Maths Week 5 Graded Assignments IIT Madras

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1. A graph is shown in the below figure,

\circ

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(s)

The graph can be of a function, because it passes the vertical line test.

The graph represents the graph of either even function or odd function.

The graph represents the graph of neither even function nor odd function.

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 passes the horizontal line test but fails the vertical line test.

Accepted Answers:

The graph can be of a function, because it passes the vertical line test.

The graph represents the graph of neither even function nor odd function.

2. For

y = x^{n}

,where

n

is a positive integer and

x \in R

, which of the following statements are 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.

n

is an even number, then

y

is not an injective function.

n

is an odd number, then

y

is an injective function.

Accepted Answers:

n

is an even number, then

y

is not an injective function.

n

is an odd number, then

y

is an injective function.

3. If

4 m - n =

-2, then the value of

(2 ^{n} 1 6 ^{m} + 9 ^{6 m} 2 7 ^{n})

Accepted Answers:

(Type: Range) 656.33,802.18

4. 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_{o} (2 1)^{(γ t)}

where

γ

is the half-life of the material,

A_{o}

is the initial concentration of the radioactive element in the given sample,

A (t)

is the concentration of the radioactive element in the sample after time

t

. If Radium has a half-life of 2000 years and the initial concentration of Radium in a sample was 100%, then calculate the percentage of Radium in that sample after 4000 years.

Accepted Answers:

(Type: Range) 22.5,27.5

5. If

f (x) = (1 - x)^{2 1}

and

g (x) = (1 - x^{2})

, then find the domain of the composite function

g \circ f

R

((- \infty, 1] \cap [- 2, \infty)) \cup (- \infty, - 2)

[1, \infty)

R ∖ (1, \infty)

Accepted Answers:

((- \infty, 1] \cap [- 2, \infty)) \cup (- \infty, - 2)

R ∖ (1, \infty)

6. Find the domain of the inverse function of

y = x^{3} + 1

R

R ∖ {1}

[1, \infty)

R ∖ [1, \infty)

Yes, the answer is correct.
Score: 1

Accepted Answers:

R

7. If

f (x) = x^{3}

, then choose the points where the graphs of the functions

f (x)

and

f^{- 1} (x)

intersect each other?

Yes, the answer is correct.
Score: 1

Accepted Answers:

(-1,-1)

(0,0)

(1,1)

8.In a survey, the population growth in an area can be predicted according to the equation

α (T) = α_{o} (1 + 100 d)^{T}

where

d

is the percentage growth rate of population per year and

T

is the time since the initial population count

α_{o}

was taken. If in 2016, the population of Adyar was 40000.0 and the population growth rate is 2

%

per year, then what will be the approximate population of Adyar in 2021?(

T = 0

corresponds to the year 2016,

T = 1

corresponds to the year 2017 and so on..)

Accepted Answers:

(Type: Range) 39746.91,48579.5

1 point

9. An ant moves along the curve whose equation is

f (x) = x^{2} + 1

in the restricted domain

[0, \infty)

. Let a mirror be placed along the line

y = x

. If the reflection of the ant with respect to the mirror moves along the curve

g (x)

, then which of the following options is(are) correct?

g (x) = f^{- 1} (x)

g (x) = f (x)

g (x) = 2 (x - 1)

g (x) = 2 (x + 1)

Yes, the answer is correct.
Score: 1

Accepted Answers:

g (x) = f^{- 1} (x)

g (x) = 2 (x - 1)

1 point

10. Suppose a textile shop provides two different types of offers during a festival season. The first offer(

D_{1}

) is “shop for more than ₹14,999 and pay only ₹9,999″. The second offer(

D_{2}

) is “avail 30% discount on the total payable amount”. If Shalini wants to buy two dresses each of which costs more than ₹8,000 and she is given the choice to avail both offers simultaneously, then which of the following options is(are) correct?

The minimum amount she should pay after applying two offers cannot be determined because the exact values of the dresses she wanted to buy are unknown.

The minimum amount she should pay after applying the two offers simultaneously is approximately ₹

6, 999

The amount she is supposed to pay after applying

D_{2}

only is approximately ₹11,199.

The amount she is supposed to pay after applying

D_{1}

only is ₹

9, 999.

Suppose the total payable amount is ₹

17, 999

for the two dresses. In order to pay minimum amount Shalini should avail offer

D_{1}

first and offer

D_{2}

next.

Suppose the total payable amount is ₹

17, 999

for the two dresses. If Shalini avails offer

D_{2}

first, then she cannot avail offer

D_{1}

Suppose the total payable amount is ₹17,999 for the two dresses. In order to pay minimum amount Shalini should avail offer

D_{2}

first and offer

D_{1}

next.

Yes, the answer is correct.
Score: 1

Accepted Answers:

The minimum amount she should pay after applying the two offers simultaneously is approximately ₹

6, 999

The amount she is supposed to pay after applying

D_{1}

only is ₹

9, 999.

Suppose the total payable amount is ₹

17, 999

for the two dresses. In order to pay minimum amount Shalini should avail offer

D_{1}

first and offer

D_{2}

next.

Suppose the total payable amount is ₹

17, 999

for the two dresses. If Shalini avails offer

D_{2}

first, then she cannot avail offer

D_{1}

1 point

11. If

f (x) = x^{2}

and

h (x) = x - 1

, then which of the following options is(are) incorrect?

f \circ h

is not an injective function.

f (f (h (x))) \times h (x) = (x - 1)^{5}

h \circ f

is not an injective function.

There are two distinct solution for

h (h (f (x))) = 0

f \circ h

is an injective function.

f (f (h (x))) \times h (x) = (x - 1)^{4}

h \circ f

is an injective function.

Accepted Answers:

f \circ h

is an injective function.

f (f (h (x))) \times h (x) = (x - 1)^{4}

h \circ f

is an injective function.

1 point

12. Let

f (x)

g (x)

p (x)

and

q (x)

be the functions defined on

R

. Refer Figure 3 (A and B) and choose the correct option(s) from the following.

Figure 3

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 functions nor odd functions.

q (x)

could not be the inverse function of

p (x)

p (x)

q (x)

can be an even degree polynomial functions and

f (x)

can be an odd degree polynomial functions.

Yes, the answer is correct.
Score: 1

Accepted Answers:

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 functions nor odd functions.

q (x)

could not be the inverse function of

p (x)

p (x)

q (x)

can be an even degree polynomial functions and

f (x)

can be an odd degree polynomial functions.

Maths Week 5 Graded Assignments IIT Madras

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