Radha an aspiring landscape designer is tasked with creating

Case study from CBSE sample paper Basic math 2024 – 2025 class 10

Case study 2:- Radha an aspiring landscape designer is tasked with creating a visually captivating pool design that incorporates a unique arrangement of fountains. The challenge entails arranging the fountains in such a way that when water is thrown upwards, it forms the shape of a parabola. The graph of one such parabola is given below.

The height of each fountain rod above water level is 10 cm. The equation of the downward-facing parabola representing the water fountain is given by

𝑝(𝑥) = −𝑥² + 5𝑥 − 4.

Based on the above information, answer the following questions:

(i) Find the zeroes of the polynomial p(x) from the graph          [1 marks]

(ii) Find the value of x at which water attains maximum height.      [1 marks]

(iii)(A) If h is the maximum height attained by the water stream from the water level of the pool, then find the value of h.           [2 marks]

OR

(iii)(B) At what point(s) on x- axis, the height of water above x- axis is 2 m?        [2 marks]

Solution:- (i) Given equation, 𝑝(𝑥) = −𝑥² + 5𝑥 − 4

P(x) = -[x² – 5x + 4]

⇒ P(x) = -[x² – 4x – x + 4]

⇒ P(x) = -[x(x – 4) – 1(x – 4)]

⇒ P(x) = -[(x – 4)(x – 1)]

For the zeros of the polynomial, P(x) =0

-[(x – 4)(x – 1)] = 0

x = 4 and x = 1

Hence, zeroes of the polynomial are 4 and 1

(ii) 𝑝(𝑥) = −𝑥² + 5𝑥 − 4

P(x) = -[x² – 5x + 4]

⇒

⇒

⇒

Since,

⇒

Maximum value of P(x) is at x = 5/2 m

(iii) The maximum value of P(x) above x axis = 9/4

The height of each fountain rod above water level is 10 cm = 0.1 m

If h is the maximum height attained by the water stream from the water level of the pool, then find the value of h = 9/4m + 0.1

= 2.25 + 0.1 = 2.35 m

OR

(iii) The height of water above x- axis is 2 m

𝑝(𝑥) = −𝑥² + 5𝑥 − 4

⇒ 2 = −𝑥² + 5𝑥 − 4

⇒ 𝑥² – 5𝑥 + 6 = 0

⇒ 𝑥² – 3𝑥 – 2x +6 = 0

⇒ x(x – 3) – 2(x – 3) = 0

⇒ (x – 3)(x – 2) = 0

⇒ x = 3 or x = 2

When x =  2 m and 3 m then The height of water above x- axis is 2 m

Case study 1:- A group of students conducted a survey to find out about the preferred mode of transportation to school among their classmates. They surveyed 200 students from their school. The results of the survey are as follows:

120 students preferred to walk to school.

25% of the students preferred to use bicycles.

10% of the students preferred to take the bus.

Remaining students preferred to be dropped off by car.

Based on the above information, answer the following questions:

(i) What is the probability that a randomly selected student does not prefer to walk to school ?

(ii) Find the probability of a randomly selected student who prefers to walk or use a bicycle.

(iii) (A) One day 50% of walking students decided to come by bicycle. What is the probability that a randomly selected student comes to school using a bicycle on that day?

OR

(iii) (B) What is the probability that a randomly selected student prefers to be dropped off by car?

Solution:- See full solution

Case study 3:- Rinku was very happy to receive a fancy jumbo pencil from his best friend Rohan on his birthday. Pencil is a basic writing tool, when sharpened its shape is a combination of cylinder & cone as given in the picture. Cylindrical pencil with conical head is a common shape worldwide since ages. Commonly pencils are made up of wood & plastic but we should promote pencils made up of eco-friendly material (many options available in the market these days) to save environment.

The dimensions of Rinku’s pencil are given as follows:

Length of cylindrical portion is 21cm. Diameter of the base is 1 cm and height of the conical portion is 1.2 cm

Based on the above information, answer the following questions:

(i) Find the slant height of the sharpened part.        [1 marks]

(ii) Find curved surface area of sharpened part (in terms of 𝜋).      [1 marks]

(iii) (A) Find the total surface area of the pencil (in terms of 𝜋).             [2 marks]

OR

(iii)(B) The pencil’s total height decreases by 8.2 cm after sharpening it many times, what is the volume of the cylindrical part of the shortened pencil (in terms of 𝜋) ?                 [2 marks]

Solution :- See full solution