Aspire Faculty ID #18118 · Topic: JEE Main 2026 (28 January Evening Shift) · Just now
JEE Main 2026 (28 January Evening Shift)

Given below are two statements :

Statement I : The function $ f : \mathbb{R} \rightarrow \mathbb{R} $ defined by
$ f(x) = \frac{x}{1 + |x|} $ is one-one.

Statement II : The function $ f : \mathbb{R} \rightarrow \mathbb{R} $ defined by
$ f(x) = \frac{x^2 + 4x - 30}{x^2 - 8x + 18} $ is many-one.

In the light of the above statements, choose the correct answer from the options given below :

Solution

Statement I:

$ f(x) = \frac{x}{1 + |x|} $

$ f(x) = \begin{cases} \frac{x}{1 + x}, & x \ge 0 \ \frac{x}{1 - x}, & x < 0 \end{cases} $

[Image used in solution — graph]

$ f(x) $ is one-one

Statement II:

$ f(x) = \frac{x^2 + 4x - 30}{x^2 - 8x + 18} $

$ f(0) = -\frac{30}{18} = -\frac{5}{3} $

$ f(-1) = -\frac{5}{3} $

$ \Rightarrow f(0) = f(-1) $

$ \Rightarrow f(x) $ is many-one
← Previous Next →

Previous 10 Questions — JEE Main 2026 (28 January Evening Shift)

Nearest first
1
Let the arithmetic mean of $ \frac{1}{a} $ and $ \frac{1}{b} $ be $ \frac{5}{16},; a &gt; 2 $.If $ \alpha $ is such tha…
Topic: JEE Main 2026 (28 January Evening Shift)
 2
Considering the principal values of inverse trigonometric functions, the value of the expression $ \tan \left( 2\sin^{-…
Topic: JEE Main 2026 (28 January Evening Shift)
 3
Let $ f(x) = \lim_{\theta \to 0} \left( \frac{\cos \pi x - x^{(\theta)} \sin(x-1)}{1 + x^{(\theta)} (x - 1)} \right),; …
Topic: JEE Main 2026 (28 January Evening Shift)
 4
The probability distribution of a random variable $ X $ is given below :If $ E(X) = \frac{263}{15} $, then $ P(X &lt; 2…
Topic: JEE Main 2026 (28 January Evening Shift)
 5
Let $ A $ be the focus of the parabola $ y^2 = 8x $. Let the line $ y = mx + c $ intersect the parabola at two distinct…
Topic: JEE Main 2026 (28 January Evening Shift)
 6
The sum of coefficients of $ x^{499} $ and $ x^{500} $ in
Topic: JEE Main 2026 (28 January Evening Shift)
 7
Given below two statements :Statement I : $ 25^{13} + 20^{13} + 8^{13} + 3^{13} $ is divisible by $ 7 $.Statement II : …
Topic: JEE Main 2026 (28 January Evening Shift)

Next 10 Questions — JEE Main 2026 (28 January Evening Shift)

Ascending by ID
1
An ellipse has center at $ (1,-2) $, one focus at $ (3,-2) $ and one vertex at $ (5,-2) $. Then the length of its latus…
Topic: JEE Main 2026 (28 January Evening Shift)
 2
Let the ellipse $ E: \frac{x^2}{144} + \frac{y^2}{169} = 1 $ and the hyperbola$ H: \frac{x^2}{16} - \frac{y^2}{\lambda^…
Topic: JEE Main 2026 (28 January Evening Shift)
 3
Let $ P_1 : y = 4x^2 $ and $ P_2 : y = x^2 + 27 $ be two parabolas. If the area of the bounded region enclosed between …
Topic: JEE Main 2026 (28 January Evening Shift)
 4
Let the circle $ x^2 + y^2 = 4 $ intersect $ x $-axis at the points $ A(a, 0),; a &gt; 0 $ and $ B(b, 0) $. Let $ P(2\c…
Topic: JEE Main 2026 (28 January Evening Shift)
 5
Let $ [\cdot] $ denote the greatest integer function. Then$ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{12(3 + […
Topic: JEE Main 2026 (28 January Evening Shift)
 6
Let $ y = y(x) $ be the solution of the differential equation$ x \frac{dy}{dx} - y = x^2 \cot x,; x \in (0,\pi) $.If $ …
Topic: JEE Main 2026 (28 January Evening Shift)
 7
The sum of all the elements in the range of$ f(x) = \text{Sgn}(\sin x) + \text{Sgn}(\cos x) + \text{Sgn}(\tan x) + \tex…
Topic: JEE Main 2026 (28 January Evening Shift)
 8
Let $ Q(a, b, c) $ be the image of the point $ P(3, 2, 1) $ in the line$ \frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1…
Topic: JEE Main 2026 (28 January Evening Shift)
 9
Let $ P $ be a point in the plane of the vector$ \overrightarrow{AB} = 3\hat{i} + \hat{j} - \hat{k} $ and $ \overrighta…
Topic: JEE Main 2026 (28 January Evening Shift)
 10
Let $ A = { z \in \mathbb{C} : |z - 2| \le 4 } $ and $ B = { z \in \mathbb{C} : |z - 2| + |z + 2| = 5 } $. Then the…
Topic: JEE Main 2026 (28 January Evening Shift)
Ask Your Question or Put Your Review.

loading...