Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x^{2}-3 x-6}{x^{2}+2 x+4}$. Then which of the following statements is(are) TRUE?
- $f$ is decreasing in the interval $(-2,-1)$
- $f$ is increasing in the interval $(1,2)$
- $f$ is onto
- Range of $f$ is $\left[-\frac{3}{2}, 2\right]$
The Correct Option is A, B
Solution and Explanation
Step 1: Given Function
The given function is \( f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4} \), and we need to determine whether the function is increasing or decreasing in the specified intervals.
Step 2: First Derivative Calculation
To determine whether \( f(x) \) is increasing or decreasing in a given interval, we need to find the first derivative \( f'(x) \). Using the quotient rule for differentiation, we have:
\[
f'(x) = \frac{(x^2 + 2x + 4)(2x - 3) - (x^2 - 3x - 6)(2x + 2)}{(x^2 + 2x + 4)^2}
\]
Let's simplify the numerator:
\[
\text{Numerator} = (x^2 + 2x + 4)(2x - 3) - (x^2 - 3x - 6)(2x + 2)
\]
Expanding both terms:
\[
(x^2 + 2x + 4)(2x - 3) = 2x^3 - 3x^2 + 4x^2 - 6x + 8x - 12 = 2x^3 + x^2 + 2x - 12
\]
\[
(x^2 - 3x - 6)(2x + 2) = 2x^3 + 2x^2 - 6x^2 - 6x - 12x - 12 = 2x^3 - 4x^2 - 18x - 12
\]
Subtracting the two expressions:
\[
\text{Numerator} = (2x^3 + x^2 + 2x - 12) - (2x^3 - 4x^2 - 18x - 12) = 5x^2 + 20x
\]
Thus, the derivative becomes:
\[
f'(x) = \frac{5x^2 + 20x}{(x^2 + 2x + 4)^2}
\]
Step 3: Analyze the Sign of the Derivative
To determine whether the function is increasing or decreasing in an interval, we need to analyze the sign of \( f'(x) \). The denominator \( (x^2 + 2x + 4)^2 \) is always positive since the quadratic \( x^2 + 2x + 4 \) has no real roots and is always positive for all \( x \). Therefore, the sign of \( f'(x) \) depends on the numerator \( 5x^2 + 20x \), which can be factored as:
\[
f'(x) = \frac{5x(x + 4)}{(x^2 + 2x + 4)^2}
\]
The critical points are the values of \( x \) where \( f'(x) = 0 \), which occur when \( x = 0 \) or \( x = -4 \).
Step 4: Interval Analysis for Statement A
We are asked to check if \( f(x) \) is decreasing in the interval \( (-2, -1) \). To do this, we need to check the sign of \( f'(x) \) in this interval. Since the critical points are \( x = -4 \) and \( x = 0 \), we evaluate the sign of \( f'(x) \) in the interval \( (-2, -1) \). Choosing a test point, say \( x = -1.5 \), we find:
\[
f'(-1.5) = \frac{5(-1.5)(-1.5 + 4)}{( (-1.5)^2 + 2(-1.5) + 4 )^2}
\]
Simplifying:
\[
f'(-1.5) = \frac{5(-1.5)(2.5)}{(2.25 - 3 + 4)^2} = \frac{-18.75}{(3.25)^2} < 0
\]
Since \( f'(-1.5) < 0 \), the function is decreasing in the interval \( (-2, -1) \). Hence, statement (A) is true.
Step 5: Interval Analysis for Statement B
We are asked to check if \( f(x) \) is increasing in the interval \( (1, 2) \). Again, we check the sign of \( f'(x) \) in this interval. The critical points \( x = -4 \) and \( x = 0 \) are not in this interval, so we check the sign of \( f'(x) \) for \( x = 1.5 \):
\[
f'(1.5) = \frac{5(1.5)(1.5 + 4)}{( (1.5)^2 + 2(1.5) + 4 )^2}
\]
Simplifying:
\[
f'(1.5) = \frac{5(1.5)(5.5)}{(2.25 + 3 + 4)^2} = \frac{41.25}{9.25^2} > 0
\]
Since \( f'(1.5) > 0 \), the function is increasing in the interval \( (1, 2) \). Hence, statement (B) is true.
Final Answer:
Both statements (A) and (B) are correct:
- (A) \( f(x) \) is decreasing in the interval \( (-2, -1) \).
- (B) \( f(x) \) is increasing in the interval \( (1, 2) \).
Top Questions on Relations
- Let the set of all relations $ R $ on the set $ \{a, b, c, d, e, f\} $, such that $ R $ is reflexive and symmetric, and $ R $ contains exactly 10 elements, be denoted by $ S $. Then the number of elements in $ S $ is ___.
- Define a relation \( R \) on the interval \( [0, \frac{\pi}{2}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1 \). Then \( R \) is:
- The number of relations on the set $ A = \{1, 2, 3\} $ containing at most 6 elements including $ (1, 2) $, which are reflexive and transitive but not symmetric, is:
- Let $ A = \{-3, -2, -1, 0, 1, 2, 3\} $ and $ R $ be a relation on $ A $ defined by $ xRy $ if and only if $ 2x - y \in \{0, 1\} $. Let $ l $ be the number of elements in $ R $. Let $ m $ and $ n $ be the minimum number of elements required to be added in $ R $ to make it reflexive and symmetric relations, respectively. Then $ l + m + n $ is equal to:
- Let \( R \) be a relation on \( \mathbb{Z} \times \mathbb{Z} \) defined by \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5.
Then \( R \) is:
Questions Asked in JEE Advanced exam
- One of the products formed from the reaction of permanganate ion with iodide ion in neutral aqueous medium is:
- JEE Advanced - 2025
- Redox reactions
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Inverse Trigonometric Functions
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is- JEE Advanced - 2025
- Coordinate Geometry
- Let $ \mathbb{R} $ denote the set of all real numbers. Let $ a_i, b_i \in \mathbb{R} $ for $ i \in \{1, 2, 3\} $.
Define the functions $ f: \mathbb{R} \to \mathbb{R},\ g: \mathbb{R} \to \mathbb{R},\ h: \mathbb{R} \to \mathbb{R} $ by:
$$ f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\quad g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4, $$ $$ h(x) = f(x+1) - g(x+2) $$ If $ f(x) \ne g(x) $ for every $ x \in \mathbb{R} $, then the coefficient of $ x^3 $ in $ h(x) $ is:- JEE Advanced - 2025
- Polynomials
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
- JEE Advanced - 2025
- Differential equations
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
