A straight rod of length 4 units slides such that its ends \(A\) and \(B\) always lie on the \(X\) and \(Y\)-axes respectively. Then, the locus of the centroid of \(\triangle O A B\) is

The absolute value of the tangent of the difference of the angles made by the lines \(4 x^2-24 x y+11 y^2=0\) with the \(\mathrm{X}\)-axis is

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

If \(A>0, B>0\) and \(A+B=\frac{\pi}{3}\), then the maximum value of \(\tan A \tan B\) is

If \(\int \frac{1+\cos (4 x)}{\cot (x)-\tan (x)} d x=k \cos (4 x)+c\), then

The mean deviation about the median of the numbers \(2,3,5,20,17,15,13,11,9,7\) is

The sum of integers from 1 to 50 that are divisible by 2 and 3 is

The energy of a photon in a monochromatic light of wavelength \(621 \mathrm{~nm}\) matches with the band gap of a semiconducting material. Then the minimum energy required to create an electron-hole pair from the semiconductor is
[Take hc \(=1242 \mathrm{eV}-\mathrm{nm}\), where h is Planck's constant and \(\mathrm{c}\) is speed of light in vacuum]

Numerically greatest term in the expansion of \((5+3 x)^6\), when \(x=1\). is

Two cards are drawn at random from a pack of 52 playing cards. If both the cards drawn are found to be black in colour, then the probability that atleast one of them is a face card is

For a monoatomic ideal gas following the cyclic process $ABCA$ shown in the $U$ vs $\rho$ plot, identify the incorrect option:

If a function \(f\) defined by \(f(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2}, & x < 0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0\end{array}\right.\) is continuous at \(\mathrm{x}=0\), then \(\mathrm{a}=\)

If the direction ratios of two lines are given by 3lm − 4ln + mn = 0 and l + 2m + 3n = 0, then the angle between the lines is