updated exercise questions for mathematics graded assignment

Author: aryan

content/exercises/graded-assignments/mathematics-1/W10GA1.md

@@ -1,14 +1,9 @@
 ---
 title: Mathematics Week 10 Graded Assignment
+label: Week 10
 weight: 10
-tags: 
-- mathematics
 categories:
 - Mathematics Graded Assignment
-series:
-- Mathematics Graded Assignment
-excludeSearch: false
-width: wide
 ---
 
 ## 1. Multiple Choice Questions (MCQ)

content/exercises/graded-assignments/mathematics-1/W11GA1.md

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 ---
 title: Mathematics Week 11 Graded Assignment Part 1
+subtitle: Week 11 Graded Assignment 
+label: Part 1
 weight: 11
-tags: 
-- mathematics
 categories:
 - Mathematics Graded Assignment
-series:
-- Mathematics Graded Assignment
-excludeSearch: false
-width: wide
 ---
 
 ## **1. Multiple Choice Questions**

content/exercises/graded-assignments/mathematics-1/W11GA2.md

@@ -1,14 +1,10 @@
 ---
 title: Mathematics Week 11 Graded Assignment Part 2
+subtitle: Week 11 Graded Assignment
+label: Part 2
 weight: 11.5
-tags: 
-- mathematics
 categories:
 - Mathematics Graded Assignment
-series:
-- Mathematics Graded Assignment
-excludeSearch: false
-width: wide
 ---
 
 ## 1. Dijkstra's Algorithm and Unique Shortest Path

content/exercises/graded-assignments/mathematics-1/W12GA1.md

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 ---
-title: Mathematics Graded Assignment Week 12
+title: Mathematics Week 12 Graded Assignment 
+label: Week 12
 weight: 12
-tags: 
-- mathematics
 categories:
 - Mathematics Graded Assignment
-series:
-- Mathematics Graded Assignment
-excludeSearch: false
-width: wide
 ---
 
-Here are the questions and solutions from the Week 12 Graded Assignment in Mathematics:
-
 ---
 
 ### 1. Uniform Distribution Expectation and Variance

content/exercises/graded-assignments/mathematics-1/W2GA1.md

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 ---
-title: Mathematics for Data Science 1
+title: Mathematics Week 3 Graded Assignment
 weight: 3
 label: Week 3
 keywords: 

content/exercises/graded-assignments/mathematics-1/W3GA1.md

@@ -1,40 +1,254 @@
 ---
 title: Mathematics Week 5 Graded Assignment
+label: Week 5
 weight: 5
-tags: 
-- mathematics
 categories:
 - Mathematics Graded Assignment
-series:
-- Mathematics Graded Assignment
-excludeSearch: false
-width: wide
 ---
 
+---
+
+**1. Function Identification via Graph (Figure M1W8A-8.1)**
+
+*Question:*
+A graph is shown in Figure M1W8A-8.1, ◦symbol signifies that the straight line does not touch the point and the -  symbol signifies that the line touches the point.
+Choose the correct option.
+
+- The graph cannot be a function, because it fails the vertical line test.
+- The graph cannot be a function, because it passes the horizontal line test but fails the vertical line test.
+- The graph can be a function, because it passes the vertical line test.
+- The graph cannot be a function, because it passes the vertical line test but fails the horizontal line test.
+
+*Solution:*
+To check if the given graph represents a function, use the vertical line test. In Figure M1W8A-8.1, every vertical line crosses the graph only once (including both -  and ◦ as per definition). Therefore, the graph represents a function.
+
+- Correct options:
+    - The graph can be of a function, because it passes the vertical line test.
+    - The graph fails the horizontal line test.
+    - The graph represents the graph of neither even function nor odd function.
+    - The given graph is not invertible in the given domain.
+- Incorrect options:
+    - The graph cannot be of a function, because it passes the vertical line test but fails the horizontal line test.
+    - The graph cannot be of a function, because it fails the vertical line test.
+    - The graph cannot be of a function, because it passes the horizontal line test but fails the vertical line test.
+    - The graph fails the horizontal line test thus it can be an injective function.
+    - The graph represents the graph of either even function or odd function.[^1]
+
+---
+
+**2. Injectivity of Power Functions (Figures M1W8AS-8.1 and M1W8AS-8.2)**
+
+*Question:*
+For \$ y = x^n \$, where \$ n \$ is a positive integer and \$ x \in \mathbb{R} \$, which of the following statement is true?
+
+- For all values of n, y is not a one-to-one function.
+- For all values of n, y is an injective function.
+- y is not a function.
+- If n is an even number, then y is not an injective function. If n is an odd number, then y is an injective function.
+
+*Solution:*
+
+- \$ y = x^n \$ is a function for all positive integers \$ n \$.
+- If \$ n \$ is odd, the function is injective (passes the horizontal line test, see Figure M1W8AS-8.1).
+- If \$ n \$ is even, the function is not injective (see Figure M1W8AS-8.2).
+- Therefore, the correct option is: If n is even, not injective; if n is odd, injective.[^1]
+
+---
+
+**3. Algebraic Simplification**
+
+*Question:*
+If \$ 4m - n = 0 \$, then the value of
+
+$$
+\frac{16^m}{2^n} + \frac{27^n}{96^m}
+$$
+
+is
+
+*Solution:*
+Given \$ 4m - n = 0 \$,
+
+$$
+\frac{16^m}{2^n} + \frac{27^n}{96^m}
+= (2^4)^m / 2^n + (3^3)^n / (2^5 \cdot 3)^m
+= 2^{4m-n} + 3^{3n-6m}
+= 2^0 + 3^0 = 1 + 1 = 2
+$$
+
+**Answer:** 2[^1]
+
+---
+
+**4. Radioactive Decay (Half-Life Calculation)**
+
+*Question:*
+Half-life of an element is the time required for half of a given sample of radioactive element to change to another element. The rate of change of concentration is calculated by the formula \$ A(t) = A_0 (1/2)^{t/\gamma} \$, where \$ \gamma \$ is the half-life.
+If Radium has a half-life of 1600 years and the initial concentration is 100%, calculate the percentage of Radium after 2000 years.
+
+- 35%
+- 42%
+- 19%
+- 21%
+
+*Solution:*
+
+$$
+A(2000) = 100 \times (1/2)^{2000/1600} = 100 \times (1/2)^{1.25} \approx 42\%
+$$
+
+**Answer:** 42%[^1]
+
+---
+
+**5. Domain of a Composite Function**
+
+*Question:*
+If \$ f(x) = (1 - x)^{1/2} \$ and \$ g(x) = 1 - x^2 \$, find the domain of the composite function \$ g \circ f \$.
+
+- \$ \mathbb{R} \$
+- \$ (-\infty, 1] \cap [-2, \infty) \cup (-\infty, -2) \$
+- \$ [1, \infty) \$
+- \$ \mathbb{R} \setminus (1, \infty) \$
+
+*Solution:*
+
+- Domain of \$ f(x) \$: \$ x \leq 1 \$ (\$ (-\infty, 1] \$)
+- Domain of \$ g(x) \$: \$ \mathbb{R} \$, but range of \$ f(x) \$ is \$ [0, \infty) \$
+- So, domain of \$ g \circ f \$ is \$ (-\infty, 1] \$ (options 2 and 4 are correct)[^1]
+
+---
+
+**6. Domain of the Inverse Function**
+
+*Question:*
+Find the domain of the inverse function of \$ y = x^3 + 1 \$.
+
+- \$ \mathbb{R} \$
+- \$ \mathbb{R} \setminus \{1\} \$
+- \$ [1, \infty) \$
+- \$ \mathbb{R} \setminus [1, \infty) \$
+
+*Solution:*
+The range of \$ y = x^3 + 1 \$ is \$ \mathbb{R} \$, so the domain of its inverse is also \$ \mathbb{R} \$.
+**Answer:** \$ \mathbb{R} \$[^1]
+
+---
+
+**7. Intersection Points of a Function and Its Inverse**
+
+*Question:*
+If \$ f(x) = x^3 \$, then which of the following is the set of points where the graphs of \$ f(x) \$ and \$ f^{-1}(x) \$ intersect?
+
+- {(-1, 1), (0, 0), (1, -1)}
+- {(-2, -8), (1, 1), (2, 8)}
+- {(-1, -1), (0, 0), (1, 1)}
+- {(-2, -8), (0, 0), (2, 8)}
+
+*Solution:*
+Solve \$ x^3 = x \Rightarrow x(x^2 - 1) = 0 \Rightarrow x = -1, 0, 1 \$.
+So, intersection points are {(-1, -1), (0, 0), (1, 1)}[^1]
+
+---
+
+**8. Population Growth Prediction**
+
+*Question:*
+In a survey, population growth is given by \$ \alpha(T) = \alpha_0 (1 + d/100)^T \$. If in 2015, the population of Adyar was 30,000 and the growth rate is 4% per year, what will be the approximate population in 2020?
+
+- 60251
+- 71255
+- 91000
+- 36500
+
+*Solution:*
+\$ T = 5 \$, \$ \alpha(5) = 30000 \times (1.04)^5 \approx 36500 \$[^1]
+
+---
+
+**9. Reflection of a Function Across \$ y = x \$ (Figure M1W8AS-8.3)**
+
+*Question:*
+An ant moves along \$ f(x) = x^2 + 1 \$ for \$ x \in [0, \infty) \$. A mirror is placed along \$ y = x \$. If the reflection moves along \$ g(x) \$, which is/are correct?
+
+- \$ g(x) = f^{-1}(x) \$
+- \$ g(x) = f(x) \$
+- \$ g(x) = \sqrt{x-1} \$
+- \$ g(x) = \sqrt{x+1} \$
+
+*Solution:*
+The reflection is the inverse function, so \$ g(x) = f^{-1}(x) = \sqrt{x-1} \$.
+Correct options: 1 and 3[^1]
+
+---
+
+**10. Festival Discount Offers (Applied Math)**
+
+*Question:*
+A textile shop offers:
+D1: Shop for more than ₹14,999 and pay only ₹9,999.
+D2: Avail 30% discount on the total payable amount.
+If Shalini buys two dresses, each over ₹8,000, and can use both offers, which is/are correct?
+
+- The minimum amount she should pay after applying two offers cannot be determined because the exact values are unknown.
+- The minimum amount she should pay after applying both offers is approximately ₹6,999.
+- The amount after D2 only is approximately ₹11,199.
+- The amount after D1 only is approximately ₹9,999.
+- If total is ₹17,999, to pay minimum, avail D1 first, then D2.
+- If total is ₹17,999, availing D2 first, then D1 is not possible.
+- If total is ₹17,999, to pay minimum, avail D2 first, then D1.
+
+*Solution:*
+
+- If D1 first, then D2: ₹9,999 × 0.7 = ₹6,999 (minimum).
+- If D2 first, amount may fall below ₹14,999, so D1 may not be applicable.
+- D1 only: ₹9,999.
+- D2 only: ₹17,999 × 0.7 = ₹12,599 (if total is ₹17,999).
+- So, correct: 2, 4, 5, 6[^1]
+
+---
+
+**11. Injectivity and Function Operations**
+
+*Question:*
+If \$ f(x) = x^2 \$ and \$ h(x) = x-1 \$, which options are incorrect?
+
+- \$ f \circ h \$ is not injective.
+- \$ f \circ h \$ is injective.
+- \$ f(f(h(x))) \times h(x) = (x-1)^4 \$
+- \$ f(f(h(x))) \times h(x) = (x-1)^5 \$
+
+*Solution:*
+
+- \$ f \circ h = (x-1)^2 \$ is not injective.
+- \$ f(f(h(x))) \times h(x) = ((x-1)^2)^2 \times (x-1) = (x-1)^5 \$.
+- So, incorrect: 2 and 3[^1]
+
+---
+
+**12. Graphical Properties and Inverses (Figure 3)**
 
-1. 
+*Question:*
+Let \$ f(x), g(x), p(x), q(x) \$ be functions defined on \$ \mathbb{R} \$. Refer Figure 3 (A and B) and choose correct options:
 
-{{< border >}}
+- \$ g(x) \$ may be the inverse of \$ f(x) \$.
+- \$ p(x) \$ and \$ q(x) \$ are even functions but \$ f(x) \$ and \$ g(x) \$ are neither even nor odd.
+- \$ q(x) \$ could not be the inverse function of \$ p(x) \$.
+- \$ p(x), q(x) \$ can be even degree polynomials and \$ f(x) \$ can be an odd degree polynomial.
 
-To determine which statements about the function \$ y = x^n \$ (where \$ n \$ is a positive integer and \$ x \in \mathbb{R} \$) are true, we analyze its injectivity based on whether \$ n \$ is even or odd:
+*Solution:*
 
-1. **Statement 1**: "For all values of \$ n \$, \$ y \$ is not a one-to-one function."
-**False**. When \$ n \$ is odd, \$ y = x^n \$ is injective (one-to-one) over \$ \mathbb{R} \$.
-2. **Statement 2**: "For all values of \$ n \$, \$ y \$ is an injective function."
-**False**. For even \$ n \$, \$ y = x^n \$ is **not** injective (e.g., \$ x = 2 \$ and \$ x = -2 \$ both map to \$ y = 16 \$ when \$ n = 4 \$).
-3. **Statement 3**: "\$ y \$ is not a function."
-**False**. \$ y = x^n \$ is a valid function, as every \$ x \in \mathbb{R} \$ maps to exactly one \$ y \$.
-4. **Statement 4**: "If \$ n \$ is an even number, then \$ y \$ is not an injective function."
-**True**. For even \$ n \$, distinct inputs like \$ x \$ and \$ -x \$ produce the same output (e.g., \$ 2^2 = (-2)^2 = 4 \$), violating injectivity.
-5. **Statement 5**: "If \$ n \$ is an odd number, then \$ y \$ is an injective function."
-**True**. For odd \$ n \$, \$ y = x^n \$ preserves the sign and uniqueness of inputs (e.g., \$ 2^3 = 8 \$, \$ (-2)^3 = -8 \$), ensuring injectivity.
+- \$ f(x) \$ and \$ g(x) \$ are symmetric across \$ y = x \$, so \$ g(x) \$ may be inverse of \$ f(x) \$.
+- \$ p(x) \$ and \$ q(x) \$ are symmetric about the y-axis, so they are even functions.
+- \$ q(x) \$ is not symmetric across \$ y = x \$, so cannot be inverse of \$ p(x) \$.
+- End behaviors suggest \$ p(x), q(x) \$ can be even degree polynomials, \$ f(x) \$ can be odd degree.
+- All options (a), (b), (c), (d) are correct[^1]
 
 ---
 
-**Final Answer:**
+**Note:** For all questions involving graphs, the referenced figures (e.g., Figure M1W8A-8.1, M1W8AS-8.1, etc.) are described in the solutions, but the actual images are not included in this text extraction. The reasoning is based on their descriptions in the PDF.[^1]
 
-$\boxed{ Statements  4 and  5 are  true. }$
+<div style="text-align: center">⁂</div>
 
-$\boxed{3}$
+[^1]: week-5.pdf
 
-{{< /border >}}

content/exercises/graded-assignments/mathematics-1/W5GA1.md

@@ -1,5 +1,6 @@
 ---
 title: Mathematics Week 6 Graded Assignment 
+label: Week 6
 weight: 6
 categories:
 - Mathematics Graded Assignment

content/exercises/graded-assignments/mathematics-1/W6GA1.md

@@ -1,15 +1,10 @@
 ---
 title: Week 7 - Sequence and Limits
+label: Week 7
 weight: 7
-tags: 
-- mathematics
 categories:
 - Mathematics Graded Assignment
 - Week-7-Sequence-and-Limits
-series:
-- Mathematics Graded Assignment
-excludeSearch: false
-width: wide
 ---
 
 ## 1. Multiple Choice/Statement Analysis