Chapter 9

Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\mathrm{△}OPQ$ is $3\sqrt{2}$, then which of the following is (are) the coordinates of $P$ ? (A) $(4,2\sqrt{2})$ (B) $(9,3\sqrt{2})$ (C) $(\frac{1}{4},\frac{1}{\sqrt{2}})$ (D) $(1,\sqrt{2})$

Let $y(x)$ be a solution of the differential equation $(1+e^x)y^{\prime }+y{e}^x=1.$ If $y(0)=2$, then which of the following statements is (are) true? (A) $y(-4)=0$ (B) $y(-2)=0$ (C) $y(x)$ has a critical point in the interval $(-1,0)$ (D) $y(x)$ has no critical point in the interval $(-1,0)$

Let $f(x)=\sin (\frac{\pi }{6}\sin (\frac{\pi }{2}\sin x))$ for all $x\in \mathbb{R}$ and $g(x)=\frac{\pi }{2}\sin x$ for all $x\in \mathbb{R}$. Let $(f\circ g)(x)$ denote $f(g(x))$ and $(g\circ f)(x)$ denote $g(f(x))$. Then which of the following is (are) true? (A) Range of $f$ is $[-\frac{1}{2},\frac{1}{2}]$ (B) Range of $f\circ g$ is $[-\frac{1}{2},\frac{1}{2}]$ (C) $\underset{x\to 0}{lim}\frac{f(x)}{g(x)}=\frac{\pi }{6}$ (D) There is an $x\in \mathbb{R}$ such that $(g\circ f)(x)=1$

Let $\mathrm{△}PQR$ be a triangle. Let $\overrightarrow{a}=\overrightarrow{QR},\overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}$. If $|\overrightarrow{a}|=12,|\overrightarrow{b}|=4\sqrt{3}$ and $\overrightarrow{b},\overrightarrow{c}=24$, then which of the following is (are) true? (A) $\frac{|\overrightarrow{c}{|}^2}{2}-|\overrightarrow{a}|=12$ (B) $\frac{|\overrightarrow{c}{|}^2}{2}+|\overrightarrow{a}|=30$ (C) $|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{c}\times \overrightarrow{a}|=48\sqrt{3}$ (D) $\overrightarrow{a}\cdot \overrightarrow{b}=-72$

Column I (A) In ${\mathbb{R}}^2$, if the magnitude of the projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3}\hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha =2+\sqrt{3}\beta$, then possible value $(\mathrm{s})$ of $|\alpha |$ is (are) (B) Let $a$ and $b$ be real numbers such that the function $$ f(x)=\left\{$$\begin{array}{rl}-3a{x}^2-2,& x<1\\ bx+a^2,& x\ge 1\end{array}$$\right. $$ is differentiable for all $x\in \mathbb{R}$. Then possible value(s) of $a$ is (are) (C) Let $\omega \ne 1$ be a complex cube root of unity. If ${(3-3\omega +2{\omega }^2)}^{4n+3}+$ ${(2+3\omega -3{\omega }^2)}^{4n+3}+{(-3+2\omega +3{\omega }^2)}^{4n+3}=0$ then possible value(s) of $n$ is (are) (D) Let the harmonic mean of two positive real numbers $a$ and $b$ be $4.$ If $q$ is a positive real number such that $a,5,q,b$ is an arithmetic progression, then the value(s) of $|q-a|$ is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 $(\mathrm{T})5$