The combined equation of the pair of lines passing through the origin and making an angle $\dfrac{\pi}{4}$ with the line $3x + y - 6 = 0$ is...

Line $l : x + y = 4$ intersects the circle $x^2 + y^2 - 2x - 2y = 2$ at points $A$ and $B$. If C is the center of the circle, then the area of $\triangle ABC$ is...

If $\lim\limits_{x \to 0}\dfrac{45^x - 9^x - 5^x + 1}{(k^x - 1)(3^x - 1)} = 2$, then the value of $k$ is ...

If the truth value of the compound statement $[(p \leftrightarrow q) \wedge (q \to r) \wedge \sim r] \to (p \wedge \sim q)$ is false, then the truth values of the statement patterns $(p \to q) \leftrightarrow (q \to r)$ and $\sim(p \vee r) \to (q \wedge p)$ are, respectively ...