If $2x + 3y + 12 = 0$ and $x - y + 4\lambda = 0$ are conjugate with respect to the parabola $y^2 = 8x$,then $\lambda$ is equal to

Consider an ionic solid $M X$ with $NaCl$ structure. Construct a new structure $( Z )$ whose unit cell is constructed from the unit cell of $M X$ following the sequential instructions given below. Neglect the charge balance.

$(i)$ Remove all the anions ( $X$ ) except the central one

$(ii)$ Replace all the face centered cations $(M)$ by anions $(X)$

$(iii)$ Remove all the corner cations $(M)$

$(iv)$ Replace the central anion ( $X$ ) with cation $(M)$

The value of $\left(\frac{\text { number of anions }}{\text { number of cations }}\right)$ in $Z$ is. . . . .

Which of the following reagents reacts differently with $HCHO$,$CH_3CHO$,and $CH_3COCH_3$?

One hundred identical coins,each with probability $p$ of showing heads,are tossed once. If $0 < p < 1$ and the probability of heads showing on $50$ coins is equal to that of heads showing on $51$ coins,then the value of $p$ is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. $A$ vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$,whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$,is given by

$A$ bag $X$ contains $2$ white and $3$ black balls and another bag $Y$ contains $4$ white and $2$ black balls. One bag is selected at random and a ball is drawn from it. Then,the probability for the ball chosen to be white is: