Continuity and Differentiability

Here is the complete ML Aggarwal Class 12 Solutions of Chapter Test for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. Is the function defined by \( f(x)=x^{2}-\sin x+5 \) continuous at \( x=\pi \) ? Solution: We observe that the function \( f(x)=x^{2}-\sin x+5 […]

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.16 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. Examine for continuity the function \[ f(x)= \begin{cases} |x|, & \text{if } x \leq 0,\\[6mm] x, & \text{if } 0 < x <

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.15 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. (i) Verify Lagrange’s mean value theorem for the function \( f(x)=x^{2} \) in the interval \([2,4]\). (NCERT) Also, find the value of \(

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.14 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question (1)(i): Verify Rolle’s theorem for the function \[ f(x) = x^{2} – 5x + 6 \quad \text{in } [1,4] \] and find the value

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.13 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. (i) Find the second order derivative of \( x^{20} \). Solution: We begin by differentiating the function: \[ \frac{d}{dx} \left( x^{20} \right) =

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.12 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. (i) Find \(\frac{dy}{dx}\) when \(x = a t^2\) and \(y = 2a t\). Solution: We differentiate both \(x\) and \(y\) with respect to

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.11 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. (i) Differentiate \( (x+3)^{2}(x+4)^{3}(x+5)^{4} \) with respect to \( x \). Solution: Let \[ y = (x+3)^{2}(x+4)^{3}(x+5)^{4}. \] Taking the natural logarithm of

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.10 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. (i) Differentiate \( e^{x} + 3 \sin x \) with respect to \( x \). Solution: We differentiate each term individually: \[ \frac{d}{dx}\left(e^{x}\right)

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.9 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. (i) Evaluate \(\lim_{x \to \infty} \frac{1}{(x-3)^2}\). Solution: We note that as \( x \to \infty \), the term \(\,(x-3)^2\) grows without bound. Hence,

Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.8 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding. Question: 1. (i) Find \(\frac{dy}{dx}\) for \(x – y = \pi\). Solution: Differentiating both sides of the equation \[ x – y = \pi \]