A solution curve of differential equation $\left(x^2+xy+4x+2y+4\right)\frac{dy}{dx}-y^2=0, x>\mathrm{0 }$ passes through the point (1, 3). Then the solution curve -

Let $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

The value of the integral \(\int_{-\pi / 2}^{\pi / 2}\left(x^{2}+\ln \frac{\pi+x}{\pi-x}\right) \cos x d x\) is

Let \(S_n=\sum_{k=0}^n \frac{n}{n^2+k n+k^2}\) and \(T_n=\sum_{k=0}^{n-1} \frac{n}{n^2+k n+k^2}\), for \(n=1,2,3, \ldots\), then

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then

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

Let $\widehat{u}=u_1\widehat{i}+u_2\widehat{j}+u_3\widehat{k}$ be a unit vector in ${\mathbb{R}}^3$ and $\widehat{w}=\frac{1}{\sqrt{6}} \left(\widehat{i}+ \widehat{j}+2\widehat{k}\right).$ Given that there exists a vector $\overrightarrow{v}$ in ${\mathbb{R}}^3$ such that $\left|\widehat{u}\times \overrightarrow{v}\right|=1$ and $\widehat{w} . \left(\widehat{u}\times \overrightarrow{v}\right)=1.$ Which of the following statement(s) is (are) correct ?

Paragraph:
Let \(A_1, G_1, H_1\) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For \(n \geq 2\), let \(A_{n-1}\) and \(H_{n-1}\) has arithmetic, geometric and harmonic means as \(A_n, G_n, H_n\), respectively.
Question:
Which of the following statements is correct?

A line L : $y=\mathit{\text{m}}x+3$ meets $y-\mathrm{a}\mathrm{x}\mathrm{\text{is}}$ at E(0,3) and the arc of the parabola $y^2=16x\text{,}0\le y\le 6$ at the point $F\left(x_0,y_0\right)$.The tangent to the parabola at $F\left(x_0,y_0\right)$ intersects the y-axis at $G\left(0,y_1\right)$.The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	m =	P.	$\frac{1}{2}$
B.	Maximum area of ΔEFG is	Q.	4
C.	y0​ =	R.	2
D.	y1 =	S.	1

Paragraph:
Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion?

In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle $\mathrm{\theta }$ with the horizontal floor. The coefficient of friction between the wall and the ladder is ${\mathrm{\mu }}_1$ and that between the floor and the ladder is ${\mathrm{\mu }}_2$ . The normal reaction of the wall on the ladder is ${\mathrm{N}}_1$ and that of the floor is ${\mathrm{N}}_2$ If the ladder is about to slip, then

Let \(X\) and \(Y\) be two events such that \(P(X \mid Y)=\frac{1}{2}\), \(P(Y / X)=\frac{1}{3}\) and \(P(X \cap Y)=\frac{1}{6}\). Which of the following is (are) correct?

A current carrying wire heats a metal rod. The wire provides a constant power $\left(\mathrm{P}\right)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $\left(\mathrm{T}\right)$ in the metal rod changes with time $\left(\mathrm{t}\right)$ as:
$\mathrm{T}\left(\mathrm{t}\right)={\mathrm{T}}_0\left(1+\mathrm{\beta }{\mathrm{t}}^{1/4}\right)$
where $\mathrm{\beta }$ is a constant with appropriate dimension while ${\mathrm{T}}_0$ is a constant with dimension of temperature. The heat capacity of the metal is: