The number of roots of the equation, \((81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30\) in the interval \([0, \pi]\), is equal to

If \(\tan \theta=\frac{a}{b}\), then \(b \cos 2 \theta+a \sin 2 \theta=\)

The general solution of the differential equation
\(
y(1+\log x)\left(\frac{d x}{d y}\right)
\) \(-x \log x=0\) is

The function to be maximized is given by \(\mathrm{Z}=3 x+2 y\). The feasible region for this function is the shaded region given below, then the linear constraints for this region are given by

\(y=m x+\frac{2}{m}\) is the general solution of

The statement pattern $\left(p\wedge q\right)\wedge \left[~r\vee \left(p\wedge q\right)\right]\vee \left(~p\wedge q\right)$ is equivalent to ________

If \(\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \quad \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\), \(\overline{\mathrm{c}}=\mathrm{c}_1 \hat{i}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\) and \(\left[\begin{array}{lll}3 \overline{\mathrm{a}}+\overline{\mathrm{b}} & 3 \overline{\mathrm{~b}}+\overline{\mathrm{c}} & 3 \overline{\mathrm{c}}+\overline{\mathrm{a}}\end{array}\right]\) \(=\lambda\left|\overline{\mathrm{a}} \cdot \hat{\mathrm{i}} \overline{\mathrm{a}} \cdot \hat{\mathrm{j}} \overline{\mathrm{a}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{b}} \cdot \hat{\mathrm{i}} \overline{\mathrm{b}} \cdot \hat{\mathrm{j}} \overline{\mathrm{b}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{c}} \cdot \hat{\mathrm{i}} \overline{\mathrm{c}} \cdot \hat{\mathrm{j}} \overline{\mathrm{c}} \cdot \hat{\mathrm{k}}\right|\) then the value of \(\lambda\) is

Which of the following benzylic alcohol is tertiary alcohol?

\(\int_0^1 \sqrt{\frac{1-x}{1+x}} d x=\)

Identify the compound amongst the following whose 0.1 M aqueous solution has the highest boiling point.

An ellipse has \(O B\) as semi-minor axis, \(S\) and \(S^{\prime}\) are foci and angle \(S B S^{\prime}\) is a right angle. Then the eccentricity of the ellipse is

If \(\bar{a}=\hat{j}-\hat{k}\) and \(\bar{c}=\hat{i}-\hat{j}-\hat{k}\), then the vector \(\bar{b}\) satisfying \(\bar{a} \times \bar{b}+\bar{c}=\overline{0}\) and \(\bar{a} \cdot \bar{b}=3\) is

In Balmer series, wavelength of the \(2^{\text {nd }}\) line is ' \(\lambda_1\) ' and for Paschen series, wavelength of the \(1^{\text {st }}\) line is ' \(\lambda_2\) ', then the ratio ' \(\lambda_1\) ' to ' \(\lambda_2\) ' is