UPSC NDA I 2022 Mathematics Solved Paper

Concept:The mean of grouped data is calculated using the formula: $\bar{x} = \frac{\sum f_i x_i}{N}$ where $x_i$ is the class mark and $f_i$ is the frequency.Explanation:Step 1: Use total frequency $N = 120$ to form the first equation.$17 + (p+q) + 32 + (p-3q) + 19 = 120$$\Rightarrow 2p - 2q = 52$$\Rightarrow p - q = 26$ ... (1)Step 2: Find class marks for each interval.$0-20$: $\frac{0+20}{2}=10$$20-40$: $\frac{20+40}{2}=30$$40-60$: $\frac{40+60}{2}=50$$60-80$: $\frac{60+80}{2}=70$$80-100$: $\frac{80+100}{2}=90$Step 3: Apply the mean formula $\bar{x} = 50$.$50 = \frac{17(10) + (p+q)(30) + 32(50) + (p-3q)(70) + 19(90)}{120}$Multiply both sides by 120:$6000 = 170 + 30p + 30q + 1600 + 70p - 210q + 1710$Simplify:$6000 = 100p - 180q + 3480$$2520 = 100p - 180q$Divide by 10:$252 = 10p - 18q$ ... (2)Step 4: Solve equations (1) and (2).Multiply (1) by 10: $10p - 10q = 260$Subtract from (2): $(10p - 18q) - (10p - 10q) = 252 - 260$$\Rightarrow -8q = -8$$\Rightarrow q = 1$Substitute into (1): $p - 1 = 26$$\Rightarrow p = 27$Answer:$p = 27$, which corresponds to option C.