Maths Week 4 Graded Assignments IIT Madras

Maths week 4 graded assignment Complete Solutions Are Discussed In This Blog We Hope This Might Help You All In Matching Answers . Or For Some Others Reasons Not Able To Complete Graded Assignments

There are some questions which have functions with discrete valued domains (such as day, month, year etc). For simplicity, we treat them as continuous functions.

For NAT type question, enter only one right answer, even if you get multiple answers for that particular question.

Notations:

$R$ = Set of real numbers $Q$ = Set of rational numbers
$Z$ = Set of integers
$N$ = Set of natural numbers

The set of natural numbers includes 0.

1 point

Let $f : R \to R$ and $g : R \to R$ be two functions, defined as $f (x) = x^{3} - 8 x^{2} + 7$ and $g (x) = - 2 f (x)$ respectively. Choose the correct option(s) from the following.

$g$ has two turning points and $y -$ coordinate of only one turning point is negative.

$g$ has two turning points and there are no turning points with positive $y -$ coordinate.

$f$ is strictly increasing in $[10, \infty)$ .

$f$ is strictly decreasing in $[10, \infty)$ .

Accepted Answers:

$g$ has two turning points and $y -$ coordinate of only one turning point is negative.

$f$ is strictly increasing in $[10, \infty)$ .

1 point

2. Which among the following function first increases and then decreases in all the intervals $(- 4, - 3) and (- 1, 2) and (5, 6)$ ?

$10000 1 (x + 1)^{2} (x - 2) (x + 3)^{2} (x + 4) (x - 5) (x - 6)^{2}$

$10000 1 (x + 1)^{2} (x - 2) (x + 3) (x + 4) (x - 5)^{2} (x - 6)^{2}$

$10000 - 1 (x + 1)^{2} (x - 2) (x + 3)^{2} (x + 4)^{2} (5 - x) (x - 6)^{2}$

$10000 1 (x + 1)^{2} (x - 2) (x + 3) (x + 4) (x - 5)^{2} (x - 6)^{2} (x + 7)$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$10000 1 (x + 1)^{2} (x - 2) (x + 3)^{2} (x + 4) (x - 5) (x - 6)^{2}$

$10000 - 1 (x + 1)^{2} (x - 2) (x + 3)^{2} (x + 4)^{2} (5 - x) (x - 6)^{2}$

1 point

3. Which of the following is/are correct ?

Fig – 1

Fig – 2

Fig – 1 and Fig – 2 are polynomials of $x$ .

The polynomial of $x$ that represents the Fig – 1 have minimum 6 Zeros

The polynomial of $x$ that represent the Fig – 2 have minimum 3 Zeros

Fig – 2 represents the function of $x$ .

Fig – 2 is not a polynomial of $x$ .

The polynomial of $x$ that represents the Fig – 1 have 3 distinct Zeros

Accepted Answers:

The polynomial of $x$ that represents the Fig – 1 have minimum 6 Zeros

Fig – 2 is not a polynomial of $x$ .

The polynomial of $x$ that represents the Fig – 1 have 3 distinct Zeros

1 point

4. Consider a polynomial function $p (x) = - (x^{2} - 16) (x - 3)^{2} (2 - x)^{2} (x + 9)$ . Choose the set of correct options.

$p (x)$ first increases then decreases in the interval $(2, 3)$

$p (x)$ is strictly increasing when $x \in (- \infty, - 9)$

There are exactly 6 points on $p (x)$ where the slope of the tangent is 0.

Total number of turning points of $p (x)$ are 7.

Accepted Answers:

$p (x)$ first increases then decreases in the interval $(2, 3)$

There are exactly 6 points on $p (x)$ where the slope of the tangent is 0.

1 point

5. The height of a roller coaster at a given time is modeled by the polynomial function $h (t) = (- 0.01 t^{3} + 0.35 t^{2} - 3.5 t + 10) (t + 5)^{2} (t - 5) (t + 1) (2 - t)^{3}$ , where t represents time in seconds. Find the turning point(s) of the height function.

Yes, the answer is correct.
Score: 1

Accepted Answers:

1 point

6. Choose the correct options based on the data given in 5th question.

The roller coaster will first go up and then go down in the interval $(- 5, - 1)$ .

The roller coaster will first go down and then go up in the interval $(10, 20)$ .

The roller coaster will first go up and then go down in the interval $(- 1, - 2)$ .

The roller coaster will first go up and then go down in the interval $(2, 5)$ .

Yes, the answer is correct.
Score: 1

Accepted Answers:

The roller coaster will first go up and then go down in the interval $(- 5, - 1)$ .

The roller coaster will first go down and then go up in the interval $(10, 20)$ .

The roller coaster will first go up and then go down in the interval $(2, 5)$ .

7. An ant named $B$ , wants to climb an uneven cliff and reach its anthill (i.e., home of ant). On its way home, $B$ makes sure that it collects some food. A group of ants have reached the food locations which are at $x -$ intercepts of the function $f (x) = (x^{2} - 20) ((x - 7)^{3} - 1)$ . As ants secrete pheromones (a form of signals which other ants can detect and reach the food location), $B$ gets to know the food location. Then the sum of the $x$ -coordinates of all the food locations is

Yes, the answer is correct.
Score: 1

Accepted Answers:

(Type: Numeric) 8

1 point

8. The Ministry of Road Transport and Highways wants to connect three aspirational districts with two roads $r_{1}$ and $r_{2}$ . Two roads are connected if they intersect. The shape of the two roads $r_{1}$ and $r_{2}$ follows polynomial curve $f (x) = (x - 19) (x - 17)^{2}$ and $g (x) = - (x - 19) (x - 17)$ respectively. What will be the $x -$ coordinate of the third aspirational district, if the first two are at $x -$ intercepts of $f (x)$ and $g (x)$ .

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 16

1 point

A company is manufacturing custom-made boxes with a volume represented by the polynomial

V (x) = 4 x^{3} + 16 x^{2} + 17 x + 5

. If the length of each box is represented by the polynomial

L (x) = x + 1

, given that width and height of the box are

W (x)

and

h (x)

respectively. Also given that for all

x > 0

h (x) > W (x)

, the Degree of polynomials

h (x)

and

W (x)

is 1.
Use this information to answer the following questions 9 and 10

1 point

9. Polynomial expression which represents width of the box $W (x)$

$(2 x + 1)$

$(2 x + 5)$

$(4 x^{2} + 12 x + 5)$

$(3 x + 4)$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$(2 x + 1)$

1 point

10. Polynomial expression which represents area of the base.

$(4 x^{2} + 12 x + 3)$

$(4 x^{2} + 12 x + 5)$

$(2 x^{2} + 6 x + 5)$

$(2 x^{2} + 3 x + 1)$

Yes, the answer is correct.
Score: 1

Accepted Answers:

$(2 x^{2} + 3 x + 1)$

A telecommunications company observes that the number of monthly subscribers (S) for its internet service can be modeled by the polynomial function

S (t) = t^{3} - 2 t^{2} - 24 t + 5000

, where t represents the number of months since the service was launched.
Use this information to answer the following questions 11 and 12.

11. The company aims to reach a milestone of 5000 subscribers. How many months will it take to achieve this goal based on the polynomial function? (Note: $t > 0$ )