Statistics
Linear Transform Y=aX+b — Sum of b Values
nta_pyq_2026_jan
Grade Class 11
Question:
Let $X=\{x\in\mathbb{N}:1\leq x\leq19\}$ and for some $a,b\in\mathbb{R}$, $Y=\{ax+b:x\in X\}$. If the mean and variance of the elements of $Y$ are 30 and 750, respectively, then the sum of all possible values of $b$ is
Step-by-Step Solution
Key Concept: Mean of $X=\tfrac{1+19}{2}=10$. Variance of $X=\tfrac{19^2-1}{12}=30$. Mean of $Y=10a+b=30$. Variance of $Y=a^2\cdot30=750\Rightarrow a^2=25\Rightarrow a=\pm5$.
Sum of $b$ values $=60$.
<div class="key-concept"><strong>Key Concept:</strong> Mean of $X=\tfrac{1+19}{2}=10$. Variance of $X=\tfrac{19^2-1}{12}=30$. Mean of $Y=10a+b=30$. Variance of $Y=a^2\cdot30=750\Rightarrow a^2=25\Rightarrow a=\pm5$.</div>
<div class="trap-box"><strong>Trap:</strong> $a=5$: $b=30-50=-20$. $a=-5$: $b=30+50=80$. Sum $=-20+80=60$.</div>
Correct Answer: 1