The demand function of a monopoly is given by $x = 100 - 4p$. Find the quantity at which the MR will be zero. [Application]
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Application of Calculus - ISC Class 12 Mathematics Questions with Answers, Page 2
69 past-paper questions on Application of Calculus from ISC Class 12 Mathematics papers (2026-2017), newest first, in full. Questions 21-40 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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A utensil manufacturer produces '$x$' dinner sets per week and sells each set at ₹ $p$, where $x = \frac{600 - p}{8}$. The cost of production of '$x$' sets is ₹ $\left(\frac{x^2}{2} + 78x + 2000\right)$.
(a)[1.0]
Write the revenue function.
(b)[1.0]
Write the profit function.
(c)[2.0]
Calculate the number of dinner sets to be produced and sold per week to ensure maximum profit.
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The fuel cost per hour for running a ship is proportional to the square of the speed generated in knots. The fuel cost is ₹ 75/h at 10 knots and the fixed charges amount to ₹ 1000/h.
(a)
Given that the fuel cost per hour is k times the square of the speed, the ship generates in km per hour, then what is the value of ‘k’?
(b)
What will be the cost per unit distance?
(c)
Determine the most economical speed to run the ship.
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To launch a new product in his company, Mr. Rajesh spends ₹ 1 lakh on the infrastructure and the variable cost of the product is estimated as ₹ 150/- per unit. The sale price per unit is fixed as ₹ 200/-. Additionally, Mr Rajesh also spends ₹ 0.5 per unit squared for marketing.
Find the profit function. Hence, draw an inference regarding the breakeven point.
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The cost function $C(x)$ of producing '$x$' quantities of a product is given by $C(x) = 500x^2 + 2500x + 5000$. If each unit of the product is sold at ₹ 6000 then, calculate
(a)[2.0]
Break-even point
(b)[2.0]
Marginal cost and show that the marginal cost increases as the output '$x$' increases.
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The total revenue received from the sale of '$x$' units of a product is $R(x) = 3x^2 + 36x + 5$. Then, the actual revenue for selling the $10^{\text{th}}$ item will be:
- (a)$27$
- (b)$90$
- (c)$93$
- (d)$33$
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The total cost of producing and marketing $x$ units of bulbs by a wholesaler is given by $C(x) = \frac{1}{3}x^2 + e^{2x} + 3e$.
What will be the average cost of producing 3 units of bulbs?
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The cost function of a commodity is $C(x) = 200 + 20x - \frac{1}{2} x^2$ (in rupees). Find the range where AC falls. [Application]
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Observe the graph shown below of Cost, Revenue versus Quantity.
Statement 1: A denotes Profit area, B denotes Break-even point and C shows Loss area.
Statement 2: A denotes Loss area, B denotes Average of total revenue and total cost and C shows Profit area.
- (a)Statement 1 is true and Statement 2 is false.
- (b)Statement 2 is true and Statement 1 is false.
- (c)Both the statements are true.
- (d)Both the statements are false.
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The fixed cost of a new product is ₹ 200 and the variable cost per unit is ₹ $x$. If the demand function $p(x) = 30$, then what are the value(s) of $x$ that result in a loss given that $x$ units of the product are sold?
- (a)$10 < x < 20$
- (b)$x > 20$
- (c)$x < 10$
- (d)$x < 10\text{ or } x > 20$
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The Average Cost of producing '$x$' units of commodity is given by:
$AC = \frac{x}{200} + 30 + \frac{5000}{x}$
(a)[1.0]
Find the Cost function.
(b)[1.0]
Find the Marginal Cost function.
(c)[1.0]
Find the Marginal Average Cost function.
(d)[1.0]
Verify that $\frac{d}{dx}(AC) = \frac{MC - AC}{x}$
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The demand function for a product is given by $p = 1500 - 9x - 2x^2$
(a)[1.0]
Find the revenue function.
(b)[1.0]
Write the marginal revenue.
(c)[2.0]
Calculate the level of output at which the total revenue is Maximum.
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In Z Square Mall in Kanpur, there is a space to keep 300 cars and the entry fees per car is ₹ 20. It is estimated that if the entry fee is decreased by ₹ 5, then 50 additional cars can be adjusted in the same parking.
Justify that the Marginal Revenue (MR) decreases at a higher rate than the Average Revenue (AR).
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In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
(i)[1.0]
A company sells hand towels at ₹ 100 per unit. The fixed cost for the company to manufacture hand towels is ₹ 35,000 and variable cost is estimated to be 30% of total revenue. What will be the total cost function for manufacturing hand towels?
- (a)$35000 + 3x$
- (b)$35000 + 30x$
- (c)$35000 + 100x$
- (d)$35000 + 10x$
(ii)[1.0]
If the correlation coefficient of two sets of variables $(X, Y)$ is $-\frac{3}{4}$, which one of the following statements is true for the same set of variables?
- (a)Only one of the two regression lines has a negative coefficient.
- (b)Both regression coefficients are positive.
- (c)Both regression coefficients are negative.
- (d)One of the lines of regression is parallel to the x-axis.
(iii)[1.0]
If the total cost function is given by $C = x + 2x^3 - \frac{7}{2}x^2$, find the Marginal Average Cost function (MAC).
(iv)[1.0]
The equations of two lines of regression are $4x + 3y + 7 = 0$ and $3x + 4y + 8 = 0$. Find the mean value of $x$ and $y$.
(v)[1.0]
The manufacturer of a pen fixes its selling price at ₹ 45, and the cost function is $C(x) = 30x + 240$. The manufacturer will begin to earn profit if he sells more than 16 pens. Why? Give one reason.
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The Average Cost function associated with producing and marketing $x$ units of an item is given by $AC = x + 5 + \frac{36}{x}$.
(a)
Find the Total Cost function.
(b)
Find the range of values of $x$ for which Average Cost is increasing.
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A monopolist’s demand function is $x = 60 - \frac{p}{5}$. At what level of output will marginal revenue be zero?
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The fixed cost of a product is ₹ 30,000 and its variable cost per unit is ₹ 800. If the demand function is $p(x) = 4500 - 100x$, find the break-even values.
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In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
(i)[1.0]
Which relation is correct for breakeven point where $R(X)$ is revenue function and $C(x)$ is cost function?
- (a)$R(X) > C(X)$
- (b)$R(X) < C(X)$
- (c)$R(X) = C(X)$
- (d)$R(X) = 2C(X)$
(ii)[1.0]
If correlation coefficient $r = 0$, then regression lines are
- (a)parallel to each other.
- (b)not mutually perpendicular.
- (c)parallel to coordinate axis.
- (d)overlapping lines.
(iii)[1.0]
Average revenue of a commodity is given by $AR(x) = a + \frac{b}{x}$. Find the demand function when marginal revenue is zero.
(iv)[1.0]
If $R(X) = 36x + 3x^2 + 5$ then find actual revenue from selling 50th item.
(v)[1.0]
For a given bi-variant distribution, the mean of variable $x = 4$ and the mean of variable $y = 6$. Find the point of intersection of two regression lines.
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For the revenue function $R(x) = \frac{bx}{a+x}$, show that the marginal revenue function is increasing for all $b < 0$ and $a > 0$.
Given: $R(x) = \frac{bx}{a+x}$ where $b < 0$ and $a > 0$
To show: The marginal revenue function is increasing for all $b < 0$ and $a > 0$.
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The average revenue function is given by $AR = 25 - \frac{x}{4}$. Find total revenue function and marginal revenue function.
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