Tangent to the ellipse \(\frac{x^{2}}{32}+\frac{y^{2}}{18}=1\) having slope \(-\frac{3}{4}\) meet the coordinate axes in \(A\) and \(B\). Find the area of the \(\Delta A O B\), where \(O\) is the origin.

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

The shaded figure given below is the solution set for the linear inequations. Choose the correct option

The equation of the line passing through \((1,2,3)\) and perpendicular to the lines
\(x-1=\frac{y+2}{2}=\frac{z+4}{4}\) and \(\frac{x-1}{2}=\frac{y-2}{2}=z+3\) is

In a triangle ABC , with usual notations if \(a=4, \mathrm{~b}=8, \angle \mathrm{C}=60^{\circ}\), then the value of \(\angle \mathrm{B}\) and the ratio \(\cos \mathrm{A}: \cos \mathrm{C}\) respectively are,

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are 27 gms of certain substance and 3 hours later it is found that \(8 \mathrm{gms}\) are left, then the amount left after one more hour is

The value of \(\int \frac{\mathrm{d} x}{(x+1)^{3 / 4}(x-2)^{5 / 4}}\) is equal to

Which among the following elements has highest electronegativity?

In case of rotational dynamics, which one of the following statements is correct?
\([\vec{\omega}=\) angular velocity, \(\vec{v}=\) linear velocity
\(\overrightarrow{\mathrm{r}}=\) radius vector, \(\vec{\alpha}=\) angular acceleration
\(\overrightarrow{\mathrm{a}}=\) linear acceleration, \(\overrightarrow{\mathrm{L}}=\) angular momentum
\(\overrightarrow{\mathrm{p}}=\) linear momentum, \(\vec{\tau}=\) torque,
\(\overrightarrow{\mathrm{f}}=\) centripetal force \(]\)

Which of the following is obtained by hydrogenation of benzoyl chloride in presence of \(\mathrm{Pd}\) on \(\mathrm{BaSO}_{4}\) ?

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

Select the CORRECT statement.

Which of the following sugar is present in nucleotide of RNA?