The negation of   $\sim S \vee (\sim r\wedge S)$ is  equivalent to

A) $S\wedge \sim r$

B) $S\wedge (r\wedge\sim S)$

C) $S\vee (r\vee\sim S)$

D) $S\wedge r$


Let $\tan ^{-1}y= \tan ^{-1} x+\tan ^{-1}\left(\frac{2x}{1-x^{2}}\right)$ where  $|x|<\frac{1}{\sqrt{3}}$. Then, the value of y is

A) $\frac{3x-x^{3}}{1-3x^{2}}$

B) $\frac{3x+x^{3}}{1-3x^{2}}$

C) $\frac{3x-x^{3}}{1+3x^{2}}$

D) $\frac{3x+x^{3}}{1+3x^{2}}$


 If the angles of elevation of the top of a tower from three  collinear points A, B and C on a line leading  to the foot of the tower are 30° , 45° and 60° respectively , then the ratio AB:BC is

A) $\sqrt{3}:1$

B) $\sqrt{3}:\sqrt{2}$

C) $1:\sqrt{3}$

D) 2:3


The mean of the data set comprising of 16 observations in 16. If one of the observation valued 16 is deleted and three new observations valued 3,4 and 5 are added to the data, then the mean of the resultant data is

A) 16.8

B) 16.0

C) 15.8

D) 14.0


If 12 identical  balls are to be placed in 3 identical boxes, then the probability  that one of the boxes contains exactly 3 balls, is

A) $\frac{55}{3}\left(\frac{2}{3}\right)^{11}$

B) $\frac{55}{3}\left(\frac{2}{3}\right)^{10}$

C) $220\left(\frac{1}{3}\right)^{12}$

D) $22\left(\frac{1}{3}\right)^{11}$