GATE 2025 AE question paper PDF and answer key

Each region carries the sum of the numbers of the shapes containing it (e.g. rectangle ∩ ellipse = 1 + 3 = 4, as shown). P lies in all three shapes: $1 + 2 + 3 = 6$. Q is in the triangle and ellipse: $2 + 3 = 5$. R is in the rectangle and triangle: $1 + 2 = 3$.

Area $= L \times W$ scales directly with $W$, so it rises by exactly 10%. The perimeter $2(L + W)$ and diagonal $\sqrt{L^2 + W^2}$ rise by less, and the shape need not become a square.

With a nose-up pitch rate the tail moves downward at $q\,l_t$, adding an upward relative wind: its angle of attack and lift increase (A, C). That extra tail lift produces a nose-down moment opposing the pitch rate — pitch damping, a more stabilising contribution (B).

$y' + \dfrac{2y}{x} = 2$ has integrating factor $x^2$: $(x^2y)' = 2x^2$, so $x^2y = \tfrac{2}{3}x^3 + C$. $y(1) = 2/3$ gives $C = 0$, so $y = \tfrac{2x}{3}$ and $y(3) = 2$.

Equilateral truss of side 1 m, 50 N horizontal at C, pin at A, roller at B. Moments about A: $R_B \times 1 = 50 \times 0.866$, so $R_B = 43.3$ N. At B: $F_{BC}\sin 60° = 43.3$ gives $F_{BC} = 50$ N, and horizontal equilibrium gives $F_{AB} = F_{BC}\cos 60° = 25$ N.