If the function $f(x)= \begin{cases} \frac{\log _e\left(1-x+x^2\right)+\log_e\left(1+x+x^2\right)}{\sec x-\cos x}, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\} \\k, \,\,\,\,\, x=0\end{cases}$ is continuous at $x=0$, then $k$ is equal to :
- 1
- -1
- e
- 0
The Correct Option is A
Solution and Explanation
The correct option is (A): 1
Top Questions on Relations and functions
- Let \[ L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{-1} \quad {and} \quad L_2 : \frac{x+1}{1} = \frac{y-2}{2} = \frac{z-2}{2} \] be two lines. Let \( L_3 \) be a line passing through the point \( (\alpha, \beta, \gamma) \) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \) where \( 5x - 11y - 8z = 1 \), then \( 5x - 11y - 8z \) equals:
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- Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos\left(\frac{\pi}{3} - x\right) \cdot \cos\left(\frac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x, \quad x \in R \text{ be } [\alpha, \beta]. \] Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is:
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- The relation \( R = \{(x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is:
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- Let the position vectors of three vertices of a triangle be \( \overrightarrow{p} = 4\hat{i} + \hat{j} - 3\hat{k} \), \( \overrightarrow{q} = -5\hat{i} + 2\hat{j} + 3\hat{k} \), and \( \overrightarrow{r} = -5\hat{i} + 3\hat{j} + 2\hat{k} \). Then \( \alpha + 2\beta + 5\gamma \) is equal to:
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Questions Asked in JEE Main exam
If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
