The area (in square units) of the triangle formed by the lines \(6 x^2+13 x y+6 y^2=0\) and \(x+2 y+3=0\) is

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

\(\mathrm{A}(1,15), \mathrm{B}(3,-12), \mathrm{C}(6,12)\) are three consecutive turning points of a continuous curve \(y=f(x)\). If \(f(x)=0\) only for \(x=\alpha\) and \(x=\beta\), then \(|\beta-\alpha| < \)

The solution of the equation \([\sin x+\cos x]^{1+\sin 2 x}=2\), where \(-\pi \leq x \leq \pi\) is

The maximum possible number of real roots of the equation \(x^{\frac{5}{5}}-6 x^2-4 x+5=0\) is

Let \(\mathrm{X}\) - axis be the transverse axis and \(\mathrm{Y}\)-axis be the conjugate axis of a hyperbola \(\mathrm{H}\). Let \(\mathrm{x}^2+\mathrm{y}^2=16\) be the director circle of \(\mathrm{H}\). If the perpendicular distance from the centre of \(\mathrm{H}\) to its latus rectum is \(\sqrt{34}\) then \(\mathrm{a}+\mathrm{b}=\)

The equation of bisectors of the angle between the lines represented by $3x^2-5xy+4y^2=0$ is

Two electric circuits \(A\) and \(B\) are shown in the figure. The ratio of power factor of circuit \(B\) to that of circuit \(A\) is

Identify the correct statements from the following :
I. Tendency to form halide hydrates gradually increases from Be to Ba down the group.
II. Tendency to form stable super oxides increases from \(\mathrm{Li}\) to \(\mathrm{Cs}\) down the group.
III. Low solubility of \(\mathrm{LiF}\) is due to its high lattice energy.
IV. Solubility of carbonates of group-2 elements increases down the group.

A body thrown vertically up to reach its maximum height in \(t\) second. The total time from the time of projection to reach a point at half of its maximum height while returning (in second) is

For a positive real number \(p\), if the perpendicular distance from a point \(-\overline{\mathrm{i}}+\mathrm{p} \overline{\mathrm{j}}-3 \overline{\mathrm{k}}\) to the plane \(\overline{\mathrm{r}} \cdot(2 \overline{\mathrm{i}}-3 \overline{\mathrm{j}}+6 \overline{\mathrm{k}})=7\) is 6 units, then \(\mathrm{p}=\)

In a \(p-n\) junction diode, the thickness of deplection layer is \(2 \times 10^{-6} \mathrm{~m}\) and barrier potential is \(0.3 \mathrm{~V}\). The intensity of the electric field at the junction is

The interval in which \(y=\ln (\ln (x)), x>1\) is decreasing is