exponent rule, $7^{-2} = \frac{1}{7^2} = \frac{1}{49}$.

Key Concept: Negative exponents, 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: 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: 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: 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: 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 Subtraction Rule

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

Key Concept: 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) 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: Negative Exponents

d) $\frac{1}{7^{4}}$
[Solution Description]
A negative exponent represents the reciprocal of the base raised to the positive exponent:
$7^{-4} = \frac{1}{7^{4}}$

Key Concept: Zero Exponents, Real-world Context

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

Key Concept: 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

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

Key Concept: Negative 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 correct according to the exponent rule for zero exponents. The reason provides the general rule that any non-zero number raised to the power of zero equals one ($n^0 = 1$), which correctly explains why $5^0 = 1$. Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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: 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: Large Numbers - Naming Conventions

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

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: 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, 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

c) $4.5 \times 10^3$
[Solution Description]
To convert 4,500 to scientific notation:
1. Move the decimal point after the first non-zero digit: 4.5
2. Count how many places the decimal moved from its original position (from after the last zero to after 4): 3 places.
3. Thus, $4,500 = 4.5 \times 10^3$.

Key Concept: Scientific Notation Conversion

b) 9,040
[Solution Description]
To convert $9.04 \times 10^3$ to standard form, multiply the coefficient $9.04$ by $10^3$ (which is 1000). This gives $9.04 \times 1000 = 9040$.

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, Magnitude and Precision

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

Key Concept: 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

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 Comparison

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

Key Concept: Scientific Notation

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: Exponential vs. Linear Growth, Scientific Notation

a) $6.4 \times 10^9$ m
[Solution Description]
Each fold doubles the thickness, so the thickness after $n$ folds is given by $T = T_0 \times 2^n$, where $T_0$ is the initial thickness.
Initial thickness $T_0 = 0.1$ mm = $0.1 \times 10^{-3}$ m = $1 \times 10^{-4}$ m
Thickness after 46 folds:
$T = 1 \times 10^{-4} \times 2^{46}$
Calculate $2^{10} = 1024 \approx 10^3$
So, $2^{40} = (2^{10})^4 \approx (10^3)^4 = 10^{12}$
And $2^{46} = 2^{40} \times 2^6 = 10^{12} \times 64 = 6.4 \times 10^{13}$
Now, $T = 1 \times 10^{-4} \times 6.4 \times 10^{13} = 6.4 \times 10^9$ m
The distance to the Moon is $3.84 \times 10^8$ m, which is less than $6.4 \times 10^9$ m, confirming the claim.

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

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

Key Concept: 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: 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: 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: Age calculation, Weight estimation

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

Key Concept: Scientific notation, Large numbers

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

Key Concept: Weight-Based Donation Calculations

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

Key Concept: 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: Exponential Growth

c) 1.024 cm
[Solution Description]

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

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

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

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

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

Key Concept: Linear Growth vs. Exponential Growth

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

Key Concept: 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, 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: 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: 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: 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 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: Standard Form

c) 7
[Solution Description]
To express 2 crores in standard form:
$2 \, \text{crores} = 2 \times 1,00,00,000 = 2 \times 10^7$
Here, the exponent is 7.

Key Concept: Scientific Notation and Magnitude

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

Key Concept: Powers of 10, Scientific notation

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

Key Concept: Scientific Notation

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

Key Concept: Exponential Growth

c) 29th day
[Solution Description]
Since the number of lotuses doubles every day, the pond must have been half-covered one day before it was fully covered.
Therefore, if the pond is fully covered on the 30th day, it was half-covered on the 29th day.

Key Concept: Linear vs Exponential Growth, 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 Growth

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

Key Concept: Linear vs Exponential Growth

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: 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: Understanding Large Numbers

b) $3.6 \times 10^6$
[Solution Description]
To convert 3,600,000 into scientific notation, we need to express it as a product of a number between 1 and 10 and a power of 10.
First, move the decimal point from its original position after the last zero to just after the first non-zero digit (3). This gives us 3.6.
Then, count how many places the decimal moved: it moved 6 places to the left. Thus, the exponent is 6.
So, the scientific notation for 3,600,000 is $3.6 \times 10^6$.

Key Concept: Population and Demographics, Scientific Notation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the world population is approximately $8 \times 10^9$ and highlights its importance in resource planning, which is true based on the syllabus. The Reason explains that scientific notation simplifies the representation and comparison of large numbers, which is also correct. However, the Reason does not directly explain why the population figure is essential for resource planning—it only describes the utility of scientific notation. Thus, both statements are true, but the Reason does not correctly explain the Assertion.

Key Concept: Population Counting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the global population is approximately $8 \times 10^9$, which is true as per the given syllabus context. The reason explains that large numbers like $8 \times 10^9$ are used for resource allocation and urban planning, which is also true as mentioned in the syllabus. Moreover, the reason correctly explains why the assertion is significant, as it highlights the practical application of such large numbers in real-world scenarios like governance and planning. Thus, both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Scientific Notation, Exponent Significance

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

Key Concept: Scientific Notation, 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.

Key Concept: Exponents and Large Numbers

c) $1 \times 10^7$
[Solution Description]
One crore is written numerically as $1,00,00,000$.
Step 1: The coefficient is $1$ (since it already lies between 1 and 10).
Step 2: The exponent is the number of zeros + 1 (for the last non-zero digit): $7$.
Thus, one crore in scientific notation is $1 \times 10^7$.

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 ($10^{12}$).