CUET Mathematics Applied Mathematics > Calculus PYQs

In a company, the cost of producing one item is ₹100, if the item is sold at ₹150 and ₹50,000 is the fixed cost of the company, then the break-even point is:

Let C = C(x) be the total cost of producing x units of a product and AC is the average cost and MC is the marginal cost, then Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) AC decreases with x | (I) when MC = AC | | (B) AC increases with x | (II) when MC < AC | | (C) AC doesn't change with x | (III) when MC = 0 | | (D) C is constant. | (IV) when AC < MC | Choose the correct answer from the options given below:

The marginal cost (MC) of producing x units of an item in a factory is given by $MC=0.006x^2-0.001x+4$. If the fixed cost is ₹1000, then the cost of producing 200 units will be :

The demand function for a commodity is $p = 45 + x - x^2$, then the consumer's surplus at equilibrium price $p_0 = 25$ will be:

Given that the total cost function for x units of a commodity is $C(x) = \dfrac{x^3}{4} + 4x^2 - 8x + 20$. Then the marginal cost when $x=4$ is:

Which of the following statements are true? (A) The break-even point is the level of production where the revenue from sales is equal to the cost of production and marketing. (B) If R is the total revenue collected by a company when it sells $x$ units of a product at a price of p per unit, then R is given by $R=px$. (C) Let $C=C(x)$ be the total cost of producing and marketing $x$ units of a product, then marginal cost, $MC=\frac{dC}{dx}$. (D) If AC is the average cost, then $AC=\frac{x}{C}$. Choose the correct answer from the options given below:

If $R(x)$ is the total revenue received and $C(x)$ is the total cost incurred in the production of x units of a commodity, then the profit function $P(x)$

The second order derivative of the function $y=\log(4x^3+6)$ with respect to 'x' will be:

Match the LIST-I with LIST-II | | LIST-I (Differential equation) | | LIST-II (Degree) | |---|---|---|---| | A. | $\left(\dfrac{dy}{dx}\right)^2 + \dfrac{1}{\dfrac{dy}{dx}} = 5$ | I. | $2$ | | B. | $\dfrac{d^3y}{dx^3} = \left\{3 + \left(\dfrac{dy}{dx}\right)^2\right\}^{3/2}$ | II. | $4$ | | C. | $\dfrac{d^2y}{dx^2} = \left(\dfrac{dy}{dx}\right)^{5/4}$ | III. | $1$ | | D. | $y\dfrac{d^2x}{dy^2} = y^2 + 1$ | IV. | $3$ | Choose the correct answer from the options given below:

$\int_1^4 x^3\log x\, dx$ is equal to:

The area of the region bounded by curves $y=x^2$ and line $y=4$ is:

The relation between 'marginal cost (MC)' and 'average cost (AC)' of producing 'x' units of a product is

For the function $f(x)=2x^3-21x^2+36x-5$, which of the following statements are correct ? A. $x=2$ is a point of local maxima. B. $x=6$ is a point of local minima. C. The local maximum value is 12. D. The local maximum value is –94. Choose the correct answer from the options given below:

If $y = e^{\frac{1}{2}\log(1+ \tan^2 x)}$, then $\frac{d^2y}{dx^2}$ is equal to:

The supply function of a commodity is P = $x^3+2x+18$. When 4 units of commodity are sold , then producer surplus is:

The demand function P for maximising a profit monopolist is given by P=274-x², while the marginal cost is 4+3x for x units of commodity. The consumer surplus is

If the total cost and total revenue of a company that produces and sells $x$ units of a particular product are $C(x) = 5x + 350$ and $R(x) = 50x - x^2$ respectively, then which of the following is/are the breakeven values: (A) $x = 10$ (B) $x = 25$ (C) $x = 45$ (D) $x = 30$ Choose the correct answer from the options given below:

A company is selling a certain commodity $x$. The demand function for the commodity is linear. The company can sell $2000$ units when the price is ₹$8$ per unit and it can sell $3000$ units when the price is ₹$4$ per unit. The Marginal revenue at $x=5$ is:

If $x = 2at, y = at^2$, where 'a' is a constant, then $\frac{d^2 y}{dx^2}$ at $x = 2$ is :

The interval in which where the function $f(x) = x^3 - 3x^2 + 4x + 1, x \in R$ is increasing in, is :

The area bounded by the curve $y = x^2$ between $x = 0$ and $x = \pi$ in the first quadrant is :

The order and degree Of the differential equation $\frac{d^2 y}{dx^2} + 2 e^{-x} = 0$, respectively are

the general solution of the differential equation $(1 + y) dx - 2x dy = 0$ is :

If $v = \frac{4}{3} \pi r^3$, at what rate is cubic / unit sec is increasing when $r = 10$, and $\frac{dr}{dt} = 0.01$ ?

The value of the integral $\int_{-3}^{3} (x^3 - x) dx$ is :

$\int \frac{dx}{x^{n+1} - x}\,dx$

The second order derivative of which of the following functions is $5^x$ ?

The degree of the differential equation $\left(1 - \left(\frac{dy}{dx}\right)^2\right)^{3/2} = k \frac{d^2 y}{dx^2}$ is :

If $C(x) = x^3 - \frac{5}{2} x^2 + 10$ represents the total cost of producing x unit by car manufacturing company. The slope of the marginal cost curve at $x = 3$, will be

If x is real, the minimum value of $f(x) = x^2 - 8x + 20$ is :

If $y = x^3 \log x$, then $\frac{d^2 y}{dx^2}$ is equal to :

The value of the integral $\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx$ is :

Area of the region bounded by the curve $x^2 = 4y$, $x$ - axis and $x = 3$ is

The value of the integral $I = \int_{-1}^{1} (x + x^3 + x^5) dx$ is :

If $(x+ 1) e^y = 1$ , then :

Match List - I with list- II | List-I Equation of curves | List - II Slope of tangent at x = 2 | |---|---| | A. $Y = x^3 - x$ | 8 | | B. $Y = (x-2)^2$ | 2/3 | | C. $Y = 2x^2 + 3$ | 11 | | D. $Y = \sqrt{4x + 1} - 7$ | 0 | Choose the correct option below :

The value of a for which the function $f(x) = a^x$ is increasing on R are given by :

Match List - I with list- II | List-I , Differential equation | List - II , Degree | |---|---| | A. $\left(\frac{dy}{dx}\right)^3 + yx = 0$ | 2 | | B. $e^{\frac{dy}{dx}} + y^2 + y'' = 0$ | 1 | | C. $Xyy'' + x(y')^2 - yy' = 0$ | Not defined | | D. $(Y'')^2 + y = 0$ | 3 | Choose the correct option below :

Any function f(x) is an increasing function in [a,b] if : (A) $x_1, x_2 \in [a, b], f(x_1) \geq f(x_2)$ if $x_1 < x_2$ (B) $x_1, x_2 \in [a, b], f(x_1) \geq f(x_2)$ if $x_1 > x_2$ (C) (A) $x_1, x_2 \in [a, b], f(x_1) \leq f(x_2)$ if $x_1 < x_2$ (D) (A) $x_1, x_2 \in [a, b], f(x_1) < f(x_2)$ if $x_1 > x_2$

If the cost function and the profit function for a company is given by $C = 10 - 0.3x^2$ and $P = 0.3x^2 + 2x - 10$ respectively , where X represent units of output, then the revenue function is given by :

A monopolist's Demand function is $x = 70 - \frac{P}{2}$, the revenue at $x = 5$ will be :

If $y = \log_3(\log_3 x)$, then $\frac{dy}{dx}$

If $t = e^{2x}$ and $\log_e t^2$, then $\frac{d^2y}{dx^2}$ is :

If $x = at^2$ and $y = a^3 t^3$, then $\frac{d^2 y}{dx^2}$ will be :

The minimum value of $f(x) = 4x^3 - 48x + 105$ in the interval [1,3] is :

For a manufacturer, total cost function is given by $C = \frac{x^2}{25} + 2x$. Which of the following are correct ? A. 2.6 is the marginal cost when 5 units are produced. B. $\frac{2x}{25} + 2$ is the marginal cost function. C. $\frac{x}{25} + 2$ is the average cost function. D. 2.4 is the marginal cost when 5 units are produced. E. $\frac{x}{25} + 1$ is the average cost function. Choose the correct answer from the options given below:

Two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum are :

The total cost of producing $x$ generators is given by TC = $x^3 - 60x^2 + 1500x + 2000$. The Marginal Cost (MC), when $x = 10$ units is:

If $y = \frac{\log x}{x^2}$, then $\frac{d^2y}{dx^2}$ is equal to

If $x^{2/3} + y^{2/3} = a^{2/3}$, then $\frac{dy}{dx}$ is equal to :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | If $y = x^2 - 8$ and $\frac{dy}{dx} = 0$, then $x = ?$ | (I) | 1 | | (B) | If $p(x) = 3x + 1$, then $R(x)$ at $x = 2$ | (II) | 0 | | (C) | If $y = x^3$, then $\frac{dy}{dx}$ at $x = -1$ | (III) | 14 | | (D) | If $C(x) = 100 + 5x$, $R(x) = 102 + 3x$, then break-even point | (IV) | 3 | Choose the **correct** answer from the options given below :

Let $f : R \to R$ be a function defined as $f(x) = 2x^3 - 21x^2 + 36x - 20$, then : (A) maximum value of $f(x)$ is $-3$ (B) minimum value of $f(x)$ is $-128$ (C) maximum value exists at $x = 6$ (D) minimum value exists at $x = 1$ Choose the **correct** answer from the options given below :

If $y = 3e^{2x} + 2e^{3x}$, then which one of the following is true ?

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | For break-even point | (I) | $< 0$ | | (B) | For maxima $\frac{d^2y}{dx^2}$ | (II) | $\frac{dy}{dx} = 0$ | | (C) | For points of maxima/minima | (III) | $R(x) - C(x)$ | | (D) | $P(x) = $ Profit function | (IV) | $R(x) = C(x)$ | Choose the **correct** answer from the options given below :

If $x = 2t^2 + 3, y = 3t^2 + 6t + 5$, then the value of $\frac{d^2y}{dx^2}$ is :

Let $f : R \to R$ be defined such that $f(x) = 16x^2 - 16x + 12$ (A) Maximum value of $f(x)$ is 8 (B) Minimum value of $f(x)$ is 8 (C) Minimum value of $f(x)$ is 16 (D) No maximum value of $f(x)$ Choose the correct answer from the options given below :

If $C(x) = ax^2 - bx - c$ represents the total cost function then the slope of the tangent to the marginal cost curve at the point $(x, y)$ is :

The demand function of a monopolist is given by $p = 1500 - 2x - x^2$, then value of marginal revenue when $x = 20$ is :

Match List - I with List - II. | List - I (Functions) | List - II (Maximum value) | |---|---| | (A) $f(x) = -x^2, x \in (-\infty, \infty)$ | (I) 8 | | (B) $f(x) = -x^2 + 1, x \in (-\infty, \infty)$ | (II) 7 | | (C) $f(x) = x + 1, x \in [0, 6]$ | (III) 1 | | (D) $f(x) = x^3, x \in [0, 2]$ | (IV) 0 | Choose the correct answer from the options given below :

If $x^3 + y^3 = xy$, then $\frac{dy}{dx}$ is equal to :

If $x = 6t^2$, $y = \frac{6}{t^2}$, then $\frac{d^2 y}{dx^2}$ is equal to :

If $y = \log\left(\frac{x^5}{e^5}\right)$, then $\frac{d^2y}{dx^2}$ is,

The point on the curve $y^2 = 16x$ for which the y-coordinate is changing 2 times as fast as the x-coordinate is :

The total cost function for $x$ units of a commodity is given by $C(x) = \frac{25x^3}{3} - 75x^2 + 48x + 34$. The output $x$ at which the marginal cost is minimum is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) The minimum value of $f(x) = 8x^2 - 4x + 7$ is | (I) 48 | | (B) The maximum value of $f(x) = x + \frac{1}{x}$, $x < 0$ is | (II) 13 | | (C) The maximum slope of the curve $y = -2x^3 + 6x^2 + 7x + 26$ is | (III) $-2$ | | (D) The minimum value of $f(x) = x^2 + \frac{128}{x}$ is | (IV) $\frac{13}{2}$ | Choose the correct answer from the options given below :

A product costs the manufacturer ₹ 20 per unit. The demand function is given by $p(x) = 1000 - 20x$, then the quantity for maximum profit is :

If $x = 3at^2$, $y = 3at^4$ then $\frac{dy}{dx}$ is :

The point on the straight line $3x + 4y = 8$, which is closest to the origin is:

If the sum of two positive numbers is 25 and their product is maximum when divided in the ratio of cubes of one and squares of the other, then the numbers are:

If $f(x) = a \log x + \frac{b}{x} + x$ has its extreme values at $x = -1$ and $x = 3$, then $(a, b)$ is equal to:

If the cost function $C(x)$ of producing $x$ units of a commodity is given as $C(x) = x^3 - 60x^2 + 13x + 50$, then the level of output for which the marginal cost is minimum is

If $x = \log t$ and $y = \frac{1}{t^2}$, then $\frac{d^2 y}{d x^2}$ is equal to

If $y = \log_e \left(\frac{x^3}{e^3}\right)$, then $\frac{d^2 y}{d x^2}$ is equal to