## 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.  ## (d) x – 1/2  x 2 ## (d) 10  ## (d) 6 cm ## (d) 10 cm ## (d) 1  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
2. Serial order wise
3. Case Based Questions (MCQ)

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) 