The cost function is $C(x) = 3x^2 - 6x + 5$. Find the average cost when $x = 2$.
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Application of Calculus - ISC Class 12 Mathematics Questions with Answers, Page 3
69 past-paper questions on Application of Calculus from ISC Class 12 Mathematics papers (2026-2017), newest first, in full. Questions 41-60 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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The total cost function for $x$ units is given by $C(x) = \sqrt{6x + 5} + 2500$. Show that the marginal cost decreases as the output $x$ increases.
Given: $C(x) = \sqrt{6x + 5} + 2500$
To show: $\text{Marginal cost decreases as the output } x \text{ increases}$
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The total revenue function $R(x) = x + 2x^3 - 3\cdot 5x^2$. Find the point where marginal revenue curve cuts the co-ordinate axis.
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If the demand function is given by $p = 1500 - 2x - x^2$ then find the marginal revenue when $x = 10$:
- (a)$1160$
- (b)$1600$
- (c)$1100$
- (d)$1200$
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A company sells its product for ₹ 20 per unit. Fixed costs for the company is ₹ 45,000 and variable costs is estimated to run 25% of total revenue. If $x$ denotes number of units produced, then what will be the total cost function?
- (a)$45000 + 5x$
- (b)$15000 + 4x$
- (c)$45000 + 2x$
- (d)$4500 + 20x$
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The demand function for a certain commodity is given by $p = 4000 - 100x$. What will be the total revenue from the sale of 3 units?
- (a)11,100
- (b)1000
- (c)4500
- (d)2000
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A company sells $x$ packets of biscuits each day at ₹ 10 a packet. The cost of manufacturing these packets is ₹ 5 per packet plus a fixed daily overhead cost of ₹ 700. What will be the profit function?
- (a)$6x - 400$
- (b)$5x - 700$
- (c)$10x - 500$
- (d)$5x - 10$
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The cost function of a firm is given by $C(x) = 3x^2 - 2x + 6$. The average cost of the firm at $x = 3$ is:
- (a)11
- (b)17
- (c)9
- (d)27
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The demand function for a certain product is represented by the equation: $p = ax^2 + bx + c$ where $x$ is the number of units demanded and $p$ is the price per unit.
Answer the following questions by choosing the correct option:
(i)[2.0]
The revenue function $R(x)$ is:
- (a)$ax^3 + bx^2 + cx$
- (b)$ax + b + \frac{c}{x}$
- (c)$ax^3 + bx^2 + cx + d$
- (d)$2ax + b$
(ii)[2.0]
The marginal revenue $MR(x)$ is:
- (a)$a - \frac{c}{x^2}$
- (b)$3ax^2 + 2bx + c$
- (c)$3ax^3 + 2bx^2 + c$
- (d)$2a$
(iii)[2.0]
The slope of the marginal revenue is:
- (a)0
- (b)$6ax + 2b$
- (c)$\frac{2c}{x^3}$
- (d)$9ax^2 + 4bx$
(iv)[2.0]
Values of $x$, for which marginal revenue increases is:
- (a)$x > -\frac{b}{3a}$
- (b)$x < -\frac{b}{3a}$
- (c)$x = -\frac{b}{3a}$
- (d)$x \le -\frac{b}{3a}$
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The total cost function for a production is given by $C(x) = \frac{3}{4}x^2 - 7x + 27$.
Find the number of units produced for which $\text{M.C.} = \text{A.C.}$
(M.C. = Marginal Cost and A.C. = Average Cost.)
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[5×1] In sub-parts (i) and (ii) choose the correct options and in sub-parts (iii) to (v), answer the questions as instructed.
(i)
The demand function of a monopolist is given by $x = 100 - 4p$. The quantity at which $\text{MR} = 0$ will be:
- (a)25
- (b)10
- (c)50
- (d)30
(ii)
If the lines of regression are parallel to coordinate axes, then the coefficient of correlation is:
- (a)1
- (b)0
- (c)$-1$
- (d)½
(iii)
Find the marginal cost function (MC), if the cost function is:
$C(x) = \frac{x^3}{3} + 5x^2 - 16x + 2$.
(iv)
If revenue function $R(x) = 3x^3 + 8x - 2$, find the average revenue function.
(v)
If $\sigma_x = 3$, $\sigma_y = 4$ and $b_{xy} = \frac{1}{3}$, then find the value of correlation coefficient ($r$).
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A company produces a commodity with ₹ 24,000 as fixed cost. The variable cost estimated to be 25% of the total revenue received on selling the product, is at the rate of ₹ 8 per unit. Find the break-even point.
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The total cost function of a firm is given by $C(x) = \frac{1}{3}x^3 - 5x^2 + 30x - 15$ where the selling price per unit is given as ₹ $6$. Find for what value of $x$ will the profit be maximum.
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The marginal cost of the production of the commodity is $30 + 2x$, it is known that fixed costs are ₹ $200$, find:
(i)[2.0]
The total cost.
(ii)[2.0]
The cost of increasing output from $100$ to $200$ units.
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[3×2]
(a)[2.0]
The selling price of a commodity is fixed at ₹ $60$ and its cost function is $C(x) = 35x + 250$.
(i) Determine the profit function.
(ii) Find the break even points.
(b)[2.0]
The revenue function is given by $R(x) = 100x - x^2 - x^3$. Find:
(i) The demand function.
(ii) Marginal revenue function.
(c)[2.0]
For the lines of regression $4x - 2y = 4$ and $2x - 3y + 6 = 0$, find the mean of ‘$x$’ and the mean of ‘$y$’.
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The cost function of a product is given by $C(x) = \frac{x^3}{3} - 45x^2 - 900x + 36$ where $x$ is the number of units produced. How many units should be produced to minimise the marginal cost?
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[3×2]
(a)[2.0]
A company produces a commodity with ₹ 24,000 as fixed cost. The variable cost estimated to be 25% of the total revenue received on selling the product, is at the rate of ₹ 8 per unit. Find the break-even point.
(b)[2.0]
The total cost function for a production is given by $C(x) = \frac{3}{4}x^2 - 7x + 27$. Find the number of units produced for which M.C. = A.C. (M.C. = Marginal Cost and A.C. = Average Cost.)
(c)[2.0]
If $\bar{x} = 18$, $\bar{y} = 100$, $\sigma_x = 14$, $\sigma_y = 20$ and correlation coefficient $r_{xy} = 0.8$, find the regression equation of $y$ on $x$.
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The marginal cost function of $x$ units of a product is given by $\text{MC} = 3x^2 - 10x + 3$. The cost of producing one unit is ₹ 7. Find the total cost function and average cost function.
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The fixed cost of new product is ₹ $35000$ and the variable cost per unit is ₹ $500$. If the demand function: $p = 5000 - 100x$, find the break-even value(s)?
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A product can be manufactured at a total cost $C(x) = \frac{x^2}{100} + 100x + 40$, where $x$ is the number of units produced. The price at which each unit can be sold is given by $P = \left(200 - \frac{x}{400}\right)$. Determine the production level $x$ at which the profit is maximum. What is the price per unit and total profit at the level of production?
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