UPSC NDA 2 2021 Mathematics Solved Paper

Concept:For a parabola in the form $(y - k)^2 = 4p(x - h)$, the endpoints of the latus rectum are $(h + p, k \pm 2p)$ and the focus is $(h + p, k)$.Explanation:Given endpoints of latus rectum are $(-2, 4)$ and $(-2, -4)$.So, $h + p = -2$ (1)$k + 2p = 4$ (2)$k - 2p = -4$ (3)Add (2) and (3): $2k = 0 \Rightarrow k = 0$.Substitute in (2): $0 + 2p = 4 \Rightarrow p = 2$.From (1): $h + 2 = -2 \Rightarrow h = -4$.Focus = $(h + p, k) = (-4 + 2, 0) = (-2, 0)$.Thus statement 2 is correct.Equation of parabola: $(y - 0)^2 = 4 \times 2 (x + 4) \Rightarrow y^2 = 8x + 32$.Check $(0,0)$: $0 = 8(0) + 32$ is false, so it does not pass through the origin.Thus statement 1 is incorrect.Only statement 2 is correct.Answer:B. 2 only