Exercises
Mathematics I
Mathematics Week 5 Graded Assignment

Mathematics Week 5 Graded Assignment


Graded Assignment

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: 21

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 31

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, 61

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 31

12. Graphical Properties and Inverses (Figure 3)

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:

  • $ 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.

Solution:

  • $ 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 correct1

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

⁂

  1. week-5.pdf ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎ ↩︎

Mathematics Week 4 Graded AssignmentMathematics Week 6 Graded Assignment

More Questions

Profile SVG
Add Chart.js in HUGO 📈 Graded Assignment
Assignments

$$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \, dt$$

Profile SVG
Mathematics Week 2 Graded Assignment Graded Assignment Solution
Week 2

1. Incident and Reflected Ray through Points (with Figure M1W2Q6) Question: A incident ray is passing through the point (2, 3) makes an angle α with horizontal. The ray gets reflected at point M and passes through the point (5, 2) as shown in figure below. Fig: M1W2Q6 Choose the set of correct option(s). The equation of incident ray is −5x −3y + 19 = 0 The equation of incident ray is 3x + 2y −12 = 0 The equation of reflected ray is 5x −3y −19 = 0 The equation of reflected ray is 2x + y −12 = 0 Answer: Option a and c

Profile SVG
Mathematics Week 6 Graded Assignment Graded Assignment
Week 6

Multiple Choice Questions (MCQ) 1. If $ 18^x - 12^x - (2 \times 8^x) = 0 $, then the value of $ x $ is: Options: ln 18 ln 2 ln 18 In 18 Answer: Option 1 (Note: The options are confusing and possibly mislabeled. The intended answer is likely a specific value or expression, but as shown, it is marked as Option 1, which is not a valid solution. However, the PDF marks Option 1 as correct. This may be an error in the PDF.)

Profile SVG
Mathematics Week 8 Graded Assignment Week 8 Graded Assignment
Week 8

1. Multiple Select Questions (MSQ) Question 1: Match the given functions in Column A with the equations of their tangents at the origin $(0,0)$ in column B and the plotted graphs and the tangents in Column C, given in Table M2W2G1. Function (Column A) Tangent at $(0,0)$ (Column B) Graph (Column C) i) $f(x)=x2^{x}$ a) $y=-4x$ 1) ii) $f(x)=x(x-2)(x+2)$ b) $y=x$ 2) iii) $f(x)=-x(x-2)(x+2)$ c) $y=4x$ 3) Options: Option 1: ii) $\rightarrow$ a) $\rightarrow 1$ Option 2: i) $\rightarrow$ b) $\rightarrow 3$ Option 3: iii) $\rightarrow$ b) $\rightarrow 1$ Option 4: iii) $\rightarrow$ c) $\rightarrow 2$ Option 5: i) $\rightarrow$ a) $\rightarrow 1$

Profile SVG
Mathematics Week 10 Graded Assignment Graded Assignment
Week 10

1. Multiple Choice Questions (MCQ) Question 1: The maximum number of non-zero entries in an adjacency matrix of a simple graph having $ n $ vertices can be Options: (a) $ n^2 $ (b) $ n(n-1) $ (c) $ \frac{n(n-1)}{2} $ (d) $ n(n-1) $ Solution: The number of non-zero entries is equal to twice the number of edges for undirected graphs, but for directed simple graphs (no loops, no multiple edges), it is $ n(n-1) $. Since the question says “simple graph” (typically undirected, but the context here implies directed or maximum possible), and the solution says “sum of degrees” is $ n(n-1) $ if every vertex has degree $ n-1 $, which is only possible in a directed simple graph. Correct option: (d) $ n(n-1) $ Question 2: We have a graph $ G $ with 6 vertices. We write down the degrees of all vertices in $ G $ in descending order. Which of the following is a possible listing of the degrees? Options: (a) 6,5,4,3,2,1 (b) 5,5,2,2,1,1 (c) 5,3,3,2,2,1 (d) 2,1,1,1,1,1 Solution:

Profile SVG
Mathematics Week 11 Graded Assignment Part 1 Week 11 Graded Assignment
Part 1

1. Multiple Choice Questions Question 1 An undirected graph $G$ has 20 vertices and the degree of each vertex is at least 3 and at most 5. Which of the following statements is true regarding the graph $G$? (a) The minimum number of edges that the graph $G$ can have is 60. (b) The maximum number of edges that the graph $G$ can have is 100. (c) The maximum number of edges that the graph $G$ can have is 60. (d) The minimum number of edges that the graph $G$ can have is 30.

Profile SVG
Mathematics Week 11 Graded Assignment Part 2 Week 11 Graded Assignment
Part 2

1. Dijkstra’s Algorithm and Unique Shortest Path Question: An undirected weighted graph $ G $ is shown (not included here, but described in text). Find the set of all positive integer values of $ x $ such that if we use Dijkstra’s algorithm, there will be a unique shortest path from vertex $ a $ to vertex $ j $ that contains the edge $ (b, e) $.

Profile SVG
Mathematics Week 12 Graded Assignment Graded Assignment
Week 12

1. Uniform Distribution Expectation and Variance Question: A random variable is uniformly distributed over $[a, b]$ with expectation $e$ and variance $v$. Find $ab$. Answer: $ab = e^2 - 3v$ Solution: