# In an agricultural institute scientists do experiments

Q 3:- In an agricultural institute scientists do experiments with varieties of seeds to grow them in different environments to produce healthy plants and get more yield.

A scientist observed that a particular seed grow very fast after germination. He had recorded growth of plant since germination and he said that its growth can be defined by the function                              [CBSE   2023]

f(x) = 1/3 x³ – 4 x² + 15 x + 2, 0 ≤ x ≤ 10

Where x is the number of days the plant is exposed to sunlight.

On the basis of the above information, answer the following questions :

(i) What are the critical points of the function f(x) ?                              (2)

(ii) Using second derivative test, find the minimum value of the function.          (2)

Solution:- Since, f(x) = 1/3 x³ – 4 x² + 15 x + 2, 0 ≤ x ≤ 10

Differentiate with respect to x

For maxima and minima

x² – 8x + 15 = 0

⇒ x² – 5x – 3x + 15 = 0

⇒ x(x – 5) – 3(x – 5) = 0

⇒ (x – 5)(x – 3) = 0

Hence, x = 5, 3 ∈ [0 10]

Now the critical points are x = 5, 3

(ii) Again differentiate with respect to x

at x = 3

Now the minimum value of function at x = 3

f(x) = 1/3 x³ – 4 x² + 15 x + 2

⇒ f(3) = 20

The minimum value of the function is 20

Q 1:- In a group activity class, there are 10 students whose ages are 16, 17, 15, 14, 19, 17, 16, 19, 16 and 15 years. One student is selected at random such that each has equal chance of being choosen and age of the student is recorded.                           [CBSE  2023]

On the basis of the above information, answer the following question :

(i) Find the probability that the age of the selected student is a composite number.             (1)

(ii) Let X be the age of the selected student. What can be the value of X ?                               (1)

(iii) (a) Find the probability distribution of random variable X and hence find the mean age.             (2)

OR

(iii) (b) A student was selected at random and his age was to be greater than 15 years. Find the probability that his age is a prime number.                                (2)

Solution:- See full solution

Q 2:- A housing society wants to commission a swimming pool for its residents. For this, they have to purchase a square piece of land and dig this to such a depth that its capacity is 250 cubic metres. Cost of land is Rs 500 per square metre. The cost of digging increases with the depth and cost for the whole pool is Rs 4000 (depth)².                                      [CBSE   2024]

Suppose the side of the square plot is x metres and depth is h metres.

On the basis of the above information, answer the following questions :

(i) Write cost C(h) as a function in terms of h.                         (1)

(ii) Find critical point.                          (1)

(iii) (a) Use second derivative test to find the value of h for which cost of construction of the pool is minimum. What is the minimum cost of construction of the pool ?             (2)

OR

(iii)(b) Use first derivative test to find the depth of the pool so that cost of construction is minimum. Also, find relation between x and h for minimum cost.                                      (2)

Solution:- See full solution