Consider a triangle \(A B C\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to vertices \(A, B\) and \(C\) respectively. Suppose \(a=6, b=10\) and the area of the triangle is \(15 \sqrt{3}\). If \(\angle A C B\) is obtuse and if \(r\) denotes the radius of the incircle of the triangle, then \(r^2\) is equal to

Let $S$ be the of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$ such that $b_1, b_2, b_3\in \mathrm{ℝ}$ and the system of equations (in real variables)
$\begin{array}{c}-x+2y+5z=b_1\\ 2x-4y+3z=b_2\\ x-2y+2z=b_3\end{array}$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]ϵ S$ ?

Let $P_1: 2x + y - z = 3 \mathrm{a}\mathrm{n}\mathrm{d} P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\).

Question: Consider the statements:

\(P\) : There exists some \(x \in \mathrm{R}\) such that \(f(x)+2 x\)

\(=2\left(1+x^{2}\right)\)

\(Q\) : There exists some \(x \in \mathrm{R}\) such that \(2 f(x)+1\) \(=2 x(1+x)\)

Then

Let \(P, Q, R\) and \(S\) be the points on the plane with position vectors \(-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, 4 \hat{\mathbf{i}}, 3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) an \(\mathrm{d}-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}\) respectively. The quadrilateral \(P Q R S\) must be a

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 one of the following statements is correct?

\(\frac{d^2 x}{d y^2}\) equals

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of $D$ is

A police car with a siren of frequency \(8 \mathrm{kHz}\) is moving with uniform velocity \(36 \mathrm{~km} / \mathrm{h}\) towards a tall building which reflects the sound waves. The speed of sound in air is \(320 \mathrm{~m} / \mathrm{s}\). The frequency of the siren heard by the car driver is

A steady current \(I\) goes through a wire loop \(P Q R\) having shape of a right angle triangle with \(P Q=3 x, P R=4 x\) and \(Q R=5 x\). If the magnitude of the magnetic field at \(P\) due to this loop is \(k\left(\frac{\mu_0 I}{48 \pi x}\right)\), find the value of \(k\).

Two parallel wires in the plane of the paper are distance ${\mathrm{X}}_0$ apart. A point charge is moving with speed u between the wires in the same plane at a distance ${\mathrm{X}}_1$ from one of the wires. When the wires carry current of magnitude $\mathrm{ }\mathrm{I}$  in the same direction, the radius of curvature of the path of the point charge is ${\mathrm{R}}_1$ . In contrast, if the currents $\mathrm{ }\mathrm{I}$ in the two wires have directions opposite to each other, the radius of curvature of the path is ${\mathrm{R}}_2$ . If $\frac{{\mathrm{X}}_0}{{\mathrm{X}}_1}=3$ , value of $\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}$ is

Given, that the abundances of isotopes, \({ }_{54} \mathrm{Fe},{ }_{56} \mathrm{~F}\) and \({ }_{57} \mathrm{Fe}\) are \(5 \%\), \(90 \%\) and \(5 \%\), respectively, the atomic mass of \(\mathrm{Fe}\) is

The correct combination of names for isomeric alcohols with molecular formula ${\mathrm{C}}_4{\mathrm{H}}_{10}\mathrm{O}$ is/are