January 19, 2024

Discover comprehensive NCERT solutions for Class 12 Science Mathematics Chapter 6: Application of Derivatives, featuring straightforward, step-by-step explanations. Highly sought after by Class 12 Science students, these solutions are a valuable resource for efficiently finishing homework assignments and preparing for exams. All questions and answers from Chapter 6 of the NCERT Mathematics textbook for Class 12 Science are available here at no cost, serving as a convenient aid for students. Page No 197: Question 1: Find the rate of change of the area of a circle with respect to its radius r when (a) r = 3 cm (b) r = 4 cm ANSWER: The area of a circle (A)with radius (r) is given by, Now, the rate of change of the area with respect to its radius is given by,  Hence, the area of the circle is changing at the rate of 6Ï€ cm when its radius is 3 cm. Hence, the area of the circle is changing at the rate of 8Ï€ cm when its radius is 4 cm. Page No 197: Question 2: The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm? ANSWER: Let x be the length of a side, V be the volume, and s be the surface area of the cube. Then, V = x3 and S = 6×2 where x is a function of time t. It is given that. Then, by using the chain rule, we have: ∴ ⇒ Thus, when x = 12 cm,  Hence, if the length of the edge of the cube is 12 cm, then the surface area is increasing at the rate of cm2/s. Page No 197: Question 3: The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm. ANSWER: The area of a circle (A) with radius (r) is given by, Now, the rate of change of area (A) with respect to time (t) is given by, It is given that, ∴ Thus, when r = 10 cm, Hence, the rate at which the area of the circle is increasing when the radius is 10 cm is 60π cm2/s. Page No 197: Question 4: An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long? ANSWER: Let x be the length of a side and V be the volume of the cube. Then, V = x3. ∴ (By chain rule) It is given that, ∴ Thus, when x = 10 cm, Hence, the volume of the cube is increasing at the rate of 900 cm3/s when the edge is 10 cm long. Page No 197: Question 5: A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing? ANSWER: The area of a circle (A) with radius (r) is given by. Therefore, the rate of change of area (A) with respect to time (t) is given by, [By chain rule] It is given that. Thus, when r = 8 cm, Hence, when the radius of the circular wave is 8 cm, the enclosed area is increasing at the rate of 80π cm2/s. Page No 198: Question 6: The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference? ANSWER: The circumference of a circle (C) with radius (r) is given by C = 2πr. Therefore, the rate of change of circumference (C) with respect to time (t) is given by, (By chain rule) It is given that. Hence, the rate of increase of the circumference Page No 198: Question 7: The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle. ANSWER: Since the length (x) is decreasing at the rate of 5 cm/minute and the width (y) is increasing at the rate of 4 cm/minute, we have: and  (a) The perimeter (P) of a rectangle is given by, P = 2(x + y) Hence, the perimeter is decreasing at the rate of 2 cm/min. (b) The area (A) of a rectangle is given by, A = x⋅ y ∴ When x = 8 cm and y = 6 cm,  Hence, the area of the rectangle is increasing at the rate of 2 cm2/min. Page No 198: Question 8: A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm. ANSWER: The volume of a sphere (V) with radius (r) is given by, ∴Rate of change of volume (V) with respect to time (t) is given by,  [By chain rule] It is given that. Therefore, when radius = 15 cm, Hence, the rate at which the radius of the balloon increases when the radius is 15 cm is Page No 198: Question 9: A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm. ANSWER: The volume of a sphere (V) with radius (r) is given by. Rate of change of volume (V) with respect to its radius (r) is given by, Therefore, when radius = 10 cm, Hence, the volume of the balloon is increasing at the rate of 400π cm2. Page No 198: Question 10: A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall? ANSWER: Let y m be the height of the wall at which the ladder touches. Also, let the foot of the ladder be x maway from the wall. Then, by Pythagoras theorem, we have: x2 + y2 = 25 [Length of the ladder = 5 m] Then, the rate …

NCERT Solutions for Class 12 Science Maths Chapter 6 – Application Of Derivatives Read More »

Explore comprehensive NCERT solutions for Class 12 Science Mathematics Chapter 5: Continuity and Differentiability, featuring clear step-by-step explanations. Widely favored by Class 12 Science students, these solutions are invaluable for efficiently completing homework assignments and exam preparation. All questions and answers from Chapter 5 of the NCERT Mathematics textbook for Class 12 Science are presented here at no cost, serving as a valuable resource for students. Page No 159: Question 1: Prove that the functionis continuous at ANSWER: Therefore, f is continuous at x = 0 Therefore, f is continuous at x = −3 Therefore, f is continuous at x = 5 Page No 159: Question 2: Examine the continuity of the function. ANSWER: Thus, f is continuous at x = 3 Page No 159: Question 3: Examine the following functions for continuity. ANSWER: It is evident that f is defined at every real number k and its value at k is k − 5. It is also observed that,  Hence, f is continuous at every real number and therefore, it is a continuous function. (b) The given function is For any real number k ≠ 5, we obtain Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function. (c) The given function is For any real number c ≠ −5, we obtain Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function. (d) The given function is  This function f is defined at all points of the real line. Let c be a point on a real line. Then, c < 5 or c = 5 or c > 5 Case I: c < 5 Then, f (c) = 5 − c Therefore, f is continuous at all real numbers less than 5. Case II : c = 5 Then,  Therefore, f is continuous at x = 5 Case III: c > 5 Therefore, f is continuous at all real numbers greater than 5. Hence, f is continuous at every real number and therefore, it is a continuous function. Page No 159: Question 4: Prove that the function is continuous at x = n, where n is a positive integer. ANSWER: The given function is f (x) = xn It is evident that f is defined at all positive integers, n, and its value at n is nn. Therefore, f is continuous at n, where n is a positive integer. Page No 159: Question 5: Is the function f defined by continuous at x = 0? At x = 1? At x = 2? ANSWER: The given function f is  At x = 0, It is evident that f is defined at 0 and its value at 0 is 0. Therefore, f is continuous at x = 0 At x = 1, f is defined at 1 and its value at 1 is 1. The left hand limit of f at x = 1 is, The right hand limit of f at x = 1 is, Therefore, f is not continuous at x = 1 At x = 2, f is defined at 2 and its value at 2 is 5. Therefore, f is continuous at x = 2 Page No 159: Question 6: Find all points of discontinuity of f, where f is defined by ANSWER: The given function f is It is evident that the given function f is defined at all the points of the real line. Let c be a point on the real line. Then, three cases arise. (i) c < 2 (ii) c > 2 (iii) c = 2 Case (i) c < 2 Therefore, f is continuous at all points x, such that x < 2 Case (ii) c > 2 Therefore, f is continuous at all points x, such that x > 2 Case (iii) c = 2 Then, the left hand limit of f at x = 2 is, The right hand limit of f at x = 2 is, It is observed that the left and right hand limit of f at x = 2 do not coincide. Therefore, f is not continuous at x = 2 Hence, x = 2 is the only point of discontinuity of f. Page No 159: Question 7: Find all points of discontinuity of f, where f is defined by ANSWER: The given function f is The given function f is defined at all the points of the real line. Let c be a point on the real line. Case I: Therefore, f is continuous at all points x, such that x < −3 Case II: Therefore, f is continuous at x = −3 Case III: Therefore, f is continuous in (−3, 3). Case IV: If c = 3, then the left hand limit of f at x = 3 is, The right hand limit of f at x = 3 is, It is observed that the left and right hand limit of f at x = 3 do not coincide. Therefore, f is not continuous at x = 3 Case V: Therefore, f is continuous at all points x, such that x > 3 Hence, x = 3 is the only point of discontinuity of f. Page No 159: Question 8: Find all points of discontinuity of f, where f is defined by ANSWER: The given function f is It is known that, Therefore, the given function can be rewritten as The given function f is defined at all the points of the real line. Let c be a point on the real line. Case I: Therefore, f is continuous at all points x < 0 Case II: If c = 0, then the left hand limit of f at x = 0 is, The right hand limit of f at x = 0 is, It is observed that the left and right hand limit of f at x = 0 do not coincide. Therefore, f is not continuous at x = 0 Case III: Therefore, f is continuous at all points x, such that x > 0 Hence, x = 0 is the only point of discontinuity of f. Page No 159: Question 9: Find all points of discontinuity of f, where f is defined by ANSWER: The given function f is It is known that, Therefore, the given function can be rewritten as Let c be any real number. Then,  Also, Therefore, the given function is a continuous function. Hence, the given function has no point of discontinuity. Page No 159: Question 10: Find all points of discontinuity of f, where f is defined by ANSWER: The given function f is The given function f is defined at all the points of the real line. Let c be a point on the real line. Case I: Therefore, f is continuous at all points x, such that x < 1 Case II: The left hand limit of f at x = 1 is, The right hand limit of f at x = 1 is, Therefore, f is continuous at x = 1 Case III: Therefore, f is continuous at all points x, such that x > 1 Hence, the given function f has no point of discontinuity. Page No 159: Question 11: Find all points of discontinuity of f, where f is defined by ANSWER: The given function f is The given function f is defined at all the points of the real line. Let c be a point on the real line. Case I: Therefore, f is continuous at all points x, such that x < 2 Case II: Therefore, f is continuous at x = 2 Case III: Therefore, f is continuous at all points x, such that x > 2 Thus, the given function f is continuous at every point on the real line. Hence, f has no point of discontinuity. Page No 159: Question 12: Find all points of discontinuity of f, where f is defined by ANSWER: The given function f is The given function f is defined at all the points of the real line. Let c be a point …

NCERT Solutions for Class 12 Science Maths Chapter 5 – Continuity And Differentiability Read More »