Angle between the tangents to the circle \(x^2+y^2=25 x^2\) from the point \((1,7)\) is

If \(\bar{x}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \bar{y}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) and \(\overline{\mathrm{z}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) where \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors, then value of \(\bar{x} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})+\bar{y} \cdot(\overline{\mathrm{~b}}+\overline{\mathrm{c}})+\overline{\mathrm{z}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})\) is

The differential equation of the family of curves \(\mathrm{y}=e^{x}(\mathrm{~A} \cos x+\mathrm{B} \sin x)\), where
\(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constants is

If the distance between the plane \(\mathrm{A} x-2 y+\mathrm{z}=\mathrm{d}\) and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\) units, then \(|d|\) is

The equation of a curve passing through \((1,0)\) and having slope of tangent at any point \((\mathrm{x}, \mathrm{y})\) of the curve as \(\frac{\mathrm{y}-1}{x^2+x}\) is

If $x=\mathrm{s}\mathrm{i}\mathrm{n}\theta ,y={\mathrm{s}\mathrm{i}\mathrm{n}}^3\theta$ then $\frac{d^{2}y}{d{x}^2}$ at $\theta$ = $\frac{\pi }{2}$ is ….

The focal distance of a point \(P\) on the parabola \(y^{2}=12 x\) if the ordinate of \(P\) is 6, is

The molecule in which central atom does NOT undergo \(\mathrm{sp}^3\) hybridisation is

What is produced by sensitized helper T-cells?

In Young's double slit experiment, the resultant intensity of light at a point on the screen is 'I' when the path difference is \(\lambda^{\prime}\) '. When the path difference is \(\frac{\lambda}{4}\), the intensity at a point will be \(\left(\lambda=\right.\) wavelength of light, \(\cos 180^{\circ}=-1, \cos 45^{\circ}=\frac{1}{\sqrt{2}}\) )

Let two non-collinear vectors \(\hat{a}\) and \(\hat{b}\) form an acute angle. A point \(\mathrm{P}\) moves, so that at any time \(t\) the position vector \(\overline{\mathrm{OP}}\), where \(\mathrm{O}\) is origin, is given by \(\hat{a} \sin t+\hat{b} \cos t\), when \(P\) is farthest from origin \(\mathrm{O}\), let \(\mathrm{M}\) be the length of \(\overline{\mathrm{OP}}\) and \(\hat{\mathrm{u}}\) be the unit vector along \(\overline{\mathrm{OP}}\), then

How many moles of urea are present in \(5.4 \mathrm{~g}\) ? (Molar mass \(=60\) )

Two blocks of masses 6 kg and 4 kg are placed in contact with each other on a smooth surface as shown. If a force of 5 N is applied on a heavier block, the force on the lighter block is