Sets, Relations & Functions
Functions
nta_abhyas_2025
Grade 11
Question:
If $P = \{1,2,3,4,5\}$ and $Q = \{a, b, c\}$, then the number of onto functions from $P$ to $Q$ is
$P = Q$
$P \subset Q$
$P - Q = \{0\}$
$Q \subset P$
Step-by-Step Solution
Key Concept: Analyze the range of a function of the form $\lambda + \cos x$ to determine when it can be surjective onto $\mathbb{R}$.
For $f(x) = \lambda + \cos x$ to be onto from $\mathbb{R}$ to $\mathbb{R}$, the range must be all of $\mathbb{R}$. Since $\cos x \in [-1, 1]$, the range of $f$ is $[\lambda - 1, \lambda + 1]$. For this to equal $\mathbb{R}$, we need $\lambda - 1 = -\infty$ and $\lambda + 1 = \infty$, which is impossible. Therefore $Q \subset P$ where $P$ is the set where $f$ is onto.
Correct Answer: 4