Paragraph:
Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0.
Question:
The number of matrices in \(A\) is

The value of $\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{x^2\cos x}{1+e^x} dx$ is equal to

Let \(f(x)=x^2\) and \(g(x)=\sin x\) for all \(x \in R\). Then, the set of all \(x\) satisfying \((\) fogogof \()(x)=(\operatorname{gogof})(x)\), where \((f \circ g)(x)=f(g(x))\) is

Tangents are drawn from the point \((17,7)\) to the circle \(x^2+y^2=169\).
Statement I The tangents are mutually perpendicular.
Statement II The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is \(x^2+y^2=338\)

Paragraph:
There are \(n\) urns each containing \((n+1)\) balls such that the ith urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_i\) be the event of selecting ith urn, \(i=1,2,3, \ldots, n\) and \(W\) denotes the event of getting a white balls.Question:
If \(P\left(u_i\right) \propto i\), where \(i=1,2,3, \ldots, n\), then \(\lim _{n \rightarrow \infty} P(W)\) is equal to

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)
Then the inradius of the triangle $ABC$ is

Paragraph:
Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The coordinates of \(A\) and \(B\) are

In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a mean separation of \(0.8 \mathrm{~mm}\). The distance between the slits at time \(t\) is given by \(d=(0.8+0.04 \sin \omega t) \mathrm{mm}\), where \(\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}\). The distance of the screen from the slits is \(1 \mathrm{~m}\) and the wavelength of the light used to illuminate the slits is \(6000 Ã…\). The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point \(O\).

The maximum speed in \(\mu \mathrm{m} / \mathrm{s}\) at which the \(8^{\text {th }}\) bright fringe will move is ________ .

Let \(\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\). A vector coplanar to \(\mathbf{a}\) and \(\mathbf{b}\) has a projection along c of magnitude \(\frac{1}{\sqrt{3}}\), then the vector is

A loop carrying current \(I\) lies in the \(x-y\) plane as shown in the figure. The unit vector \(\hat{k}\) is coming out of the plane of the paper. The magnetic moment of the current loop is

Let S be the set of all twice differentiable functions f from $\mathrm{ℝ}$ to $\mathrm{ℝ}$ such that $\frac{d^{2}f}{d{x}^2}\left(x\right)>0$ for all $x\in \left(-1,1\right)$. For $f\in S$, let $X_f$ be the number of points $x\in \left(-1,1\right)$ for which $f\left(x\right)=x$. Then which of the following statements is(are) true?

Consider the vectors
\(\vec{x}=\hat{i}+2 \hat{j}+3 \hat{k}, \quad \vec{y}=2 \hat{i}+3 \hat{j}+\hat{k}, \quad \text { and } \quad \vec{z}=3 \hat{i}+\hat{j}+2 \hat{k}\)
For two distinct positive real numbers \(\alpha\) and \(\beta\), define
\(\vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z}, \quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x}, \quad \text { and } \quad \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}\)
If the vectors \(\vec{X}, \vec{Y}\), and \(\vec{Z}\) lie in a plane, the value of \(\alpha+\beta-3\) is ______ .

In the following reaction sequence, the major product \(\mathbf{P}\) is formed.

Glycerol reacts completely with excess \(\mathbf{P}\) in the presence of an acid catalyst to form \(\mathbf{Q}\). Reaction of \(\mathbf{Q}\) with excess \(\mathrm{NaOH}\) followed by the treatment with \(\mathrm{CaCl}_2\) yields Ca-soap \(\mathbf{R}\), quantitatively. Starting with one mole of \(\mathbf{Q}\), the amount of \(\mathbf{R}\) produced in gram is ________.
[Given, atomic weight: \(\mathrm{H}=1, \mathrm{C}=12,\) \(\mathrm{~N}=14, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{Cl}=35,\) \(\mathrm{Ca}=40\)]