If \(0 < x < \frac{\pi}{2}\) then the maximum area (in sq. units) of the triangle whose vertices are \((0,0)\), \((x, \cos x)\) and \(\left(\sin ^3 x, 0\right)\) is

If tangents are drawn to the ellipse \(x^2+2 y^2=2\), then the locus of the midpoints of the intercepts made by the tangents between the coordinate axes is

\(\cos ^2 10^{\circ}+\cos ^2 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}\) is equal to

Three numbers are chosen from 1 to 30. The probability that they are not three consecutive numbers is

Let \(\bar{a}=4 \bar{i}+3 \bar{j}\) and \(\bar{b}\) be two perpendicular vectors in the XOY-plane. A vector \(\bar{c}\) in the same plane and having projections 1 and 2 respectively on \(\bar{a}\) and \(\bar{b}\) is

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

If the tangent drawn at the point \(\left(\mathrm{x}_1, \mathrm{y}_1\right), \mathrm{x}_1, \mathrm{y}_1 \in \mathrm{~N}\) on the curve \(y=x^4-2 x^3+x^2+5 x\) passes through origin, then \(x_1+y_1=\)

If the wave number of radiation emitted for the electron transition from an excited state to ground state of hydrogen is \(\frac{5 \mathrm{x}}{36} \mathrm{~m}^{-1}\), the wave number of radiation absorbed for the electron transition from the above excited state to next immediate excited state in \(\mathrm{m}^{-1}\) is:

The function \(f(x)=\left\{\begin{array}{ll}\frac{2}{5-x}, & x \lt 3 \\ 5-x, & x \geq 3\end{array}\right.\) is

If \(1, \omega, \omega^2, \ldots \omega^{10}\) are 11 th roots of unity, then product of these roots is 1

The efficiency of an ideal Carnot engine working between temperatures \(T_1\) and \(T_2\) is \(1 / 3\). If the temperature of the sink is reduced by \(40 \%\), then its efficiency will be

If \(A\) is a square matrix of order 3, then \(\left|\operatorname{adj}\left(\operatorname{adj} A^2\right)\right|\) is equal to

The position of the direct image obtained at $O ,$ , when a monochromatic beam of light is passed through a plane transmission grating at normal incidence is shown in figure. The diffracted images $\mathrm{A},\mathrm{B}$ and $\mathrm{C}$ correspond to the first, second and third order diffraction. When the source is replaced by another source of shorter wavelength