UPSC NDA I 2025 Mathematics Paper

Concept:Use the binomial probability formula and the given relation to form a quadratic equation. Solve for k, keeping only the value between 0 and 1.Explanation:For a binomial distribution with n = 6 and success probability p = k, the probability mass function is:$P(X=x) = \binom{6}{x} k^{x} (1-k)^{6-x}$.Given: $9P(X=4) = P(X=2)$.Compute each probability:$P(X=4) = \binom{6}{4} k^{4}(1-k)^{2} = 15k^{4}(1-k)^{2}$.$P(X=2) = \binom{6}{2} k^{2}(1-k)^{4} = 15k^{2}(1-k)^{4}$.Substitute into the equation:$9 \times 15k^{4}(1-k)^{2} = 15k^{2}(1-k)^{4}$.Cancel 15: $9k^{4}(1-k)^{2} = k^{2}(1-k)^{4}$.Divide both sides by $k^{2}(1-k)^{2}$ (valid since k>0 and k<1):$9k^{2} = (1-k)^{2}$.Expand the right side: $9k^{2} = 1 - 2k + k^{2}$.Rearrange: $8k^{2} + 2k - 1 = 0$.Solve using quadratic formula: $k = \frac{-2 \pm \sqrt{4 + 32}}{16} = \frac{-2 \pm 6}{16}$.So $k = \frac{4}{16} = \frac{1}{4}$ or $k = \frac{-8}{16} = -\frac{1}{2}$.Since probability must be between 0 and 1, reject the negative value.Answer:$k = \frac{1}{4}$, which corresponds to option C.