Here is the complete ML Aggarwal Class 12 Solutions of Exercise – 5.3 for Chapter 5 – Continuity and Differentiability. Each question is solved step by step for better understanding.
Question:
1. Is \( \tan x \) a continuous function?
Solution:
\( \tan x \) is continuous on its domain
(where \( \cos x \neq 0 \)).
Final Answer:
\( \tan x \) is continuous on its domain.
Question:
2. State which of the following functions are continuous:
(i) \( \sin x + \cos x \)
(ii) \( \sin x – \cos x \)
(iii) \( \sin x \cdot \cos x \)
Solution:
Since \( \sin x \) and \( \cos x \) are continuous for all \( x \),
their sum, difference, and product are also continuous for all \( x \).
Final Answer:
All three functions are continuous for all \( x \).
Question:
3. Is the function \( f(x)=2-3x+|x| \) a continuous function?
Solution:
The function \( f(x)=2-3x+|x| \) can be split into two parts:
\[
g(x)=2-3x \quad \text{and} \quad h(x)=|x|.
\]
The function \( g(x)=2-3x \) is a polynomial, and since all polynomials are continuous for all \( x \), \( g(x) \) is continuous everywhere.
The function \( h(x)=|x| \) is defined as:
\[
|x|=\begin{cases}
x, & \text{if } x \geq 0, \\
-x, & \text{if } x < 0,
\end{cases}
\]
and it is continuous for all \( x \) because both branches are continuous and they match at \( x=0 \) (since \(\lim_{x\to 0^-}|x| = 0 = \lim_{x\to 0^+}|x|\)).
Since both \( g(x) \) and \( h(x) \) are continuous, their sum, \( f(x)=g(x)+h(x) \), is also continuous for all \( x \).
Final Answer:
\( f(x)=2-3x+|x| \) is continuous for all \( x \).
Question:
4. Is the function \( f(x)=|\sin x| \) a continuous function?
Solution:
The function \( \sin x \) is continuous for all \( x \).
The absolute value function \( |x| \) is continuous for all \( x \).
Since \( |\sin x| \) is the composition of two continuous functions,
it is continuous for all \( x \).
Final Answer:
\( f(x)=|\sin x| \) is continuous for all \( x \).
Question:
5. Is the function \( f(x)=\cos|x| \) a continuous function?
Solution:
The function \( |x| \) is continuous for all \( x \).
The cosine function \( \cos x \) is continuous for all \( x \).
Since \( f(x)=\cos|x| \) is the composition of \( \cos x \) with \( |x| \),
and the composition of continuous functions is continuous,
\( f(x)=\cos|x| \) is continuous for all \( x \).
Final Answer:
\( f(x)=\cos|x| \) is continuous for all \( x \).
Question:
6. (i) Prove that \( \frac{2x^3 – 7x^2 + 3}{(x-1)(x+3)} \) is continuous.
Solution:
The numerator \(2x^3 – 7x^2 + 3\) and the denominator \((x-1)(x+3)\) are polynomials, which are continuous everywhere.
Thus, the function is continuous for all \( x \) except where the denominator equals zero, i.e., at \( x = 1 \) and \( x = -3 \).
Final Answer:
The function is continuous for \( x \neq 1 \) and \( x \neq -3 \).
Question:
6. (ii) Prove that \( |\sec x+\tan x| \) is continuous.
Solution:
The functions \(\sec x\) and \(\tan x\) are continuous on their common domain (where \(\cos x \neq 0\)).
Their sum, \(\sec x+\tan x\), is continuous on this domain, and taking the absolute value (a continuous operation) preserves continuity.
Final Answer:
\( |\sec x+\tan x| \) is continuous on its domain (where \(\cos x \neq 0\)).
Question:
7. (i) Examine the continuity of \( \cos(x^2) \).
Solution:
The function \( x^2 \) is a polynomial and hence continuous everywhere.
The cosine function \( \cos x \) is also continuous for all \( x \).
Since the composition of continuous functions is continuous,
\( \cos(x^2) \) is continuous for all \( x \).
Final Answer:
\( \cos(x^2) \) is continuous for all \( x \).
Question:
7. (ii) Examine the continuity of \( \sin(3x^2 – 5) \).
Solution:
The function \( 3x^2 – 5 \) is a polynomial and is continuous for all \( x \).
The sine function \( \sin x \) is continuous for all \( x \).
Since the composition of continuous functions is continuous,
\( \sin(3x^2 – 5) \) is continuous for all \( x \).
Final Answer:
\( \sin(3x^2 – 5) \) is continuous for all \( x \).
Question:
7. (iii) Examine the continuity of \( \tan^{-1}(2x^2 + 3) \).
Solution:
The function \( 2x^2 + 3 \) is a polynomial and is continuous for all \( x \).
The arctangent function \( \tan^{-1} x \) is continuous for all real numbers.
Since the composition of continuous functions is continuous,
\( \tan^{-1}(2x^2 + 3) \) is continuous for all \( x \).
Final Answer:
\( \tan^{-1}(2x^2 + 3) \) is continuous for all \( x \).