A store has been selling calculators at Rs 350 each

Q 3:- A store has been selling calculators at Rs 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units (x) sold. The relation between the price and quantity sold is given by the demand function $p = 450 - \frac{1}{2} x$ . [CBSE 2024]

A store has been selling calculators at Rs 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units (x) sold.

Based on the above information, answer the following questions:

(i) Determine the number of units (x) that should be sold to maximise the revenue R(x) = xp(x). Also, verify result. (2)

(ii) What rebate in price of calculator should the store give to maximise the revenue ? (2)

Solution:- (i) Given, R(x) = xp(x)

$R(x) = x(450 - \frac{1}{2} x)$ .

⇒ $R(x) = 450 x - \frac{1}{2} x^2$ . . . (i)

Now differentiate the function with respect to x

$\dfrac{d}{dx} R(x) = 450 - \dfrac{1}{2}.2x$

⇒ $\dfrac{d}{dx} R(x) = 450 - x$ . . . (ii)

For maxima and minima $\dfrac{d}{dx} R(x) = 0$

450 – x = 0

⇒ x = 450

Again differentiate with respect to x

$\dfrac{d^2}{dx^2}R(x) = -1 < 0$

Thus, R(x) is maximum when x = 450 units

(ii) Price of calculator when x = 450 units

$p(x) = 450 - \frac{1}{2} x$

⇒ $p(450) = 450 - \dfrac{1}{2}.450$

⇒ $p(450) = 450 - 225= 225$

The rebate in price of calculator should the store give to maximise the revenue

= 350 – 225 = Rs 125

Q 1:- An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at O(0, 0, 0) and the three stars have their locations at the points D, A and V having position vectors $2\hat{i} + 3\hat{j} + 4\hat{k}, 7\hat{i} + 5\hat{j} + 8\hat{k}$ and $-3\hat{i} + 7\hat{j} + 11\hat{k}$ respectively.

Based on the above information, answer the following question:

(i) How far is the star V from star A ? (1)

(ii) Find a unit vector in the direction of $\vec{DA}$ . (1)

(iii)(a) Find the measure of ∠VDA . (2)

(iii) (b) What is the projection of vector $\vec{DV}$ on vector $\vec{DA}$ ? (2)

Solution :- See full solution

Q 2: Rohit Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit’s selection is 1/5, Jaspreet’s selection is 1/3 and Alia’s selection is 1/4. The event of selection is independent of each other. [CBSE 2024]

Based on the above information, answer the following question :

(i) What is the probability that at least one of them is selected ? (1)

(ii) Find $P(G/\overline{H})$ where G is the event of Jaspreet’s selection and $\bar{H}$ denotes the event that Rohit is not selected. (1)