VITEEE 2016 :: Mathematics :: General questions

• Mathematics - General questions

A box contains 20 identical balls of which 10 are blue and 10 are green. The balls are drawn at random from the box one at a time with replacement. The probability that a blue ball is drawn 4 th time on the 7 th draw is 

Answer:

Explanation:

Probability  of getting a blue ball at any draw  =p= $\frac{10}{20}=\frac{1}{2}$

 P[ getting a blue ball 4th time in 7 th draw ]=P[ getting 3 blue balls in 6 draw]Λ P[ a blue-ball in the 7th draw]

$=^{6}C_{3}\left(\frac{1}{2}\right)^{3}\left(\frac{1}{2}\right)^{3}.\frac{1}{2}$

 =  $\frac{6\times5\times4}{1\times2\times3}(\frac{1}{2})^{7}=20\times\frac{1}{32\times4}=\frac{5}{32}$

 Which of the following is INCORRECT for the hyperbola   $x^{2}-2y^{2}-2x+8y-1=0$

Answer:

Explanation:

The equation of the hyperbola  is 

 $x^{2}-2y^{2}-2x+8y-1=0$

 or   $ (x-1)^{2}-2(y-2)^{2}+6=0$

 or   $\frac{(x-1)^{2}}{-6}+\frac{(y-2)^{2}}{3}=1$

or  $\frac{(y-2)^{2}}{3}-\frac{(x-1)^{2}}{6}=1$     .....(i)

 or  $\frac{Y^{2}}{3}-\frac{X^{2}}{6}=1$    

 where X =x-1  and Y= y-2 ....(ii)

$\therefore$  The centre =(0,0,) in X-Y co-ordinates 

$\therefore$  The centre = (1,2) in the x-y co-ordinates using (ii)

 If the transverse axis be length 2a, than a  = $\sqrt{3}$, since in the equation (i) the tansverse axis is parallel to the y-axis

 If the conjugate axis is of length 2b n, then b= $\sqrt{6}$

But   $b^{2}=a^{2}(e^{2}-1)$

$\therefore$   $6=3(e^{2}-1), \therefore e^{2}=3$   or  $ e=\sqrt{3}$

The  length of the transverse axis =2$\sqrt{3}$

 The length of the conjugate axis =2$\sqrt{6}$

 Latus rectum =   $\frac{2b^{2}}{a}=\frac{2\times6}{\sqrt{3}}=4\sqrt{3}$

While shuffling a pack of 52 playing cards, 2 are accidentally dropped. The probability that the missing cards to be of different colours is 

Answer:

Explanation:

 There are 26 red cards and 26 black cards 

i.e, total  number of cards =52

 P(both cards of different colours)

= P(B)P(R)+P(R)P(B)

  =  $\frac{26}{52}\times\frac{26}{51}+\frac{26}{52}\times\frac{26}{51}=2\left(\frac{26}{52}\times\frac{26}{51}\right)=\frac{26}{51}$

A variable plane remains at constant distance p from the origin .If it meets coordinate axes at points A , B, C  then the locus of the centroid  of $\triangle$  ABC is  

Answer:

Explanation:

Let A = (a,0,0) , B= (0,b,0) C= (0,0,c) then equation of the plane is 

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

 Its distance from the origin,

 $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}}$  ......(i)

 If (x,y,z) the centroid of $\triangle$ ABC, then

$x=\frac{a}{3},y=\frac{b}{3},z=\frac{c}{3}$   .....(ii)

eliminating a,b,c from (i) and (ii) required locus is 

 $x^{-2}+y^{-2}+z^{-2}=9p^{-2}$

If A and B are two matrices such than rank of A= m and rank of B= n, then

Answer:

Explanation:

$\because$    rank (AB )  $\leq$ rank (A)

 and rank (A B) $\leq$ rank (B) 

 therefore rank (A B ) $\leq$  min (rank A, rank B)

• Mathematics - General questions