Application Of Derivatives

01

PYQ 2022

easy

mathematics ID: cuet-ug-

A cable network provider in a small town has 500 subscriber and he used to collect Rs. 300 per month from each subscriber. He proposes o increase the monthly charges and it is believed from the past experience that for every increase of Rs. 1, one subscriber will discontinue the service. Based on the above information answer the following question: What is the increase in the changes per subscriber that yields maximum revenue?

1

100

2

200

3

300

4

400

02

PYQ 2023

medium

mathematics ID: cuet-ug-

The total cost function for x units of a commodity is given by . The output at which the marginal cost is minimum is:

1

6

2

3

3

10

4

5

03

PYQ 2023

medium

mathematics ID: cuet-ug-

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:

1

25 units

2

50 units

3

49 units

4

04

PYQ 2023

medium

mathematics ID: cuet-ug-

The total cost of a firm is given by c(x)= The level of output at which marginal cost is minimum is

1

8

2

15

3

2

4

12

05

PYQ 2023

medium

mathematics ID: cuet-ug-

The price per unit of a commodity produced by a company is given by P = 92 - 2x 2, where x is the quantity demanded. The marginal revenue of producing 3 units of such a commodity shall be :

1

28

2

38

3

26

4

44

06

PYQ 2023

medium

mathematics ID: cuet-ug-

If the cost function and revenue of x unit of an item are given by C(x)=25x 2 -x and R(x)=4x. Then the number of item to be produced to have maximum profit is,

1

2

3

4

07

PYQ 2023

medium

mathematics ID: cuet-ug-

Match List I with List II LIST I LIST II A . The minimum value of is I . 48 B . The maximum value of is II . 13 C . The maximum slope of the cure is III . -2 D . The minimum value of is IV . Choose the correct answer from the options given below:

1

(A)-(I), (B)-(III), (C)-(II), (D)-(IV)

2

(A)-(IV), (B)-(III), (C)-(II), (D)-(I)

3

(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

4

(A)-(III), (B)-(IV), (C)-(II), (D)-(I)

08

PYQ 2025

medium

mathematics ID: cuet-ug-

The interval, on which the function f(x) = x2e-x is increasing, is equal to

1

(-,)

2

(-, 2) (2,)

3

(-2, 0)

4

(0, 2)

09

PYQ 2025

medium

mathematics ID: cuet-ug-

If the maximum value of the function f(x) =, x > 0 occurs at x = a, then a2f''(a) is equal to

1

2

3

4

10

PYQ 2025

medium

mathematics ID: cuet-ug-

The function f(x) = tanx - x

1

is a decreasing function on [0,)

2

is an increasing function on [0,)

3

is a constant function

4

is neither increasing nor decreasing function on [0,)

11

PYQ 2025

medium

mathematics ID: cuet-ug-

$Match List-I with List-II List-I List-II (A) The minimum value of (I) 4 (B) The maximum value of (II) 10 (C) The minimum value of (III) 3 (D) The maximum value of (IV) 5 Choose the correct answer from the options given below:$

1

$(A) - (I), (B) - (II), (C) - (III), (D) - (IV)$

2

$(A) - (III), (B) - (II), (C) - (I), (D) - (IV)$

3

$(A) - (III), (B) - (I), (C) - (IV), (D) - (II)$

4

$(A) - (III), (B) - (IV), (C) - (I), (D) - (II)$

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept: We need to find the minimum or maximum value for each function in List-I by analyzing their properties. This involves understanding the range of basic functions like squares, absolute values, and trigonometric functions. Step 3: Detailed Explanation: (A) f(x) = : The term is a square, so its minimum value is 0. This occurs when . Therefore, the minimum value of the entire function is . This matches with (III) . (B) f(x) = : (Assuming a typo in the OCR, which likely missed the negative sign before the absolute value). The term is always greater than or equal to 0. Its minimum value is 0. Therefore, the term has a maximum value of 0. The maximum value of the entire function is . This matches with (I) . (C) f(x) = sin(2x) + 6: The range of the sine function, sin(2x), is [-1, 1]. The minimum value of sin(2x) is -1. Therefore, the minimum value of the entire function is . This matches with (IV) . (D) f(x) = : The term has a minimum value of 0. Therefore, the term has a maximum value of 0. The maximum value of the entire function is . This matches with (II) . Step 4: Final Answer: The correct matching is (A) - (III), (B) - (I), (C) - (IV), (D) - (II). This corresponds to option (3).$

List-I	List-II
(A) The minimum value of $f(x)%20%3D%20(2x%20-%201)%5E2%20%2B%203$	(I) 4
(B) The maximum value of $f(x)%20%3D%20-%7Cx%20%2B%201%7C%20%2B%204$	(II) 10
(C) The minimum value of $f(x)%20%3D%20%5Csin(2x)%20%2B%206$	(III) 3
(D) The maximum value of $f(x)%20%3D%20-(x%20-%201)%5E2%20%2B%2010$	(IV) 5

12

PYQ 2025

medium

mathematics ID: cuet-ug-

$Match List-I with List-II List-I List-II (A) The minimum value of (I) 4 (B) The maximum value of (II) 10 (C) The minimum value of (III) 3 (D) The maximum value of (IV) 5 Choose the correct answer from the options given below:$

1

$(A) - (I), (B) - (II), (C) - (III), (D) - (IV)$

2

$(A) - (III), (B) - (II), (C) - (I), (D) - (IV)$

3

$(A) - (III), (B) - (I), (C) - (IV), (D) - (II)$

4

$(A) - (III), (B) - (IV), (C) - (I), (D) - (II)$

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept: We need to find the minimum or maximum value for each function in List-I by analyzing their properties. This involves understanding the range of basic functions like squares, absolute values, and trigonometric functions. Step 3: Detailed Explanation: (A) f(x) = : The term is a square, so its minimum value is 0. This occurs when . Therefore, the minimum value of the entire function is . This matches with (III) . (B) f(x) = : (Assuming a typo in the OCR, which likely missed the negative sign before the absolute value). The term is always greater than or equal to 0. Its minimum value is 0. Therefore, the term has a maximum value of 0. The maximum value of the entire function is . This matches with (I) . (C) f(x) = sin(2x) + 6: The range of the sine function, sin(2x), is [-1, 1]. The minimum value of sin(2x) is -1. Therefore, the minimum value of the entire function is . This matches with (IV) . (D) f(x) = : The term has a minimum value of 0. Therefore, the term has a maximum value of 0. The maximum value of the entire function is . This matches with (II) . Step 4: Final Answer: The correct matching is (A) - (III), (B) - (I), (C) - (IV), (D) - (II). This corresponds to option (3).$

List-I	List-II
(A) The minimum value of $f(x)%20%3D%20(2x%20-%201)%5E2%20%2B%203$	(I) 4
(B) The maximum value of $f(x)%20%3D%20-%7Cx%20%2B%201%7C%20%2B%204$	(II) 10
(C) The minimum value of $f(x)%20%3D%20%5Csin(2x)%20%2B%206$	(III) 3
(D) The maximum value of $f(x)%20%3D%20-(x%20-%201)%5E2%20%2B%2010$	(IV) 5

13

PYQ 2025

hard

mathematics ID: cuet-ug-

$The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = . The marginal cost when 10 items are produced is:$

1

$537.1$

2

$441.15$

3

$1575$

4

$875.25$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: Marginal cost is the rate of change of the total cost with respect to the number of units produced. In calculus, this is represented by the first derivative of the total cost function, C(x). The marginal cost at a specific production level (x=10) is found by evaluating the derivative at that point. Step 2: Key Formula or Approach: Marginal Cost (MC) = . Step 3: Detailed Explanation: The given total cost function is: First, we find the derivative of C(x) with respect to x to get the marginal cost function, MC(x). Using the power rule for differentiation (): Now, we need to find the marginal cost when 10 items are produced, so we substitute x = 10 into the MC(x) function. Step 4: Final Answer: The marginal cost when 10 items are produced is Rs. 537.1.$

14

PYQ 2025

easy

mathematics ID: cuet-ug-

$The rate of change of area of a circle with respect to its circumference when radius is 4cm, is$

1

2

3

4

Official Solution

Correct Option: (2)

$Step 1: Understanding the Concept: We are asked to find the derivative of the area with respect to the circumference, which is represented as . This is a related rates problem where both and are functions of the radius . Step 2: Key Formula or Approach: Let be the area and be the circumference of a circle with radius . We can use the chain rule to find : Step 3: Detailed Explanation: First, we find the derivatives of and with respect to : Now, we can find using the chain rule: The problem asks for this rate of change when the radius is 4 cm. Substituting cm: The units of this rate are area units divided by length units, which is cm /cm. So the result should just be cm. However, the options provided all have units cm /cm, which is a bit redundant but guides us to the numerical answer. The final numerical value is 4. Step 4: Final Answer: The rate of change of the area with respect to its circumference is 4 cm 2 /cm .$

15

PYQ 2025

easy

mathematics ID: cuet-ug-

$The function f(x) = tanx - x$

1

$is a decreasing function on [0,)$

2

$is an increasing function on [0,)$

3

$is a constant function$

4

$is neither increasing nor decreasing function on [0,)$

Official Solution

Correct Option: (2)

$Step 1: Understanding the Concept: To determine if a function is increasing or decreasing on an interval, we examine the sign of its first derivative on that interval. If f'(x)>0, the function is increasing. If f'(x)<0, it is decreasing. Step 2: Key Formula or Approach: 1. Find the derivative of f(x), which is f'(x). 2. Analyze the sign of f'(x) in the given interval [0,). Step 3: Detailed Explanation: The given function is f(x) = tanx - x. First, find the derivative f'(x): We know the trigonometric identity . Substituting this into the derivative expression: Now, we need to analyze the sign of f'(x) = on the interval [0,). For any value of x in this interval, tanx is defined. Since is a square, its value is always greater than or equal to zero. The derivative is zero only at x = 0 (since tan(0) = 0). For all other values of x in (0,), tanx is positive, so is strictly positive. Since f'(x) 0 on the interval [0,) and is not identically zero over any subinterval, the function f(x) is increasing on this interval. Step 4: Final Answer: The function f(x) = tanx - x is an increasing function on [0,)}.$

16

PYQ 2025

medium

mathematics ID: cuet-ug-

$The slope of the normal to the curve y = at x = 1 is:$

1

$4$

2

$-4$

3

$1/2$

4

$-1/4$

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: The slope of the tangent to a curve at a point is given by the value of its derivative () at that point. The normal line is perpendicular to the tangent line at that same point. The slopes of two perpendicular lines are negative reciprocals of each other. Step 2: Key Formula or Approach: 1. Find the derivative of the curve's equation: . 2. Evaluate the derivative at the given x-value to find the slope of the tangent at that point. 3. The slope of the normal is the negative reciprocal of the tangent's slope: . Step 3: Detailed Explanation: The equation of the curve is given by: First, we find the derivative of y with respect to x: This expression,, gives the slope of the tangent at any point x on the curve. We need to find the slope at . So, we substitute into the derivative: The slope of the tangent line at is 4. Now, we find the slope of the normal line, which is the negative reciprocal of the tangent's slope. Step 4: Final Answer: The slope of the normal to the curve at x = 1 is -1/4.$

17

PYQ 2025

medium

mathematics ID: cuet-ug-

$If the maximum value of the function f(x) =, x > 0 occurs at x = a, then a2f''(a) is equal to$

1

2

3

4

Official Solution

Correct Option: (2)

$Step 1: Understanding the Concept: To find the maximum of a function, we find its critical points by setting the first derivative to zero. The second derivative test can confirm if a critical point is a maximum. We then evaluate the required expression. Step 2: Key Formula or Approach: Use the quotient rule for differentiation: For a maximum at, we must have and . Step 3: Detailed Explanation: Given function: . First Derivative: Using the quotient rule with and : To find the maximum, set : So, the maximum occurs at . Second Derivative: Now, differentiate using the quotient rule with and : Now evaluate : Calculate the final expression: We need to find . With : Step 4: Final Answer: The value of is .$

18

PYQ 2025

medium

mathematics ID: cuet-ug-

$The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = . The marginal cost when 10 items are produced is:$

1

$537.1$

2

$441.15$

3

$1575$

4

$875.25$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: Marginal cost is the rate of change of the total cost with respect to the number of units produced. In calculus, this is represented by the first derivative of the total cost function, C(x). The marginal cost at a specific production level (x=10) is found by evaluating the derivative at that point. Step 2: Key Formula or Approach: Marginal Cost (MC) = . Step 3: Detailed Explanation: The given total cost function is: First, we find the derivative of C(x) with respect to x to get the marginal cost function, MC(x). Using the power rule for differentiation (): Now, we need to find the marginal cost when 10 items are produced, so we substitute x = 10 into the MC(x) function. Step 4: Final Answer: The marginal cost when 10 items are produced is Rs. 537.1.$

19

PYQ 2025

medium

mathematics ID: cuet-ug-

$The interval, on which the function f(x) = x2e-x is increasing, is equal to$

1

$(-,)$

2

$(-, 2) (2,)$

3

$(-2, 0)$

4

$(0, 2)$

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: A function is increasing on an interval where its first derivative is positive, i.e., f'(x)>0. Step 2: Key Formula or Approach: We need to find the first derivative of f(x) using the product rule: . Then, we find the intervals where f'(x)>0. Step 3: Detailed Explanation: The given function is . Differentiating f(x) with respect to x using the product rule, where and : For the function to be increasing, we must have f'(x)>0. Since is always positive for all real x, the sign of f'(x) depends on the term . So we need to solve the inequality: The critical points are x = 0 and x = 2. We can analyze the sign in the intervals determined by these points: For x < 0: is negative, is positive. Product is negative. For 0 < x < 2: is positive, is positive. Product is positive. For x > 2: is positive, is negative. Product is negative. The derivative f'(x) is positive when 0 < x < 2. Step 4: Final Answer: The function f(x) is increasing on the interval (0, 2) .$

20

PYQ 2025

medium

mathematics ID: cuet-ug-

$The slope of the normal to the curve y = at x = 1 is:$

1

$4$

2

$-4$

3

$1/2$

4

$-1/4$

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: The slope of the tangent to a curve at a point is given by the value of its derivative () at that point. The normal line is perpendicular to the tangent line at that same point. The slopes of two perpendicular lines are negative reciprocals of each other. Step 2: Key Formula or Approach: 1. Find the derivative of the curve's equation: . 2. Evaluate the derivative at the given x-value to find the slope of the tangent at that point. 3. The slope of the normal is the negative reciprocal of the tangent's slope: . Step 3: Detailed Explanation: The equation of the curve is given by: First, we find the derivative of y with respect to x: This expression,, gives the slope of the tangent at any point x on the curve. We need to find the slope at . So, we substitute into the derivative: The slope of the tangent line at is 4. Now, we find the slope of the normal line, which is the negative reciprocal of the tangent's slope. Step 4: Final Answer: The slope of the normal to the curve at x = 1 is -1/4.$

21

PYQ 2025

medium

mathematics ID: cuet-ug-

$The rate of change of area of a circle with respect to its circumference when radius is 4cm, is$

1

2

3

4

Official Solution

Correct Option: (2)

$Step 1: Understanding the Concept: We are asked to find the derivative of the area with respect to the circumference, which is represented as . This is a related rates problem where both and are functions of the radius . Step 2: Key Formula or Approach: Let be the area and be the circumference of a circle with radius . We can use the chain rule to find : Step 3: Detailed Explanation: First, we find the derivatives of and with respect to : Now, we can find using the chain rule: The problem asks for this rate of change when the radius is 4 cm. Substituting cm: The units of this rate are area units divided by length units, which is cm /cm. So the result should just be cm. However, the options provided all have units cm /cm, which is a bit redundant but guides us to the numerical answer. The final numerical value is 4. Step 4: Final Answer: The rate of change of the area with respect to its circumference is 4 cm 2 /cm .$

	LIST I		LIST II
A.	The minimum value of $f(x)%3D8x%C2%B2-4x%2B7$ is	I.	48
B.	The maximum value of $f(x)%20%3D%20x%2B%5Cfrac%7B1%7D%7Bx%7D%2C%20x%20%3C%200$ is	II.	13
C.	The maximum slope of the cure $y%20%3D%20-2x%5E3%2B6x%5E2%2B7x%2B26$ is	III.	-2
D.	The minimum value of $f(x)%20%3D%20x%C2%B2%20%2B%5Cfrac%7B128%7D%7Bx%7D$ is	IV.	$%5Cfrac%7B13%7D%7B2%7D$

High-Yield Trend

Chapter Questions
21 MCQs

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Application Of Derivatives

High-Yield Trend

Chapter Questions 21 MCQs

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Chapter Questions
21 MCQs