JEE (Advanced) 2018 Paper-1

$$arg(-1-i)=$$\frac{\pi}{4}$$,where i=\sqrt{-1}$$

The function f:R $$\rightarrow (-\pi, \pi]$$, defined by f(t)= arg(−1 + it) for all t $$\in$$ R, is continuous at all points of R, where $$i=\sqrt{-1}$$

For any two non-zero complex numbers $$Z_{1} and Z_{2}$$, $$arg(\frac{Z_{1}}{Z_{2}})-arg(Z_{1})+arg_{Z_{2}}$$ is an integral multiple of $$2 \pi$$

For any three given distinct complex numbers $$Z_{1}, Z_{2}$$ and $$Z_{3}$$ the locus of the point Z satisfying the condition arg $$\left(\frac{(Z-Z_{1})(Z_{2}-Z_{3})}{(Z-Z_{3})(Z_{2}-Z_{1})}\right)=\pi$$ lies on a straight line

$$\angle QPR =45^\circ$$

The area of the triangle 𝑃𝑄𝑅 is $$25\sqrt{3} and \angle QRP$$ =$$120^\circ$$

The radius of the incircle of the triangle 𝑃𝑄𝑅 is $$10\sqrt{3}-15$$

The area of the circumcircle of the triangle 𝑃𝑄𝑅 is $$100 \pi$$

The line of intersection of $$𝑃_{1}and 𝑃_{2}$$ has direction ratios 1, 2, −1

The line $$\frac{3x-4}{9}=\frac{1-3y}{9}=\frac{Z}{3}$$ is perpendicular to the line of intersection of $$𝑃_{1} and 𝑃_{2}$$

The acute angle between $$P_{1} and P_{2}$$ is $$60^\circ$$

If $$P_{3}$$ is the plane passing through the point (4, 2, −2) and perpendicular to the line of intersection of $$𝑃_{1} and 𝑃_{2}$$, then the distance of the point (2, 1, 1) from the plane $$P_{3}$$ is $$\frac{2}{\sqrt{3}}$$

There exist r,s \in R where r<s such that 𝑓 is one-one on the open interval (r, s)

There exists $$X_{0} \in (-4,0)$$ such that$$\mid (f'(x_{0})) \mid \leq 1$$

$$\lim_{x\rightarrow\infty } f(x) =1$$

There exists $$\alpha$$ $$\in$$ (-4,4) such that $$f(\alpha)+f''(\alpha) = 0$$ and $$f'(\alpha) \neq 0$$