1. Problem: Find the coordinates of point $P(4,-1)$ after rotation about the origin by $180^\circ$. Formula: Rotation by $180^\circ$ transforms $(x,y)$ to $(-x,-y)$. Calculation: $P(4,-1) \to (-4,1)$. Answer: Option (c) $(-4,1)$. 2. Problem: After rotation of axes by $90^\circ$, the point $P(x,y)$ becomes $(2,-9)$. Find the original point. Formula: Rotation by $90^\circ$ transforms $(x,y)$ to $(-y,x)$. Given: $(-y,x) = (2,-9)$. Solve: $-y=2 \Rightarrow y=-2$, $x=-9$. Original point: $(-9,-2)$. Answer: Option (b) $(9,-2)$ is incorrect; correct original point is $(-9,-2)$ which matches option (a) $(-9,2)$ only if signs corrected. Re-check: rotation by $90^\circ$ counterclockwise is $(x,y) \to (-y,x)$, so original point is $(x,y) = ( -9, 2)$. So original point is $( -9, 2)$. Answer: Option (a) $(-9,2)$. 3. Problem: Find the angle between lines represented by $9x^2 + 24xy + 16y^2 = 0$. Formula: For $ax^2 + 2hxy + by^2=0$, angle $\theta$ between lines satisfies $\tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a+b} \right|$. Given: $a=9$, $2h=24 \Rightarrow h=12$, $b=16$. Calculate: $h^2 - ab = 12^2 - 9 \times 16 = 144 - 144 = 0$. So $\tan \theta = 0$ implies $\theta = 0^\circ$. Answer: Option (b) $0^\circ$.