Class 12 RD Sharma Solutions - Chapter 17 Increasing and Decreasing Functions - Exercise 17.1 Last Updated : 03 Jan, 2021 Comments Improve Suggest changes Like Article Like Report Question 1: Prove that the function f(x) = loge x is increasing on (0,∞). Solution: Let x1, x2 ∈ (0, ∞) We have, x1<x2 ⇒ loge x1 < loge x2 ⇒ f(x1) < f(x2) Therefore, f(x) is increasing in (0, ∞). Question 2: Prove that the function f(x) = loga (x) is increasing on (0,∞) if a>1 and decreasing on (0,∞) if 0<a<1. Solution: Case 1: When a>1 Let x1, x2 ∈ (0, ∞) We have, x1<x2 ⇒ loge x1 < loge x2 ⇒ f(x1) < f(x2) Therefore, f(x) is increasing in (0, ∞). Case 2: When 0<a<1 f(x) = loga x = logx/loga When a<1 ⇒ log a< 0 let x1<x2 ⇒ log x1<log x2 ⇒ ( log x1/log a) > (log x2/log a) [log a<0] ⇒ f(x1) > f(x2) Therefore, f(x) is decreasing in (0, ∞). Question 3: Prove that f(x) = ax + b, where a, b are constants and a>0 is an increasing function on R. Solution: We have, f(x) = ax + b, a > 0 Let x1, x2 ∈ R and x1 >x2 ⇒ ax1 > ax2 for some a>0 ⇒ ax1 + b > ax2 + b for some b ⇒ f(x1) > f(x2) Hence, x1 > x2 ⇒ f(x1) > f(x2) Therefore, f(x) is increasing function of R. Question 4: Prove that f(x) = ax + b, where a, b are constants and a<0 is a decreasing function on R. Solution: We have, f(x) = ax + b, a < 0 Let x1, x2 ∈ R and x1 >x2 ⇒ ax1 < ax2 for some a>0 ⇒ ax1 + b <ax2 + b for some b ⇒ f(x1) <f(x2) Hence, x1 > x2 ⇒ f(x1) <f(x2) Therefore, f(x) is decreasing function of R. Question 5: Show that f(x) = 1/x is a decreasing function on (0,∞). We have, f(x) = 1/x Let x1, x2 ∈ (0,∞) and x1 > x2 ⇒ 1/x1 < 1/x2 ⇒ f(x1) < f(x2) Thus, x1 > x2 ⇒ f(x1) < f(x2) Therefore, f(x) is decreasing function. Question 6: Show that f(x) = 1/(1+x2) decreases in the interval [0, ∞] and increases in the interval [-∞,0]. Solution: We have, f(x) = 1/1+ x2 Case 1: when x ∈ [0, ∞] Let x1, x2 ∈ [0,∞] and x1 > x2 ⇒ x12 > x22 ⇒ 1+x12 < 1+x22 ⇒ 1/(1+ x12 )> 1/(1+ x22 ) ⇒ f(x1) < f(x2) Therefore, f(x) is decreasing in [0, ∞]. Case 2: when x ∈ [-∞, 0] Let x1 > x2 ⇒ x12 < x22 [-2>-3 ⇒ 4<9] ⇒ 1+x12 < 1+x22 ⇒ 1/(1+ x12)> 1/(1+ x22 ) ⇒ f(x1) > f(x2) Therefore, f(x) is increasing in [-∞,0]. Question 7: Show that f(x) = 1/(1+x2) is neither increasing nor decreasing on R. Solution: We have, (x) = 1/1+ x2 R can be divided into two intervals [0, ∞] and [-∞,0] Case 1: when x ∈ [0, ∞] Let x1 > x2 ⇒ x12 > x22 ⇒ 1+x12 < 1+x22 ⇒ 1/(1+ x12 )> 1/(1+ x22 ) ⇒ f(x1) < f(x2) Therefore, f(x) is decreasing in [0, ∞]. Case 2: when x ∈ [-∞, 0] Let x1 > x2 ⇒ x12 < x22 [-2>-3 ⇒ 4<9] ⇒ 1+x12 < 1+x22 ⇒ 1/(1+ x12)> 1/(1+ x22 ) ⇒ f(x1) > f(x2) Therefore, f(x) is increasing in [-∞,0]. Here, f(x) is decreasing in [0, ∞] and f(x) is increasing in [-∞,0]. Thus, f(x) neither increases nor decreases on R. Question 8: Without using the derivative, show that the function f(x) = |x| is,(i) strictly increasing in (0,∞) (ii) strictly decreasing in (-∞,0) Solution: (i). Let x1, x2 ∈ [0,∞] and x1 > x2 ⇒ f(x1) > f(x2) Thus, f(x) is strictly increasing in [0,∞]. (ii). Let x1, x2 ∈ [-∞, 0] and x1 > x2 ⇒ -x1<-x2 ⇒ f(x1) < f(x2) Thus, f(x) is strictly decreasing in [-∞,0]. Question 9: Without using the derivative show that the function f(x) = 7x - 3 is strictly increasing function on R. Solution: f(x) = 7x-3 Let x1, x2 ∈ R and x1 >x2 ⇒ 7x1 > 7x2 ⇒ 7x1 - 3 > 7x2 - 3 ⇒ f(x1) > f(x2) Thus, f(x) is strictly increasing on R Comment More infoAdvertise with us Next Article Class 12 RD Sharma Solutions - Chapter 17 Increasing and Decreasing Functions - Exercise 17.1 V vermaman947 Follow Improve Article Tags : Technical Scripter Mathematics School Learning Class 12 RD Sharma Solutions Technical Scripter 2020 RD Sharma Class-12 Maths-Class-12 +4 More Similar Reads RD Sharma Class 12 Solutions for Maths RD Sharma Solutions for class 12 provide solutions to a wide range of questions with a varying difficulty level. With the help of numerous sums and examples, it helps the student to understand and clear the chapter thoroughly. Solving the given questions inside each chapter of RD Sharma will allow t 13 min read Chapter 1: RelationsClass 12 RD Sharma Solutions - Chapter 1 Relations - Exercise 1.1 | Set 1In mathematics, understanding the properties of relations is very fundamental to forming a strong basic. Reflexive, symmetric, and transitive relations form the cornerstone of relational algebra, which finds a huge application in various fields such as computer science, logic finding, and data scien 15+ min read Class 12 RD Sharma Solutions - Chapter 1 Relations - Exercise 1.1 | Set 2Question 11. Is it true every relation which is symmetric and transitive is also reflexive? Give reasons.Solution:We will verify this by taking an example.Consider a set A = {1, 2, 3} and a relation R on A such that R = { (1, 2), (2,1), (2,3), (1,3) }The relation R over the set A is symmetric and tr 13 min read Class 12 RD Sharma Solutions - Chapter 1 Relations - Exercise 1.2 | Set 1Question 1. Show that the relation R = {(a,b): a-b is divisible by 3;, a, b ∈ Z} is an equivalence relation. Solution: According to question, relation R = {(a,b): a-b is divisible by 3;, a, b ∈ Z} We have to show that R is an equivalence relation. (i) reflexibity: let a = z => a - a = 0 => 0 i 15 min read Class 12 RD Sharma Solutions - Chapter 1 Relations - Exercise 1.2 | Set 2Question 11. Let O be the origin . We define a relation between two points P and Q in a plane if OP = OQ . Show that the relation, so defined is an equivalence relation.Solution:Let A be the set of points on planeand let R = {(P, Q): OP = OQ} be a relation on A where O is the origin.now (i) reflexib 10 min read Chapter 2: FunctionsClass 12 RD Sharma Mathematics Solutions - Chapter 2 Functions - Exercise 2.1 | Set 1Chapter 2 of RD Sharma's Class 12 Mathematics textbook delves into the concept of functions which is foundational in the higher mathematics. Exercise 2.1 | Set 1 introduces students to the basic concepts and types of functions laying the groundwork for the more advanced topics. Understanding functio 15+ min read Class 12 RD Sharma Solutions - Chapter 2 Functions - Exercise 2.1 | Set 2Question 12: Show that the exponential function f: R → R, given by f(x) = ex is one one but not onto. What happens if the codomain is replaced by Ro+.Solution:We have f: R → R, given by f(x) = ex.Let x,y ϵ R, such that=> f(x) = f(y)=> ex = ey=> e(x-y) = 1 = e0=> x - y = 0=> x = yHence 8 min read Class 12 RD Sharma Solutions - Chapter 2 Functions - Exercise 2.2Question 1(i). Find g o f and f o g when f: R -> R and g: R -> R is defined by f(x) = 2x + 3 and g(x) = x2 + 5Solution:f: R -> R and g: R -> RTherefore, f o g: R -> R and g o f: R -> RNow, f(x) = 2x + 3 and g(x) = x2 + 5g o f(x) = g(2x + 3) =(2x + 3)2 + 5=> g o f(x) = 4x2 + 12x 13 min read Class 12 RD Sharma Solutions - Chapter 2 Functions - Exercise 2.3Question 1. Find fog and gof, if(i) f (x) = ex,g (x) = \log_exSolution:Let f: R → (0, ∞); and g: (0, ∞) → RClearly, the range of g is a subset of the domain of f.So, fog: (0, ∞) → R and we know, (fog)(x) = f(g(x))= f(\log_ex)= e^{log_ex}(fog)(x) = xClearly, the range of f is a subset of the domain o 10 min read Chapter 3: Binary OperationsClass 12 RD Sharma Solutions - Chapter 3 Binary Operations - Exercise 3.1Question 1. Determine whether the following operation define a binary operation on the given set or not:(i) ‘*’ on N defined by a * b = ab for all a, b ∈ N.(ii) ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.(iii) ‘*’ on N defined by a * b = a + b – 2 for all a, b ∈ N(iv) ‘×6‘ on S = {1, 2, 3, 4, 5 10 min read Class 12 RD Sharma Solutions - Chapter 3 Binary Operations - Exercise 3.2Question 1. Let ‘*’ be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N(i) Find 2 * 4, 3 * 5, 1 * 6Solution:We are given that a * b = L.C.M. (a, b) ⇒ 2 * 4 = L.C.M. (2, 4) = 4and, 3 * 5 = L.C.M. (3, 5) = 15now, 1 * 6 = L.C.M. (1, 6) = 6Hence, 2 * 4 = 4, 3 * 5 = 15 and 1 * 6 15+ min read Class 12 RD Sharma Solutions- Chapter 3 Binary Operations - Exercise 3.3Binary operations are fundamental concepts in mathematics and computer science that involve operations on pairs of elements. In Chapter 3 of RD Sharma’s Class 12 textbook, we explore the binary operations in detail focusing on their properties and applications. Exercise 3.3 covers practical problems 5 min read Class 12 RD Sharma Solutions - Chapter 3 Binary Operations - Exercise 3.4In this article, we will explore the solutions to Exercise 3.4 from Chapter 3 of RD Sharma's Class 12 Mathematics textbook which focuses on "Binary Operations". This chapter is essential for understanding how operations can be performed on pairs of elements within a set a fundamental concept in alge 15+ min read Class 12 RD Sharma Solutions - Chapter 3 Binary Operations - Exercise 3.5Exercise 3.5 in Chapter 3 of RD Sharma's Class 12 Mathematics textbook focuses on binary operations. This exercise likely covers advanced concepts related to binary operations, including their properties, applications, and problem-solving techniques. Students will be challenged to apply their unders 7 min read Chapter 4: Inverse Trigonometric FunctionsClass 12 RD Sharma Solutions- Chapter 4 Inverse Trigonometric Functions - Exercise 4.1The Inverse trigonometric functions are critical for the solving equations where the angle is unknown and they help in the understanding of the various trigonometric properties. These functions, denoted as sin−1(x), cos−1(x) and tan−1(x) allow us to find the angle whose trigonometric function gives 6 min read Chapter 5: Algebra of MatricesClass 12 RD Sharma Solutions - Chapter 5 Algebra of Matrices - Exercise 5.1 | Set 1Chapter 5 of RD Sharma's Class 12 Mathematics textbook focuses on the Algebra of Matrices, with Exercise 5.1 Set 1 introducing fundamental concepts and operations of matrices. This exercise covers matrix notation, types of matrices, and basic matrix operations. Students will learn to represent data 15+ min read Class 12 RD Sharma Solutions- Chapter 5 Algebra of Matrices - Exercise 5.1 | Set 2Chapter 5 of RD Sharma's Class 12 Mathematics textbook continues exploring the Algebra of Matrices, with Exercise 5.1 Set 2 building upon the foundational concepts introduced in Set 1. This set likely delves deeper into matrix operations, properties, and applications. Students will encounter more co 11 min read Class 12 RD Sharma Solutions - Chapter 5 Algebra of Matrices - Exercise 5.2 | Set 1Chapter 5 of RD Sharma's Class 12 Mathematics textbook focuses on the Algebra of Matrices. Exercise 5.2 specifically deals with operations on matrices, including addition, subtraction, and multiplication. This exercise helps students understand how to perform these operations and apply them to solve 14 min read Class 12 RD Sharma Solutions - Chapter 5 Algebra of Matrices - Exercise 5.2 | Set 2Chapter 5 of RD Sharma's Class 12 Mathematics textbook continues to explore the Algebra of Matrices. Exercise 5.2 Set 2 typically delves deeper into matrix operations, focusing on more complex problems involving matrix addition, subtraction, multiplication, and related concepts. This set often inclu 13 min read Class 12 RD Sharma Solutions - Chapter 5 Algebra of Matrices - Exercise 5.3 | Set 1Chapter 5 of RD Sharma's Class 12 Mathematics textbook on the Algebra of Matrices continues with Exercise 5.3. This exercise typically focuses on more advanced matrix operations and properties, including determinants, adjoint matrices, and inverse matrices. Set 1 of this exercise often introduces th 15+ min read Class 12 RD Sharma Solutions - Chapter 5 Algebra of Matrices - Exercise 5.3 | Set 2Exercise 5.3 Set 2 in Chapter 5 of RD Sharma's Class 12 Mathematics textbook continues the exploration of advanced matrix concepts. This set typically delves deeper into the properties of determinants, adjoint matrices, and inverse matrices. It often includes more complex problems that require stude 15+ min read Class 12 RD Sharma Solutions - Chapter 5 Algebra of Matrices - Exercise 5.3 | Set 3Exercise 5.3 Set 3 in Chapter 5 of RD Sharma's Class 12 Mathematics textbook further expands on the concepts of determinants, adjoint matrices, and inverse matrices. This set typically includes more challenging problems that require a deeper understanding of matrix properties and their interrelation 15+ min read Class 12 RD Sharma Solutions- Chapter 5 Algebra of Matrices - Exercise 5.4Chapter 5 of RD Sharma's Class 12 Mathematics textbook focuses on the Algebra of Matrices. Exercise 5.4 specifically deals with elementary operations on matrices and their properties. This exercise is crucial for understanding how matrices can be manipulated and transformed, which is fundamental in 6 min read Class 12 RD Sharma Solutions- Chapter 5 Algebra of Matrices - Exercise 5.5Exercise 5.5 in Chapter 5 of RD Sharma's Class 12 Mathematics focuses on the concept of rank of a matrix. This is a fundamental concept in linear algebra that helps in understanding the structure and properties of matrices, as well as their applications in solving systems of linear equations.Questio 5 min read Chapter 6: DeterminantsClass 12 RD Sharma Solutions - Chapter 6 - Exercise 6.1In this article, we will be going to solve the entire exercise 6.1 of our RD Sharma textbook. In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix and the linear transformations it represent 15 min read Class 12 RD Sharma Solutions - Chapter 6 Determinants - Exercise 6.2 | Set 1In Chapter 6 of RD Sharma's Class 12 Mathematics textbook determinants play a crucial role in the solving systems of linear equations and various matrix-related problems. Exercise 6.2 | Set 1 provides the practice problems designed to deepen students' understanding of the determinants and their prop 15+ min read Class 12 RD Sharma Solutions - Chapter 6 Determinants - Exercise 6.2 | Set 2Prove the following identities:Question 18. \begin{vmatrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1 \end{vmatrix} = -2Solution:Considering the determinant, we have\triangle = \begin{vmatrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 15+ min read Class 12 RD Sharma Solutions - Chapter 6 Determinants - Exercise 6.2 | Set 3Prove the following identities:Question 35. \begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{vmatrix} = 4xyzSolution:Considering the determinant, we have\triangle = \begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{vmatrix}R1 15+ min read Class 12 RD Sharma Solutions - Chapter 6 Determinants - Exercise 6.3Question 1. Find the area of the triangle with vertices at the points:(i) (3, 8), (−4, 2) and (5, −1)Solution:Given (3, 8), (−4, 2) and (5, −1) are the vertices of the triangle. We know that, if vertices of a triangle are (x1, y1), (x2, y2) and (x3, y3), then the area of the triangle is given by,A = 11 min read Class 12 RD Sharma Solutions - Chapter 6 Determinants - Exercise 6.4 | Set 1Question 1. Solve the following system of linear equations by Cramer’s rule.x – 2y = 4−3x + 5y = −7Solution:Using Cramer's Rule, we get,D=\left|\begin{array}{cc} 1 & -2 \\ -3 & 5 \end{array} \right|= 5 − 6 = −1Also, we get,D_1=\left|\begin{array}{cc} 4 & -2 \\ -7 & 5 \end{array} \rig 12 min read Class 12 RD Sharma Solutions - Chapter 6 Determinants - Exercise 6.4 | Set 2Question 17. Solve the following system of the linear equations by Cramer's rule.2x − 3y − 4z = 29−2x + 5y − z = −153x − y + 5z = −11Solution:Using Cramer's Rule, we get,D=\left|\begin{array}{cc} 2 & -3 & -4 \\ -2 & 5 & -1 \\ 3 & -1 & 5 \end{array} \right|Expanding along R1, 15+ min read Class 12 RD Sharma Solutions - Chapter 6 Determinants - Exercise 6.5Question 1. Solve each of the following system of homogeneous linear equations:x + y - 2z = 02x + y - 3z =05x + 4y - 9z = 0Solution:Given: x + y - 2z = 02x + y - 3z =05x + 4y - 9z = 0This system of equations can be expressed in the form of a matrix AX = BNow find the determinant,D=\begin{vmatrix}1 5 min read Chapter 7: Adjoint and Inverse of a MatrixClass 12 RD Sharma Solutions - Chapter 7 Adjoint and Inverse of a Matrix - Exercise 7.1 | Set 1In matrix algebra, the concepts of the adjoint and inverse of a matrix are crucial for solving systems of linear equations finding the determinants, and more. The adjoint of a matrix is a matrix obtained by transposing the cofactor matrix of the original matrix while the inverse of a matrix is a mat 12 min read Class 12 RD Sharma Solutions - Chapter 7 Adjoint and Inverse of a Matrix - Exercise 7.1 | Set 2Question 10. For the following parts of matrices verify that (AB)-1 = B-1A-1.(i) A = \begin{bmatrix}3&2\\7&5\end{bmatrix} and B = \begin{bmatrix}4&6\\3&2\end{bmatrix} Solution:To prove (AB)-1= B-1A-1We take LHSAB = \begin{bmatrix}3&2\\7&5\end{bmatrix}\begin{bmatrix}4&6\\3 9 min read Class 12 RD Sharma Solutions - Chapter 7 Adjoint and Inverse of a Matrix - Exercise 7.1 | Set 3Question 25. Show that the matrix A = \begin{bmatrix}1&0&-2\\-2&-1&2\\3&4&1\end{bmatrix} satisfies the equation A3 - A2 - 3A - I3 = 0. Hence, find A-1.Solution:Here, A = \begin{bmatrix}1&0&-2\\-2&-1&2\\3&4&1\end{bmatrix}A2 = \begin{bmatrix}1&0& 7 min read Class 12 RD Sharma Solutions - Chapter 7 Adjoint and Inverse of a Matrix - Exercise 7.2Find the inverse of each of the following matrices by using elementary row transformation(Questions 1- 16):Question 1. \begin{bmatrix}7&1\\4&-3\end{bmatrix} Solution:Here, A = \begin{bmatrix}7&1\\4&-3\end{bmatrix} A = AI Using elementary row operation⇒\begin{bmatrix}7&1\\4&-3 4 min read Chapter 8: Solutions of Simultaneous Linear EquationsClass 12 RD Sharma Solutions - Chapter 8 Solution of Simultaneous Linear Equations - Exercise 8.1 | Set 1Chapter 8 of RD Sharma's Class 12 textbook focuses on solving the simultaneous linear equations a fundamental topic in algebra. This chapter provides the methods to find the values of variables that satisfy the multiple linear equations simultaneously. Mastery of these techniques is crucial for solv 15+ min read Class 12 RD Sharma Solutions - Chapter 8 Solution of Simultaneous Linear Equations - Exercise 8.1 | Set 2Question 8. (i) If A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix} , find A−1. Using A−1, solve the system of linear equations x − 2y = 10, 2x + y + 3z = 8, −2y + z = 7.Solution:Here, A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 15+ min read Class 12 RD Sharma Solutions - Chapter 8 Solution of Simultaneous Linear Equations - Exercise 8.2Solve the following systems of homogeneous linear equations by matrix method:Question 1. 2x – y + z = 03x + 2y – z = 0x + 4y + 3z = 0Solution:Given2x – y + z = 03x + 2y – z = 0X + 4y + 3z = 0The system can be written as\begin{bmatrix} 2 & -1 & 1\\ 3 & 2 & -1\\ 1 & 4 &3 \end{b 7 min read Chapter 9: ContinuityClass 12 RD Sharma Solutions - Chapter 9 Continuity - Exercise 9.1 | Set 1Question 1. Test the continuity of the following function at the origin:  f(x)= \begin{cases}\frac{x}{|x|},&  x \neq 0 \\1,& x=0\end{cases}Solution:Given thatf(x)= \begin{cases}\frac{x}{|x|},&  x\neq0 \\1,& x=0\end{cases}   Now, let us consider LHL at x = 0\lim_{x\to0^-}f(x) =\lim_{h 8 min read Class 12 RD Sharma Solutions - Chapter 9 Continuity - Exercise 9.1 | Set 2Question 16. Discuss the continuity of the function f(x)=\begin{cases}x,&0\leq x<(\frac{1}{2}) \\(\frac{1}{2}),&x=(\frac{1}{2}) \\1-x,&(\frac{1}{2})<x\leq1\end{cases} at the point x = 1/2.Solution:Given that, f(x)=\begin{cases}x,&0\leq x<(\frac{1}{2}) \\(\frac{1}{2}),&x= 6 min read Class 12 RD Sharma Solutions - Chapter 9 Continuity - Exercise 9.1 | Set 3Question 31. If f(x)=\begin{cases}\frac{2^{x+2}-16}{4^x-16},& \text{if }x\neq2 \\k,& \text{if }x=2\end{cases} is continuous at x = 2, find k.Solution: Given that,f(x)=\begin{cases}\frac{2^{x+2}-16}{4^x-16},& \text{if }x\neq2 \\k,& \text{if }x=2\end{cases} Also, f(x) is continuous at 11 min read Class 12 RD Sharma Solutions - Chapter 9 Continuity - Exercise 9.2 | Set 1Question 1. Prove that the function f(x)=\begin{cases}\frac{sinx}{x} \ \ \ \ ,x≤0\\x+1\ \ \ ,x≤0\end{cases} is continuous everywhere.Solution:We know sin x/ x is continuous everywhere since it is the composite function of the functions sin x and x which are continuous.When x > 0, we have f(x) = x 15+ min read Class 12 RD Sharma Solutions - Chapter 9 Continuity - Exercise 9.2 | Set 2Question 8. If f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x} for x ≠π/4, find the value which can be assigned to f(x) at x = Ï€/4 so that the function f(x) becomes continuous every where in [0, Ï€/2].Solution:If x ≠π/4, tan (Ï€/4 - x) and cot2x are continuous in [0, Ï€/2]. T 11 min read Like