Paragraph:
A fair die is tossed repeatedly until a six is obtained. Let \(X\) denotes the number of tosses required.Question:
The probability that \(X \geq 3\) equals

Let $S=\left\{\mathrm{1,2},3,\dots .9\right\}$. For $k=\mathrm{1,2},\dots 5,$ 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 bag contains \(N\) balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For \(i=1,2,3\), let \(W_i, G_i\), and \(B_i\) denote the events that the ball drawn in the \(i^{\text {th }}\) draw is a white ball, green ball, and blue ball, respectively. If the probability \(P\left(W_1 \cap G_2 \cap B_3\right)=\frac{2}{5 N}\) and the conditional probability \(P\left(B_3 \mid W_1 \cap G_2\right)=\frac{2}{9}\), then \(N\) equals ________.

Let $L_1$ and $L_2$ be the following straight lines.
$L_1:\frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $L_2:\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$
Suppose the straight line $L:\frac{x-\alpha }{l}=\frac{y-1}{m}=\frac{z-\gamma }{-2}$ lies in the plane containing $L_1$ and $L_2$, and passes through the point of intersection of $L_1$ and $L_2$. If the line $L$ bisects the acute angle between the lines $L_1$ and $L_2$, then which of the following statements is/are TRUE?

Paragraph:
If a continuous \(f\) defined on the real line \(R\), assume positive and negative values in \(R\), then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive at some point and its minimum values is negative, then the equation \(f(x)=0\) has a root in \(R\).
Consider \(f(x)=k e^x-x\) for all real \(x\), where \(k\) is real constant.Question:
The line \(y=x\) meets \(y=k e^x\) for \(k \leq 0\) at

The least value of $\alpha \in R$ for which, $4\alpha x^2+\frac{1}{x} \ge 1,$ for all $x>0,$ is 

Let $f\left(x\right)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$ for all $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right).$ Then the correct expression(S) is(are)

Let $n\ge 2$ be a natural number and $f:\left[0, 1\right]\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)=\left\{\begin{array}{ccc}n\left(1-2nx\right)& \mathrm{if}& 0\le x\le \frac{1}{2n}\\ 2n\left(2nx-1\right)& \mathrm{if}& \frac{1}{2n}\le x\le \frac{3}{4n}\\ 4n\left(1-nx\right)& \mathrm{if}& \frac{3}{4n}\le x\le \frac{1}{n}\\ \frac{n}{n-1}\left(nx-1\right)& \mathrm{if}& \frac{1}{n}\le x\le 1\end{array}\right.$

If $n$ is such that the area of the region bounded by the curves $x=0, x=1, y=0$ and $y=f\left(x\right)$ is $4$, then the maximum value of the function $f$ is

The function $\mathit{\text{f}}\left(x\right)=2\left|x\right|+\left|x+2\right|-\left|\left|x+2\right|-2\left|x\right|\right|$ has a local minimum or a local maximum at $x=$

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius $1$.

(There are two questions based on $\mathrm{PARAGRAPH} " 1 "$, the question given below is one of them)
Let $\alpha$ be the area of the triangle $ABC$. Then the value of $(64\alpha {)}^2$ is

Paragraph:
Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
The sum \(V_1+V_2+\ldots+V_n\) is

A solid glass sphere of refractive index \(n=\sqrt{3}\) and radius \(R\) contains a spherical air cavity of radius \(\frac{\mathrm{R}}{2}\), as shown in the figure. A very thin glass layer is present at the point O so that the air cavity (refractive index \(n=1\)) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source \(S\) emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point O and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is \(\theta\). The value of \(\sin \theta\) is ______ [given : θ > 30°]

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors satisfying \(|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9\), then \(|2 \vec{a}+5 \vec{b}+5 \vec{c}|\) is