1 point
Let
f
:
R
→
R
f:R→R and
g
:
R
→
R
g:R→R be two functions, defined as
f
(
x
)
=
x
3
−
8
x
2
+
7
f(x)=x
3
−8x
2
+7 and
g
(
x
)
=
−
2
f
(
x
)
g(x)=−2f(x) respectively. Choose the correct option(s) from the following.
f
f is strictly increasing in
[
10
,
∞
)
[10,∞).
g
g has two turning points and
y
−
y−coordinate of only one turning point is positive.
f
f has two turning points and there are no turning points with positive
y
−
y− coordinate.
g
g has two turning points and there are no turning points with negative
y
−
y−coordinate.
1 point
Which among the following function first increases and then decreases in all the intervals
(
−
4
,
−
3
)
and
(
−
1
,
2
)
and
(
5
,
6
)
(−4,−3) and (−1,2) and (5,6)?
1
10000
(
x
+
1
)
2
(
x
−
2
)
(
x
+
3
)
(
x
+
4
)
(
x
−
5
)
2
(
x
−
6
)
2
(
x
+
7
)
10000
1
(x+1)
2
(x−2)(x+3)(x+4)(x−5)
2
(x−6)
2
(x+7)
−
1
10000
(
x
+
1
)
2
(
x
−
2
)
(
x
+
3
)
2
(
x
+
4
)
(
5
−
x
)
(
x
−
6
)
2
10000
−1
(x+1)
2
(x−2)(x+3)
2
(x+4)(5−x)(x−6)
2
1
10000
(
x
+
1
)
2
(
x
−
2
)
(
x
+
3
)
2
(
x
+
4
)
(
x
−
5
)
2
(
x
−
6
)
2
10000
1
(x+1)
2
(x−2)(x+3)
2
(x+4)(x−5)
2
(x−6)
2
−
1
10000
(
x
+
1
)
2
(
x
−
2
)
(
x
+
3
)
2
(
x
+
4
)
2
(
5
−
x
)
2
(
x
−
6
)
2
(
3
−
x
)
10000
−1
(x+1)
2
(x−2)(x+3)
2
(x+4)
2
(5−x)
2
(x−6)
2
(3−x)
1 point
Consider a polynomial function
p
(
x
)
=
−
(
x
2
−
16
)
(
x
−
3
)
2
(
2
−
x
)
2
(
x
+
9
)
p(x)=−(x
2
−16)(x−3)
2
(2−x)
2
(x+9). Choose the set of correct options.
p
(
x
)
p(x) is strictly increasing when
x
∈
(
−
∞
,
−
9
)
x∈(−∞,−9)
Total number of turning points of
p
(
x
)
p(x) are 6.
p
(
x
)
p(x) first increases then decreases in the interval
(
2
,
3
)
(2,3)
Total number of turning points of
p
(
x
)
p(x) are 7.
An ant named
B
B, wants to climb an uneven cliff and reach its anthill (i.e., home of ant). On its way home,
B
B makes sure that it collects some food. A group of ants have reached the food locations which are at
x
−
x−intercepts of the function
f
(
x
)
=
(
x
2
−
28
)
(
(
x
−
1
)
3
−
1
)
f(x)=(x
2
−28)((x−1)
3
−1). As ants secrete pheromones (a form of signals which other ants can detect and reach the food location),
B
B gets to know the food location. Then the sum of the
x
x-coordinates of all the food locations is
1 point
The Ministry of Road Transport and Highways wants to connect three aspirational districts with two roads
r
1
r
1
and
r
2
r
2
. Two roads are connected if they intersect. The shape of the two roads
r
1
r
1
and
r
2
r
2
follows polynomial curve
f
(
x
)
=
(
x
−
19
)
(
x
−
17
)
2
f(x)=(x−19)(x−17)
2
and
g
(
x
)
=
−
(
x
−
19
)
(
x
−
17
)
g(x)=−(x−19)(x−17) respectively. What will be the
x
−
x−coordinate of the third aspirational district, if the first two are at
x
−
x−intercepts of
f
(
x
)
f(x) and
g
(
x
)
g(x).
1 point
1 point
Consider a polynomial function
P
(
x
)
=
(
x
4
+
4
x
3
+
x
+
10
)
P(x)=(x
4
+4x
3
+x+10) and
Q
(
x
)
=
(
x
3
+
2
x
2
−
6
)
Q(x)=(x
3
+2x
2
−6). If
M
(
x
)
M(x) is the equation of the straight line passing through
(
2
,
Q
(
2
)
)
(2,Q(2)) and having slope 3, then find out the equation of
P
(
x
)
+
M
(
x
)
Q
(
x
)
P(x)+M(x)Q(x).
Choose the correct answer.
4
x
4
+
14
x
3
+
8
x
2
−
17
x
−
14
4x
4
+14x
3
+8x
2
−17x−14
4
x
4
+
14
x
3
−
6
x
2
−
19
x
−
34
4x
4
+14x
3
−6x
2
−19x−34
4
x
4
+
2
x
3
+
8
x
2
−
17
x
−
14
4x
4
+2x
3
+8x
2
−17x−14
4
x
4
+
2
x
3
+
8
x
2
−
18
x
−
34
4x
4
+2x
3
+8x
2
−18x−34
Find the number of distinct intersecting points (real) between the line
l
(
x
)
=
x
l(x)=x and polynomial
r
(
x
)
r(x), where
r
(
x
)
r(x) is the quotient when polynomial
f
(
x
)
=
x
4
−
13
x
−
42
f(x)=x
4
−13x−42 is divided by polynomial
g
(
x
)
=
x
2
−
x
−
6
g(x)=x
2
−x−6.
1 point
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
S(t)=t
3
−2t
2
−24t+5000, where t represents the number of months since the service was launched.
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
t>0)
1 point
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
S(t)=t
3
−2t
2
−24t+5000, where t represents the number of months since the service was launched.
Number of actual subscribers after one year; given that the polynomial function over estimates the actual subscribers by 60%
1 point
1 point
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
h(t)=(−0.01t
3
+0.35t
2
−3.5t+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.
6
7
5
8
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