UPSC 2017 CDS I Math Paper

ABCD is a square.
∴ AB = BC
⇒ AX = BY
(X is the mid-point of AB and Y is the mid-point of BC)
In ΔADX and ΔBAY,
AD = AB (Given)
∠DAX = ∠ABY (90°)
AX = BY (Proved)
∴ΔADX ≅ ΔBAY(SAS congruence rule)
⇒∠DXA = ∠AYB (CPCT)
Hence, statements 1 and 2 are correct.
Now, ∠YAB + ∠AYB = 90°
⇒ ∠PAX + ∠PXA = 90° (∠DXA = ∠AYB)
⇒ 180°– ∠APX = 90°
(Using angle sum property in ΔAPX)
⇒ ∠APX = 90°
Therefore, DX is perpendicular to AY.
Hence, statements 3 and 4 are incorrect.