Question 1 - Case Based Questions (MCQ) - Chapter 6 Class 12 Application of Derivatives

Last updated at April 16, 2024 by Teachoo

The Relation between the height of the plant (y in cm) with respect to exposure to sunlight is governed by the following equation
y = 4x – 1/2 x
^{
2
}
where x is the number of days exposed to sunlight.

Question 1

The rate of growth of the plant with respect to sunlight is ______ .

(a) 4x – 1/2 x
^{
2
}

(b) 4 – x

(c) x – 4

(d) x – 1/2 x
^{
2
}

Question 2

What is the number of days it will take for the plant to grow to the maximum height

(a) 4

(b) 6

(c) 7

(d) 10

Question 3

What is the maximum height of the plant?

(a) 12 cm

(b) 10 cm

(c) 8 cm

(d) 6 cm

Question 4

What will be the height of the plant after 2 days?

(a) 4 cm

(b) 6 cm

(c) 8 cm

(d) 10 cm

Question 5

If the height of the plant is 7/2 cm, the number of days it has been exposed to the sunlight is ______ .

(a) 2

(b) 3

(c) 4

(d) 1

Transcript

Question The Relation between the height of the plant (y in cm) with respect to exposure to sunlight is governed by the following equation y = 4x – 1/2 𝑥^2 where x is the number of days exposed to sunlight.
Question 1 The rate of growth of the plant with respect to sunlight is ______ . (a) 4𝑥 – 1/2 𝑥^2 (b) 4 – 𝑥 (c) 𝑥 – 4 (d) 𝑥 – 1/2 𝑥^2
Rate of growth of plant with respect to sunlight
= 𝒅𝒚/𝒅𝒙
= d(4𝑥 − 1/2 𝑥^2 )/𝑑𝑥
= 4 − 1/2 × 2𝑥
= 4 – 𝒙
So, the correct answer is (b)
Question 2 What is the number of days it will take for the plant to grow to the maximum height? (a) 4 (b) 6 (c) 7 (d) 10
Given
y = 4𝑥−1/2 𝑥^2
We need to find value of x, when y is maximum
Thus, we need to find Maximum value of y
Finding Maximum value of y
𝑑𝑦/𝑑𝑥 = 4 − x
Putting 𝒅𝒚/𝒅𝒙 = 0
4 − x = 0
x = 4
Finding sign of (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 )
(𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥)
= (𝑑(4 − 𝑥))/𝑑𝑥
= −1
< 0
Therefore, y is Maximum when x = 4
So, the correct answer is (a)
Question 3 What is the maximum height of the plant? (a) 12 cm (b) 10 cm (c) 8 cm (d) 6 cm
Putting x = 4 in value of y
y = 4𝑥−1/2 𝑥^2
y = 4(4)−1/2 × 4^2
y = 16 − 8
y = 8 cm
So, the correct answer is (c)
Question 4 What will be the height of the plant after 2 days? (a) 4 cm (b) 6 cm (c) 8 cm (d) 10 cm
To find Height of plant after 2 days
Putting x = 2 in value of y
y = 4𝑥−1/2 𝑥^2
y = 4(2)−1/2 × 2^2
y = 8 − 2
y = 6 cm
So, the correct answer is (b)
Question 5 If the height of the plant is 7/2 cm, the number of days it has been exposed to the sunlight is ______ . (a) 2 (b) 3 (c) 4 (d) 1
Putting y = 𝟕/𝟐 in equation of y
y = 4𝑥−1/2 𝑥^2
𝟕/𝟐 = 4𝑥−1/2 𝑥^2
Multiplying both sides by 2
7 = 8x − x2
x2 − 8x + 7 = 0
x2 − 7x − x + 7 = 0
x(x − 7) − 1 (x − 7) = 0
(x − 1) (x − 7) = 0
∴ x = 1, x = 7
Since x = 1 is option (d)
So, the correct answer is (d)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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