Let $P$ be the image of the point $( 3,1,7 )$ with respect to the plane $x - y + z = 3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { 1 }$ is (A) $x + y - 3 z = 0$ (B) $3 x + z = 0$ (C) $x - 4 y + 7 z = 0$ (D) $2 x - y = 0$
Let $a , b \in \mathbb { R }$ and $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = a \cos \left( \left| x ^ { 3 } - x \right| \right) + b | x | \sin \left( \left| x ^ { 3 } + x \right| \right)$. Then $f$ is (A) differentiable at $x = 0$ if $a = 0$ and $b = 1$ (B) differentiable at $x = 1$ if $a = 1$ and $b = 0$ (C) NOT differentiable at $x = 0$ if $a = 1$ and $b = 0$ (D) NOT differentiable at $x = 1$ if $a = 1$ and $b = 1$
Let $f : \mathbb { R } \rightarrow ( 0 , \infty )$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be twice differentiable functions such that $f ^ { \prime \prime }$ and $g ^ { \prime \prime }$ are continuous functions on $\mathbb { R }$. Suppose $f ^ { \prime } ( 2 ) = g ( 2 ) = 0 , \quad f ^ { \prime \prime } ( 2 ) \neq 0$ and $g ^ { \prime } ( 2 ) \neq 0$. If $\lim _ { x \rightarrow 2 } \frac { f ( x ) g ( x ) } { f ^ { \prime } ( x ) g ^ { \prime } ( x ) } = 1$, then (A) $f$ has a local minimum at $x = 2$ (B) $f$ has a local maximum at $x = 2$ (C) $f ^ { \prime \prime } ( 2 ) > f ( 2 )$ (D) $f ( x ) - f ^ { \prime \prime } ( x ) = 0$ for at least one $x \in \mathbb { R }$
Let $f : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ and $g : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ be functions defined by $f ( x ) = \left[ x ^ { 2 } - 3 \right]$ and $g ( x ) = | x | f ( x ) + | 4 x - 7 | f ( x )$, where $[ y ]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb { R }$. Then (A) $f$ is discontinuous exactly at three points in $\left[ - \frac { 1 } { 2 } , 2 \right]$ (B) $f$ is discontinuous exactly at four points in $\left[ - \frac { 1 } { 2 } , 2 \right]$ (C) $g$ is NOT differentiable exactly at four points in $\left( - \frac { 1 } { 2 } , 2 \right)$ (D) $g$ is NOT differentiable exactly at five points in $\left( - \frac { 1 } { 2 } , 2 \right)$
Let $a , b \in \mathbb { R }$ and $a ^ { 2 } + b ^ { 2 } \neq 0$. Suppose $S = \left\{ z \in \mathbb { C } : z = \frac { 1 } { a + i b t } , t \in \mathbb { R } , t \neq 0 \right\}$, where $i = \sqrt { - 1 }$. If $z = x + i y$ and $z \in S$, then $( x , y )$ lies on (A) the circle with radius $\frac { 1 } { 2 a }$ and centre $\left( \frac { 1 } { 2 a } , 0 \right)$ for $a > 0 , b \neq 0$ (B) the circle with radius $- \frac { 1 } { 2 a }$ and centre $\left( - \frac { 1 } { 2 a } , 0 \right)$ for $a < 0 , b \neq 0$ (C) the $x$-axis for $a \neq 0 , b = 0$ (D) the $y$-axis for $a = 0 , b \neq 0$
Let $P$ be the point on the parabola $y ^ { 2 } = 4 x$ which is at the shortest distance from the center $S$ of the circle $x ^ { 2 } + y ^ { 2 } - 4 x - 16 y + 64 = 0$. Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then (A) $S P = 2 \sqrt { 5 }$ (B) $S Q : Q P = ( \sqrt { 5 } + 1 ) : 2$ (C) the $x$-intercept of the normal to the parabola at $P$ is 6 (D) the slope of the tangent to the circle at $Q$ is $\frac { 1 } { 2 }$
Let $a , \lambda , \mu \in \mathbb { R }$. Consider the system of linear equations $$\begin{aligned}
& a x + 2 y = \lambda \\
& 3 x - 2 y = \mu
\end{aligned}$$ Which of the following statement(s) is(are) correct? (A) If $a = - 3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$ (B) If $a \neq - 3$, then the system has a unique solution for all values of $\lambda$ and $\mu$ (C) If $\lambda + \mu = 0$, then the system has infinitely many solutions for $a = - 3$ (D) If $\lambda + \mu \neq 0$, then the system has no solution for $a = - 3$
Let $\hat { u } = u _ { 1 } \hat { i } + u _ { 2 } \hat { j } + u _ { 3 } \hat { k }$ be a unit vector in $\mathbb { R } ^ { 3 }$ and $\hat { w } = \frac { 1 } { \sqrt { 6 } } ( \hat { i } + \hat { j } + 2 \hat { k } )$. Given that there exists a vector $\vec { v }$ in $\mathbb { R } ^ { 3 }$ such that $| \hat { u } \times \vec { v } | = 1$ and $\hat { w } \cdot ( \hat { u } \times \vec { v } ) = 1$. Which of the following statement(s) is(are) correct? (A) There is exactly one choice for such $\vec { v }$ (B) There are infinitely many choices for such $\vec { v }$ (C) If $\hat { u }$ lies in the $x y$-plane then $\left| u _ { 1 } \right| = \left| u _ { 2 } \right|$ (D) If $\hat { u }$ lies in the $x z$-plane then $2 \left| u _ { 1 } \right| = \left| u _ { 3 } \right|$
Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games. $P ( X > Y )$ is (A) $\frac { 1 } { 4 }$ (B) $\frac { 5 } { 12 }$ (C) $\frac { 1 } { 2 }$ (D) $\frac { 7 } { 12 }$
Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games. $P ( X = Y )$ is (A) $\frac { 11 } { 36 }$ (B) $\frac { 1 } { 3 }$ (C) $\frac { 13 } { 36 }$ (D) $\frac { 1 } { 2 }$
Let $F _ { 1 } \left( x _ { 1 } , 0 \right)$ and $F _ { 2 } \left( x _ { 2 } , 0 \right)$, for $x _ { 1 } < 0$ and $x _ { 2 } > 0$, be the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$. Suppose a parabola having vertex at the origin and focus at $F _ { 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant. If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $M Q R$ to area of the quadrilateral $M F _ { 1 } N F _ { 2 }$ is (A) $3 : 4$ (B) $4 : 5$ (C) $5 : 8$ (D) $2 : 3$