Map each axis to its primary surface: roll \(\Rightarrow\) ailerons, pitch \(\Rightarrow\) elevator (longitudinal), yaw \(\Rightarrow\) rudder (directional). When in doubt about a control surface's effect on stability, match axis-to-axis and you'll avoid confusions like (D).
Step 1: Recall the primary aerodynamic roles.
The rudder is the primary control surface for yaw about the vertical axis; it provides directional control and contributes to directional (weathercock) stability. The surfaces linked to longitudinal (pitch) stability are the horizontal tail and elevator, not the rudder.
Step 2: Evaluate each option.
(A) Adverse yaw in turns. During aileron-induced roll, the downgoing aileron increases drag on the rising wing, yawing the nose opposite to the turn (adverse yaw). Coordinated rudder input counters this yaw to keep the turn coordinated (ball centered). \(\Rightarrow\) True.
(B) Asymmetric thrust / engine failure. In multi-engine aircraft, failure of one engine creates a yawing moment toward the dead engine. Applying rudder toward the live engine (and sometimes banking a few degrees) provides the required yawing moment to maintain directional control. \(\Rightarrow\) True.
(C) Crosswind landing. Two common techniques are crab-and-decrab and sideslip. In both, rudder is used to align the longitudinal axis with the runway centerline just before or during touchdown, countering the crosswind-induced drift. \(\Rightarrow\) True.
(D) Longitudinal stability. Longitudinal stability concerns restoring moments in pitch, governed by the horizontal tail/elevator and static margin. The rudder affects directional (yaw) stability; it does not "enhance longitudinal stability." \(\Rightarrow\) False.
\[ \boxed{\text{Correct uses of the rudder: (A), (B), (C).}} \]
While taking off, the net external force acting on an airplane during the ground roll segment can be assumed to be constant. The airplane starts from rest. \( S_{LO} \) and \( V_{LO} \) are the ground roll distance and the lift-off speed, respectively. \( \alpha V_{LO} \) (\( \alpha>0 \)) denotes the airplane speed at 0.5 \( S_{LO} \). Neglecting changes in the airplane mass during the ground roll segment, the value of \( \alpha \) is _________ (rounded off to two decimal places).
A jet-powered airplane is steadily climbing at a rate of 10 m/s. The air density is 0.8 kg/m³, and the thrust force is aligned with the flight path. Using the information provided in the table below, the airplane’s thrust to weight ratio is __________ (rounded off to one decimal place).

In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:

The number of patients per shift (X) consulting Dr. Gita in her past 100 shifts is shown in the figure. If the amount she earns is ₹1000(X - 0.2), what is the average amount (in ₹) she has earned per shift in the past 100 shifts?

The equation of a closed curve in two-dimensional polar coordinates is given by \( r = \frac{2}{\sqrt{\pi}} (1 - \sin \theta) \). The area enclosed by the curve is ___________ (answer in integer).
For a three-bar truss loaded as shown in the figure, the magnitude of the force in the horizontal member AB is ____________ N (answer in integer).
