_ x ( 1 + 2 x )^x =

($1$) The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and $NONE$ of the remaining students gets the seat previously allotted to him/her is

$(A)$ $\frac{3}{40}$ $(B)$ $\frac{1}{8}$ $(C)$ $\frac{7}{40}$ $(D)$ $\frac{1}{5}$

($2$) For $i =1,2,3,4$, let $T _{ i }$ denote the event that the students $S _{ i }$ and $S _{ i +1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T _1 \cap T _2 \cap T _3 \cap T _4$ is

$(A)$ $\frac{1}{15}$ $(B)$ $\frac{1}{10}$ $(C)$ $\frac{7}{60}$ $(D)$ $\frac{1}{5}$

Give the answer or quetion ($1$) and ($2$)

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Normal at $P$, $Q$, $R$ drawn to $y^2$ = $4x$ which intersect at $(3, 0)$, then $\Delta PQR$ is -

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Two parabolas $x = cy^2 + ay + b$ and $x = cy -y^2$ touch each other at the point $(0,1)$ , then the value of $2a + b -c$ is

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The least value of k which makes the roots of the equation $x^2+ 5x + k = A0$ imaginary is:

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A father with $8$ children takes them $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once. The number of times he will go to the garden is

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Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ lie on the circles $\left|z-z_0\right|^2=4$ and $z-\left.z_0\right|^2=16$ respectively, where $z_0=1+i$. Then, the value of $100|\alpha|^2$ is.