The equation of the plane passing through the point $( 1,1,1 )$ and perpendicular to the planes $2 x + y - 2 z = 5$ and $3 x - 6 y - 2 z = 7$, is [A] $14 x + 2 y - 15 z = 1$ [B] $14 x - 2 y + 15 z = 27$ [C] $14 x + 2 y + 15 z = 31$ [D] $- 14 x + 2 y + 15 z = 3$
Let $O$ be the origin and let $P Q R$ be an arbitrary triangle. The point $S$ is such that $$\overrightarrow { O P } \cdot \overrightarrow { O Q } + \overrightarrow { O R } \cdot \overrightarrow { O S } = \overrightarrow { O R } \cdot \overrightarrow { O P } + \overrightarrow { O Q } \cdot \overrightarrow { O S } = \overrightarrow { O Q } \cdot \overrightarrow { O R } + \overrightarrow { O P } \cdot \overrightarrow { O S }$$ Then the triangle $P Q R$ has $S$ as its [A] centroid [B] circumcentre [C] incentre [D] orthocenter
How many $3 \times 3$ matrices $M$ with entries from $\{ 0,1,2 \}$ are there, for which the sum of the diagonal entries of $M ^ { T } M$ is 5 ? [A] 126 [B] 198 [C] 162 [D] 135
Let $S = \{ 1,2,3 , \ldots , 9 \}$. For $k = 1,2 , \ldots , 5$, let $N _ { k }$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N _ { 1 } + N _ { 2 } + N _ { 3 } + N _ { 4 } + N _ { 5 } =$ [A] 210 [B] 252 [C] 125 [D] 126
Three randomly chosen nonnegative integers $x , y$ and $z$ are found to satisfy the equation $x + y + z = 10$. Then the probability that $z$ is even, is [A] $\frac { 36 } { 55 }$ [B] $\frac { 6 } { 11 }$ [C] $\frac { 1 } { 2 }$ [D] $\frac { 5 } { 11 }$
If $f : \mathbb { R } \rightarrow \mathbb { R }$ is a differentiable function such that $f ^ { \prime } ( x ) > 2 f ( x )$ for all $x \in \mathbb { R }$, and $f ( 0 ) = 1$, then [A] $f ( x )$ is increasing in $( 0 , \infty )$ [B] $f ( x )$ is decreasing in $( 0 , \infty )$ [C] $f ( x ) > e ^ { 2 x }$ in $( 0 , \infty )$ [D] $f ^ { \prime } ( x ) < e ^ { 2 x }$ in $( 0 , \infty )$
Let $f ( x ) = \frac { 1 - x ( 1 + | 1 - x | ) } { | 1 - x | } \cos \left( \frac { 1 } { 1 - x } \right)$ for $x \neq 1$. Then [A] $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 0$ [B] $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ does not exist [C] $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 0$ [D] $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ does not exist
If $f ( x ) = \left| \begin{array} { c c c } \cos ( 2 x ) & \cos ( 2 x ) & \sin ( 2 x ) \\ - \cos x & \cos x & - \sin x \\ \sin x & \sin x & \cos x \end{array} \right|$, then [A] $f ^ { \prime } ( x ) = 0$ at exactly three points in $( - \pi , \pi )$ [B] $f ^ { \prime } ( x ) = 0$ at more than three points in $( - \pi , \pi )$ [C] $f ( x )$ attains its maximum at $x = 0$ [D] $f ( x )$ attains its minimum at $x = 0$
Let $O$ be the origin, and $\overrightarrow { O X } , \overrightarrow { O Y } , \overrightarrow { O Z }$ be three unit vectors in the directions of the sides $\overrightarrow { Q R } , \overrightarrow { R P }$, $\overrightarrow { P Q }$, respectively, of a triangle $P Q R$. $| \overrightarrow { O X } \times \overrightarrow { O Y } | =$ [A] $\sin ( P + Q )$ [B] $\sin 2 R$ [C] $\sin ( P + R )$ [D] $\sin ( Q + R )$
Let $p , q$ be integers and let $\alpha , \beta$ be the roots of the equation, $x ^ { 2 } - x - 1 = 0$, where $\alpha \neq \beta$. For $n = 0,1,2 , \ldots$, let $a _ { n } = p \alpha ^ { n } + q \beta ^ { n }$. FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt { 5 } = 0$, then $a = 0 = b$. If $a _ { 4 } = 28$, then $p + 2 q =$ [A] 21 [B] 14 [C] 7 [D] 12